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%%EndProlog
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1 pagesetup
12 B f
(Efficient Algorithms for Constructing Testing Sets, Covering Paths,)7 3467 1 1146 1230 t
(and Minimum Flows)2 1069 1 2345 1380 t
10 I f
(Alfred V. Aho)2 547 1 2606 1770 t
(David Lee)1 413 1 2673 1950 t
10 R f
(AT&T Bell Laboratories)2 993 1 2383 2130 t
(Murray Hill, New Jersey)3 992 1 2384 2250 t
10 I f
(ABSTRACT)2643 2630 w
10 R f
(Although testing is an essential part of program and circuit design, the area is still)14 3350 1 1330 2890 t
( considers several fundamental problems arising in)6 2042( paper)1 249( This)1 231(more an art than a science.)5 1078 4 1080 3010 t
( on)1 140(program and circuit testing, and abstracts them in terms of path-covering problems)11 3460 2 1080 3130 t
( problems are representative of important classes of graph-optimization)8 3002(graphs. These)1 598 2 1080 3250 t
( problems.)1 442(problems, and we introduce a technique called ``balancing'' to solve these)10 3158 2 1080 3370 t
( that are faster, simpler, and easier to implement than)9 2247(This technique yields algorithms)3 1353 2 1080 3490 t
( in the classes of problems are)6 1299( Included)1 415(those obtained by applying existing methods.)5 1886 3 1080 3610 t
(minimum network-flow problems and the Chinese-postman problem.)6 2773 1 1080 3730 t
10 B f
( Sets, Covering Paths, and Minimum Flows)6 1845(1. Testing)1 442 2 720 4090 t
10 R f
( test a)2 249( To)1 169( important part of the process of constructing reliable programs and circuits.)11 3127(Testing is an)2 525 4 970 4246 t
(program one would like to construct a set of inputs that exercises, if possible, each edge \(and/or node\) in)18 4320 1 720 4366 t
( such as)2 322( one would like to minimize the cost of the test,)10 1922( Furthermore,)1 576(the program flow graph at least once.)6 1500 4 720 4486 t
( programmers have attested to)4 1219( good)1 230( Many)1 288(the size of the test suite or the time required to execute the test.)13 2583 4 720 4606 t
(the difficulty of constructing comprehensive test suites [Knuth, 1984].)8 2808 1 720 4726 t
( A)1 127( set of paths such that each node is on at least one path.)13 2267(To test a logic circuit, one can sensitize a)8 1676 3 970 4882 t
( reduce the cost)3 635( To)1 165( sequential\) or a block of such elements.)7 1636(node can be a logic element \(combinational or)7 1884 4 720 5002 t
( also want to minimize the)5 1078( may)1 200( One)1 219(of testing, one would like to minimize the number of paths sensitized.)11 2823 4 720 5122 t
( Uyar and Dahbura,)3 801(total length of the paths to reduce the complexity of the test [Seth and Agrawal, 1985;)15 3519 2 720 5242 t
(1986].)720 5362 w
( directed graph,)2 631(To simplify the problem, we will represent a program flow graph or logic circuit as a)15 3439 2 970 5518 t
10 I f
(G)720 5638 w
10 S f
(=)841 5638 w
10 R f
([)945 5638 w
10 I f
(V)986 5638 w
15 S f
(\310)1104 5668 w
10 I f
({ s)1 111 1 1277 5638 t
10 R f
(,)1396 5638 w
10 I f
(t })1 100 1 1453 5638 t
10 R f
(,)1561 5638 w
10 I f
(E)1627 5638 w
10 R f
(], with)1 264 1 1696 5638 t
10 I f
(n)1988 5638 w
10 R f
(vertices and)1 482 1 2066 5638 t
10 I f
(m)2576 5638 w
10 R f
( distinguished vertex)2 839(edges. The)1 460 2 2676 5638 t
10 I f
(s)4003 5638 w
10 R f
(is called a)2 405 1 4070 5638 t
10 I f
(source)4503 5638 w
10 R f
(; it has)2 271 1 4769 5638 t
( distinguished vertex)2 849( The)1 213(in-degree zero and corresponds to a unique entry point.)8 2267 3 720 5768 t
10 I f
(t)4082 5768 w
10 R f
(is called a)2 417 1 4143 5768 t
10 I f
(sink)4594 5768 w
10 R f
(; it has)2 285 1 4755 5768 t
( vertices in)2 450( The)1 211(out-degree zero and corresponds to a unique exit point.)8 2246 3 720 5888 t
10 I f
(V)3658 5888 w
10 R f
(are called)1 390 1 3750 5888 t
10 I f
(intermediate)4170 5888 w
10 R f
(vertices.)4705 5888 w
(Each edge)1 447 1 720 6008 t
10 I f
(e)1227 6008 w
10 R f
(in)1331 6008 w
10 I f
(E)1470 6008 w
10 R f
(has a lower and an upper capacity bound)7 1878 1 1592 6008 t
10 I f
(l)3531 6008 w
10 R f
(\()3567 6008 w
10 I f
(e)3608 6008 w
10 R f
(\) and)1 238 1 3660 6008 t
10 I f
(u)3959 6008 w
10 R f
(\()4017 6008 w
10 I f
(e)4058 6008 w
10 R f
(\), respectively, where)2 930 1 4110 6008 t
(0)720 6128 w
10 S f
(\243)811 6128 w
10 I f
(l)907 6128 w
10 R f
(\()943 6128 w
10 I f
(e)984 6128 w
10 R f
(\))1036 6128 w
10 S f
(\243)1118 6128 w
10 I f
(u)1214 6128 w
10 R f
(\()1272 6128 w
10 I f
(e)1313 6128 w
10 R f
(\), and also has an associated nonnegative cost,)7 1849 1 1365 6128 t
10 I f
(cost)3239 6128 w
10 R f
(\()3408 6128 w
10 I f
(e)3449 6128 w
10 R f
(\).)3501 6128 w
( of the techniques presented here can also provide algorithms with the same time)13 3446(A modification)1 624 2 970 6284 t
( For)1 190( a cost and/or capacity lower and upper bound is assigned to each vertex.)13 2923(bounds for problems in which)4 1207 3 720 6404 t
( capacity)1 362(simplicity, however, we will consider directed graphs where only edges have associated costs and)13 3958 2 720 6524 t
(bounds.)720 6644 w
(A)970 6800 w
10 I f
(v)1078 6800 w
7 R f
(0)1133 6820 w
10 S f
(-)1192 6800 w
10 I f
(v)1263 6800 w
7 I f
(r)1318 6820 w
10 R f
(path)1389 6800 w
10 I f
(p)1597 6800 w
10 R f
(in)1683 6800 w
10 I f
(G)1797 6800 w
10 R f
(is a sequence of vertices)4 1013 1 1905 6800 t
10 I f
(v)2954 6800 w
7 R f
(0)3009 6820 w
10 R f
(, . . . ,)4 225 1 3060 6800 t
10 I f
(v)3342 6800 w
7 I f
(r)3397 6820 w
10 R f
( of edges)2 384(such that there is a sequence)5 1188 2 3468 6800 t
10 I f
(e)720 6920 w
7 R f
(0)775 6940 w
10 R f
(, . . . ,)4 225 1 826 6920 t
10 I f
(e)1108 6920 w
7 I f
(r)1163 6940 w
7 S f
(-)1206 6940 w
7 R f
(1)1256 6940 w
10 R f
(, where)1 319 1 1299 6920 t
10 I f
(e)1669 6920 w
7 I f
(i)1724 6940 w
10 S f
(=)1801 6920 w
10 R f
(\()1905 6920 w
10 I f
(v)1946 6920 w
7 I f
(i)2001 6940 w
10 R f
(,)2037 6920 w
10 I f
(v)2103 6920 w
7 I f
(i)2158 6940 w
7 S f
(+)2194 6940 w
7 R f
(1)2244 6940 w
10 R f
(\), for)1 225 1 2295 6920 t
10 I f
(i)2571 6920 w
10 S f
(=)2648 6920 w
10 R f
( . . . ,)4 200(0 ,)1 83 2 2752 6920 t
10 I f
(r)3092 6920 w
10 S f
(-)3180 6920 w
10 R f
( such a)2 329(1. For)1 290 2 3284 6920 t
10 I f
(v)3953 6920 w
7 R f
(0)4008 6940 w
10 S f
(-)4067 6920 w
10 I f
(v)4138 6920 w
7 I f
(r)4193 6940 w
10 R f
(path)4278 6920 w
10 I f
(p)4500 6920 w
10 R f
(, we define)2 490 1 4550 6920 t
10 I f
(cost)720 7090 w
10 R f
(\()889 7090 w
10 I f
(p)930 7090 w
10 R f
(\) to be)2 259 1 988 7090 t
7 I f
(i)1278 7190 w
7 S f
(=)1332 7190 w
7 R f
(0)1405 7190 w
15 S f
(S)1315 7120 w
7 I f
(r)1275 6990 w
7 S f
(-)1336 6990 w
7 R f
(1)1409 6990 w
10 I f
(cost)1452 7090 w
10 R f
(\()1621 7090 w
10 I f
(e)1662 7090 w
7 I f
(i)1717 7110 w
10 R f
( set of)2 250(\). A)1 183 2 1753 7090 t
10 I f
(covering)2214 7090 w
10 R f
(paths for)1 355 1 2591 7090 t
10 I f
(G)2974 7090 w
10 R f
(is a set of)3 389 1 3074 7090 t
10 I f
(s)3491 7090 w
10 S f
(-)3554 7090 w
10 I f
(t)3625 7090 w
10 R f
(paths such that each edge)4 1026 1 3681 7090 t
10 I f
(e)4735 7090 w
10 R f
(is tra-)1 233 1 4807 7090 t
( once and no more than)5 1024(versed at least)2 603 2 720 7290 t
10 I f
(u)2390 7290 w
10 R f
(\()2448 7290 w
10 I f
(e)2489 7290 w
10 R f
(\) times, where)2 604 1 2541 7290 t
10 I f
(u)3188 7290 w
10 R f
(\()3246 7290 w
10 I f
(e)3287 7290 w
10 R f
(\))3339 7290 w
10 S f
(\263)3421 7290 w
10 R f
( basic versions of the testing)5 1225(1. The)1 298 2 3517 7290 t
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10 R f
(- 2 -)2 166 1 2797 480 t
(problems can be couched as covering-paths problems:)6 2162 1 720 840 t
10 B f
(Problem)720 1092 w
10 I f
(CP)1111 1092 w
7 R f
(1)1244 1111 w
(\( 1 \))2 91 1 1244 1052 t
10 R f
(.)1343 1092 w
10 I f
(Minimum-cardinality covering paths)2 1471 1 1418 1092 t
10 R f
(.)2889 1092 w
(Find a set of covering paths for)6 1248 1 970 1248 t
10 I f
(G)2243 1248 w
10 R f
(such that the number of paths is minimum.)7 1711 1 2340 1248 t
10 B f
(Problem)720 1464 w
10 I f
(CP)1111 1464 w
7 R f
(2)1244 1483 w
(\( 1 \))2 91 1 1244 1424 t
10 R f
(.)1343 1464 w
10 I f
(Minimum-cost minimum-cardinality covering paths)3 2062 1 1418 1464 t
10 R f
(.)3480 1464 w
(Among the sets of minimum-cardinality covering paths, find one with minimum cost.)11 3424 1 970 1620 t
10 B f
(Problem)720 1836 w
10 I f
(CP)1111 1836 w
7 R f
(3)1244 1855 w
(\( 1 \))2 91 1 1244 1796 t
10 R f
(.)1343 1836 w
10 I f
(Minimum-cost covering paths)2 1193 1 1418 1836 t
10 R f
(.)2611 1836 w
(Find a minimum-cost set of covering paths for)7 1857 1 970 1992 t
10 I f
(G)2852 1992 w
10 R f
(.)2924 1992 w
( will be distin-)3 602( forms)1 264( These)1 294(Throughout this paper, three forms of each problem will be considered.)10 2910 4 970 2268 t
( a minimum-cardinality problem, a value of two a)8 2093(guished by a subscript, where a value of one denotes)9 2227 2 720 2388 t
( addition,)1 386( In)1 141( problem, and a value of three a minimum-cost problem.)9 2327(minimum-cost minimum-cardinality)1 1466 4 720 2508 t
( if any, are placed)4 739(we will consider several versions of each problem form depending on what constraints,)12 3581 2 720 2628 t
( version in which, for every edge)6 1370( The)1 214(upon the edge capacity bounds.)4 1292 3 720 2748 t
10 I f
(e)3630 2748 w
10 R f
(,)3674 2748 w
10 I f
(l)3733 2748 w
10 R f
(\()3769 2748 w
10 I f
(e)3810 2748 w
10 R f
(\))3862 2748 w
10 S f
(=)3952 2748 w
10 R f
(1 and)1 228 1 4056 2748 t
10 I f
(u)4318 2748 w
10 R f
(\()4376 2748 w
10 I f
(e)4417 2748 w
10 R f
(\))4469 2748 w
10 S f
(\243 \245)1 169 1 4551 2748 t
10 R f
(will be)1 285 1 4755 2748 t
( we can let)3 429( the purpose of program or logic circuit testing,)8 1897( For)1 190( 1 \).)2 124(denoted by the superscript \()4 1113 5 720 2868 t
10 I f
(l)4498 2868 w
10 R f
(\()4534 2868 w
10 I f
(e)4575 2868 w
10 R f
(\))4627 2868 w
10 S f
(=)4717 2868 w
10 R f
(1 and)1 219 1 4821 2868 t
10 I f
(u)720 2988 w
10 R f
(\()778 2988 w
10 I f
(e)819 2988 w
10 R f
(\))871 2988 w
10 S f
(= \245)1 177 1 961 2988 t
10 R f
( third version, denoted by the superscript)6 1640( A)1 123( 2 \).)2 124( version will be denoted by the superscript \()8 1761(. This)1 254 5 1138 2988 t
( where 0)2 343(\( 0 \),)2 157 2 720 3108 t
10 S f
(\243)1261 3108 w
10 I f
(l)1357 3108 w
10 R f
(\()1393 3108 w
10 I f
(e)1434 3108 w
10 R f
(\))1486 3108 w
10 S f
(\243)1568 3108 w
10 I f
(u)1664 3108 w
10 R f
(\()1722 3108 w
10 I f
(e)1763 3108 w
10 R f
(\))1815 3108 w
10 S f
(\243 \245)1 169 1 1897 3108 t
10 R f
(will be considered later.)3 959 1 2091 3108 t
(Problem)970 3264 w
10 I f
(CP)1334 3264 w
7 R f
(1)1467 3283 w
(\( 2 \))2 91 1 1467 3224 t
10 R f
( by Ntafos and Hakimi [1979] in the context of program testing, but if one)14 2980(was studied)1 469 2 1591 3264 t
( the test, then solutions to Problems)6 1432(wants to minimize the cost of)5 1190 2 720 3384 t
10 I f
(CP)3368 3384 w
7 R f
(2)3501 3403 w
(\( 2 \))2 91 1 3501 3344 t
10 R f
(and)3626 3384 w
10 I f
(CP)3796 3384 w
7 R f
(3)3929 3403 w
(\( 2 \))2 91 1 3929 3344 t
10 R f
( Special)1 345(are also needed.)2 641 2 4054 3384 t
( have been considered by Krause, Smith, and Goodwin [1973], Gabow, Mahesh-)11 3345(cases of these problems)3 975 2 720 3504 t
(wari, and Osterweil [1976], and Ntafos and Hakimi [1979].)8 2370 1 720 3624 t
( problems on directed graphs with arbitrary capac-)7 2042(Here we consider the more general covering-paths)6 2028 2 970 3780 t
( Section 4 we show that these covering-paths problems are equivalent to)11 2931( In)1 137( bounds.)1 343(ity upper)1 363 4 720 3900 t
10 I f
(positive-flow)4523 3900 w
10 R f
(problems, which are special cases of)5 1567 1 720 4020 t
10 I f
(minimum-flow)2334 4020 w
10 R f
( way to solve these)4 852( One)1 239(problems [Even, 1979].)2 989 3 2960 4020 t
( maximum-flow problems as in Ford and Fulk-)7 1901(minimum-flow problems is to reduce them to a sequence of)9 2419 2 720 4140 t
( find an algorithm for Prob-)5 1127( this approach, we can)4 901( Following)1 465(erson [1962], Lawler [1976], or Even [1979].)6 1827 4 720 4260 t
(lem)720 4380 w
10 I f
(CP)949 4380 w
7 R f
(1)1082 4399 w
(\( 1 \))2 91 1 1082 4340 t
10 R f
(with cost)1 418 1 1260 4380 t
10 I f
(O)1756 4380 w
10 R f
(\()1836 4380 w
10 I f
(mn)1877 4380 w
10 R f
(log)2040 4380 w
10 I f
(n)2209 4380 w
10 R f
(\), and algorithms for Problems)4 1436 1 2267 4380 t
10 I f
(CP)3781 4380 w
7 R f
(2)3914 4399 w
(\( 1 \))2 91 1 3914 4340 t
10 R f
(and)4091 4380 w
10 I f
(CP)4313 4380 w
7 R f
(3)4446 4399 w
(\( 1 \))2 91 1 4446 4340 t
10 R f
(with cost)1 417 1 4623 4380 t
10 I f
(O)720 4500 w
10 R f
(\()800 4500 w
10 I f
(n)841 4500 w
7 R f
(2)902 4460 w
10 R f
(\()953 4500 w
10 I f
(m)994 4500 w
10 S f
(+)1115 4500 w
10 I f
(n)1219 4500 w
10 R f
(log)1310 4500 w
10 I f
(n)1479 4500 w
10 R f
(\) log)1 210 1 1537 4500 t
10 I f
(n)1788 4500 w
10 R f
(\) using the methods of Tarjan [1983] and Galil and Tardos [1986].)11 2651 1 1846 4500 t
(Instead, we use)2 634 1 970 4656 t
10 I f
(balancing)1640 4656 w
10 R f
( first reduce the minimum-)4 1119( We)1 200(to solve these minimum-flow problems.)4 1645 3 2076 4656 t
( pre-)1 189(flow problems to minimum-cost maximum-flow problems, but the reductions are different from those)12 4131 2 720 4776 t
( balancing, the reduced flow)4 1177( Using)1 300( [Ford and Fulkerson, 1962; Lawler, 1976; Even, 1979].)8 2330(viously used)1 513 4 720 4896 t
( as)1 133(problem has an obvious bound on the flow value, and efficient algorithms can be devised, such)15 4187 2 720 5016 t
( our approach is intuitive and easy to implement.)8 2047( Furthermore,)1 585(minimum-cost augmentation and scaling.)3 1688 3 720 5136 t
( feel it is worthwhile to)5 995( We)1 201( presentation of this approach.)4 1258(To our knowledge, there has been no formal)7 1866 4 720 5256 t
(explore this method and to apply it to practical problems, since the ideas involved are simple.)15 3732 1 720 5376 t
( been well)2 420(We also present methods for minimizing both the flow value and the cost, which have not)15 3650 2 970 5532 t
( also be used for)4 674( can)1 169(studied. Balancing)1 775 3 720 5652 t
10 I f
(circulation)2368 5652 w
10 R f
( special cases, we present algorithms for)6 1640(problems. As)1 563 2 2837 5652 t
( example, we obtain an)4 999( For)1 208(covering-paths and postman-tour problems.)3 1799 3 720 5772 t
10 I f
(O)3770 5772 w
10 R f
(\()3850 5772 w
10 I f
(n)3891 5772 w
10 R f
(\()3949 5772 w
10 I f
(m)3990 5772 w
10 S f
(+)4111 5772 w
10 I f
(n)4215 5772 w
10 R f
(log)4306 5772 w
10 I f
(n)4475 5772 w
10 R f
( \()1 41(\) log)1 210 2 4533 5772 t
10 I f
(m / n)2 166 1 4792 5772 t
10 R f
(\) \))1 74 1 4966 5772 t
(algorithm for the)2 679 1 720 5892 t
10 I f
(Chinese-postman)1425 5892 w
10 R f
( algorithm was)2 594( this problem, the best previously published)6 1753(problem. For)1 548 3 2145 5892 t
10 I f
(O)720 6012 w
10 R f
(\()800 6012 w
10 I f
(n)841 6012 w
7 R f
(5)902 5972 w
10 R f
(\) [Papadimitriou, 1976], and an)4 1258 1 953 6012 t
10 I f
(O)2237 6012 w
10 R f
(\()2317 6012 w
10 I f
(mn)2358 6012 w
10 R f
(log)2521 6012 w
10 I f
(n)2690 6012 w
10 R f
( be obtained by reducing the problem to a)8 1680(\) algorithm can)2 612 2 2748 6012 t
( by scaling [Gabow and Tar-)5 1154(minimum-cost flow problem using balancing and solving the reduced problem)9 3166 2 720 6132 t
( versions of the covering-paths)4 1279( Several)1 361( is faster than either of these methods.)7 1598( algorithm)1 425( Our)1 216(jan, 1987].)1 441 6 720 6252 t
(problems and the postman-tour problems can also be solved with the same run time.)13 3363 1 720 6372 t
( example, we show that Problem)5 1329( For)1 194( study these problems on mixed graphs.)6 1615(We also)1 328 4 970 6528 t
10 I f
(CP)4466 6528 w
7 R f
(1)4599 6547 w
(\()4599 6488 w
7 I f
(j)4638 6488 w
7 R f
(\))4663 6488 w
10 R f
(remains)4724 6528 w
(polynomial, but Problems)2 1037 1 720 6648 t
10 I f
(CP)1782 6648 w
7 I f
(i)1915 6667 w
7 R f
(\()1915 6608 w
7 I f
(j)1954 6608 w
7 R f
(\))1979 6608 w
10 R f
(,)2010 6648 w
10 I f
(i)2060 6648 w
10 S f
(=)2137 6648 w
10 R f
( and)1 169( ,)1 33( 3)1 91(2 ,)1 83 4 2241 6648 t
10 I f
(j)2642 6648 w
10 S f
(=)2719 6648 w
10 R f
( become NP-complete.)2 912( 2,)1 116(1 ,)1 83 3 2823 6648 t
( we spe-)2 350( Then)1 264( problems using balancing.)3 1104(We begin by providing algorithms for the minimum-flow)7 2352 4 970 6804 t
(cialize the algorithms to)3 1026 1 720 6924 t
10 I f
(positive-flow)1792 6924 w
10 R f
( equivalent to the covering-paths problems.)5 1831(problems, which are)2 854 2 2355 6924 t
(Finally, we consider several versions of minimum-circulation and postman-tour problems.)9 3610 1 720 7044 t
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3 pagesetup
10 R f
(- 3 -)2 166 1 2797 480 t
10 B f
( Problems)1 430(2. Minimum-Flow)1 797 2 720 840 t
10 R f
(A)970 996 w
10 I f
(preflow f)1 368 1 1076 996 t
10 R f
(on a directed graph)3 794 1 1478 996 t
10 I f
(G)2306 996 w
10 S f
(=)2427 996 w
10 R f
([)2531 996 w
10 I f
(V)2572 996 w
15 S f
(\310)2690 1026 w
10 I f
({ s)1 111 1 2863 996 t
10 R f
(,)2982 996 w
10 I f
(t })1 100 1 3039 996 t
10 R f
(,)3147 996 w
10 I f
(E)3213 996 w
10 R f
( from)1 229(] is a function)3 579 2 3282 996 t
10 I f
(E)4125 996 w
10 R f
(to nonnegative inte-)2 819 1 4221 996 t
( define)1 274(gers. We)1 379 2 720 1126 t
10 I f
(cost)1398 1126 w
10 R f
(\()1567 1126 w
10 I f
(f)1624 1126 w
10 R f
(\) as)1 141 1 1676 1126 t
7 I f
(e)1842 1226 w
7 S f
(\316)1878 1226 w
7 I f
(E)1933 1226 w
15 S f
(S)1864 1156 w
10 I f
(f)2032 1126 w
10 R f
(\()2076 1126 w
10 I f
(e)2117 1126 w
10 R f
(\))2169 1126 w
10 I f
(cost)2218 1126 w
10 R f
(\()2387 1126 w
10 I f
(e)2428 1126 w
10 R f
(\). The)1 263 1 2480 1126 t
10 I f
(balancing index)1 641 1 2768 1126 t
10 R f
(of a vertex)2 426 1 3434 1126 t
10 I f
(v)3885 1126 w
10 R f
(in)3954 1126 w
10 I f
(G)4057 1126 w
10 R f
(with a preflow)2 582 1 4154 1126 t
10 I f
(f)4761 1126 w
10 R f
(is)4814 1126 w
10 S f
(b)1937 1386 w
10 R f
(\()2000 1386 w
10 I f
(v)2041 1386 w
10 R f
(,)2093 1386 w
10 I f
(f)2167 1386 w
10 R f
(\))2219 1386 w
10 S f
(=)2309 1386 w
7 R f
(\()2413 1486 w
7 I f
(v)2441 1486 w
7 R f
(,)2477 1486 w
7 I f
(w)2500 1486 w
7 R f
(\))2552 1486 w
7 S f
(\316)2586 1486 w
7 I f
(E)2641 1486 w
15 S f
(S)2504 1416 w
10 I f
(f)2741 1386 w
10 R f
(\()2785 1386 w
10 I f
(v)2826 1386 w
10 R f
(,)2878 1386 w
10 I f
(w)2944 1386 w
10 R f
(\))3019 1386 w
10 S f
(-)3109 1386 w
7 R f
(\()3213 1486 w
7 I f
(u)3241 1486 w
7 R f
(,)3281 1486 w
7 I f
(v)3304 1486 w
7 R f
(\))3340 1486 w
7 S f
(\316)3374 1486 w
7 I f
(E)3429 1486 w
15 S f
(S)3298 1416 w
10 I f
(f)3529 1386 w
10 R f
(\()3573 1386 w
10 I f
(u)3614 1386 w
10 R f
(,)3672 1386 w
10 I f
(v)3738 1386 w
10 R f
(\))3790 1386 w
(We define)1 412 1 720 1646 t
10 I f
(value)1157 1646 w
10 R f
(\()1381 1646 w
10 I f
(f)1438 1646 w
10 R f
(\) to be)2 255 1 1490 1646 t
10 S f
(b)1770 1646 w
10 R f
(\()1833 1646 w
10 I f
(s)1874 1646 w
10 R f
(,)1921 1646 w
10 I f
(f)1995 1646 w
10 R f
(\).)2047 1646 w
(A)970 1802 w
10 I f
(flow)1075 1802 w
10 R f
(on)1281 1802 w
10 I f
(G)1414 1802 w
10 R f
(is a preflow)2 487 1 1519 1802 t
10 I f
(f)2039 1802 w
10 R f
(such that)1 366 1 2100 1802 t
10 I f
(l)2500 1802 w
10 R f
(\()2536 1802 w
10 I f
(e)2577 1802 w
10 R f
(\))2629 1802 w
10 S f
(\243)2711 1802 w
10 I f
(f)2815 1802 w
10 R f
(\()2859 1802 w
10 I f
(e)2900 1802 w
10 R f
(\))2952 1802 w
10 S f
(\243)3034 1802 w
10 I f
(u)3130 1802 w
10 R f
(\()3188 1802 w
10 I f
(e)3229 1802 w
10 R f
(\) for all)2 317 1 3281 1802 t
10 I f
(e)3632 1802 w
10 R f
(in)3710 1802 w
10 I f
(E)3822 1802 w
10 R f
(and such that the balancing)4 1123 1 3917 1802 t
(index)720 1922 w
10 S f
(b)967 1922 w
10 R f
(\()1030 1922 w
10 I f
(v)1071 1922 w
10 R f
(,)1123 1922 w
10 I f
(f)1197 1922 w
10 R f
(\) is zero for all intermediate vertices)6 1446 1 1249 1922 t
10 I f
(v)2720 1922 w
10 R f
(in)2789 1922 w
10 I f
(V)2892 1922 w
10 R f
(.)2953 1922 w
(We can now consider the following minimum-flow problems:)7 2478 1 970 2078 t
10 B f
(Problem)720 2294 w
10 I f
(F)1111 2294 w
7 R f
(1)1177 2313 w
(\( 0 \))2 91 1 1177 2254 t
10 R f
(.)1276 2294 w
10 I f
(Minimum flow)1 581 1 1351 2294 t
10 R f
(.)1932 2294 w
(Find a flow for)3 602 1 970 2450 t
10 I f
(G)1597 2450 w
10 R f
(with the minimum flow value.)4 1214 1 1694 2450 t
10 B f
(Problem)720 2666 w
10 I f
(F)1111 2666 w
7 R f
(2)1177 2685 w
(\( 0 \))2 91 1 1177 2626 t
10 R f
(.)1276 2666 w
10 I f
(Minimum-cost minimum flow)2 1172 1 1351 2666 t
10 R f
(.)2523 2666 w
(Among the minimum flows for)4 1250 1 970 2822 t
10 I f
(G)2245 2822 w
10 R f
(, find one with minimum cost.)5 1209 1 2317 2822 t
10 B f
(Problem)720 3038 w
10 I f
(F)1111 3038 w
7 R f
(3)1177 3057 w
(\( 0 \))2 91 1 1177 2998 t
10 R f
(.)1276 3038 w
10 I f
(Minimum-cost flow)1 775 1 1351 3038 t
10 R f
(.)2126 3038 w
(Find a flow for)3 602 1 970 3194 t
10 I f
(G)1597 3194 w
10 R f
(with minimum cost.)2 804 1 1694 3194 t
( for)1 148( indicates that there are no constraints on the capacity bounds on the edges; that is,)15 3385( 0 \))2 99(The superscript \()2 688 4 720 3410 t
(each edge)1 395 1 720 3530 t
10 I f
(e)1140 3530 w
10 R f
(, 0)1 100 1 1184 3530 t
10 S f
(\243)1325 3530 w
10 I f
(l)1421 3530 w
10 R f
(\()1457 3530 w
10 I f
(e)1498 3530 w
10 R f
(\))1550 3530 w
10 S f
(\243)1632 3530 w
10 I f
(u)1728 3530 w
10 R f
(\()1786 3530 w
10 I f
(e)1827 3530 w
10 R f
(\))1879 3530 w
10 S f
(\243 \245)1 169 1 1961 3530 t
10 R f
(.)2130 3530 w
( these minimum-flow problems using balancing when-)6 2217(In the next section, we provide algorithms for)7 1853 2 970 3686 t
( approach has been to)4 920( conventional)1 556( The)1 220(ever they have a feasible solution [Lawler, 1976; Even, 1979].)9 2624 4 720 3806 t
( second maximum-flow problem,)3 1356(reduce a minimum-flow problem to a maximum-flow problem, then to a)10 2964 2 720 3926 t
( the other)2 403( On)1 185( this has a feasible solution, optimize it by solving a third maximum-flow problem.)13 3488(and if)1 244 4 720 4046 t
( optimal solution directly by solving a maximum-flow problem on a)10 2822(hand, with balancing we can find an)6 1498 2 720 4166 t
(modified graph.)1 638 1 720 4286 t
10 B f
( for Minimum-Flow Problems)3 1279(3. Algorithms)1 608 2 720 4526 t
10 R f
( to preflow-minimization problems,)3 1469(We first show that the minimum-flow problems are equivalent)8 2601 2 970 4682 t
( so-called)1 385(which can be reduced to maximum-flow problems on a)8 2215 2 720 4802 t
10 I f
(balancing graph)1 680 1 3345 4802 t
10 R f
(with only capacity upper)3 990 1 4050 4802 t
( make the reduction conceptually clear.)5 1578( problems are introduced to)4 1102( Balancing-preflow)1 799(bounds on the edges.)3 841 4 720 4922 t
( modification of the original)4 1183(For practical purposes, one can construct the balancing graphs by a simple)11 3137 2 720 5042 t
(graph, and then solve the final maximum-flow problems directly.)8 2608 1 720 5162 t
10 B f
( Problems)1 430(3.1. Balancing-Preflow)1 994 2 720 5402 t
10 R f
(A preflow)1 416 1 970 5558 t
10 I f
(f)1420 5558 w
7 R f
(0)1459 5578 w
10 R f
(is)1536 5558 w
10 I f
(basic)1637 5558 w
10 R f
(if for every)2 466 1 1882 5558 t
10 I f
(e)2382 5558 w
10 R f
(in)2460 5558 w
10 I f
(E)2572 5558 w
10 R f
(,)2633 5558 w
10 I f
(f)2693 5558 w
7 R f
(0)2732 5578 w
10 R f
(\()2783 5558 w
10 I f
(e)2824 5558 w
10 R f
(\))2876 5558 w
10 S f
(=)2966 5558 w
10 I f
(l)3070 5558 w
10 R f
(\()3106 5558 w
10 I f
(e)3147 5558 w
10 R f
(\), where)1 336 1 3199 5558 t
10 I f
(l)3570 5558 w
10 R f
(\()3606 5558 w
10 I f
(e)3647 5558 w
10 R f
(\) is the capacity lower bound on)6 1341 1 3699 5558 t
(edge)720 5678 w
10 I f
(e)947 5678 w
10 R f
( intermediate vertices)2 885( partition the set of)4 811(. We)1 227 3 991 5678 t
10 I f
(V)2952 5678 w
10 R f
(into three sets:)2 609 1 3051 5678 t
10 I f
(S)3698 5678 w
10 R f
(\(surplus\),)3786 5678 w
10 I f
(B)4204 5678 w
10 R f
(\(balanced\), and)1 627 1 4303 5678 t
10 I f
(D)4968 5678 w
10 R f
( intermediate vertex)2 806(\(deficient\). An)1 616 2 720 5798 t
10 I f
(v)2171 5798 w
10 R f
(is in)1 174 1 2244 5798 t
10 I f
(S)2447 5798 w
10 R f
(,)2497 5798 w
10 I f
(B)2551 5798 w
10 R f
(, or)1 137 1 2612 5798 t
10 I f
(D)2778 5798 w
10 R f
(depending on whether)2 895 1 2879 5798 t
10 S f
(b)3804 5798 w
10 R f
(\()3867 5798 w
10 I f
(v)3908 5798 w
10 R f
(,)3960 5798 w
10 I f
(f)4034 5798 w
7 R f
(0)4073 5818 w
10 R f
(\) is less than, equal to,)5 916 1 4124 5798 t
( preflow)1 343( A)1 130(or greater than zero, respectively.)4 1366 3 720 5918 t
10 I f
(f)2591 5918 w
7 I f
(b)2630 5938 w
10 R f
(is)2705 5918 w
10 I f
(balancing)2804 5918 w
10 R f
(if)3236 5918 w
10 I f
(f)3329 5918 w
7 I f
(b)3368 5938 w
10 R f
(\()3419 5918 w
10 I f
(e)3460 5918 w
10 R f
(\))3512 5918 w
10 S f
(\243)3594 5918 w
10 I f
(u)3690 5918 w
10 R f
(\()3748 5918 w
10 I f
(e)3789 5918 w
10 R f
(\))3841 5918 w
10 S f
(-)3931 5918 w
10 I f
(l)4035 5918 w
10 R f
(\()4071 5918 w
10 I f
(e)4112 5918 w
10 R f
(\) for all edges)3 572 1 4164 5918 t
10 I f
(e)4768 5918 w
10 R f
(in)4844 5918 w
10 I f
(E)4954 5918 w
10 R f
(,)5015 5918 w
(and)720 6038 w
10 S f
(b)903 6038 w
10 R f
(\()966 6038 w
10 I f
(v)1007 6038 w
10 R f
(,)1059 6038 w
10 I f
(f)1133 6038 w
7 I f
(b)1172 6058 w
10 R f
(\))1223 6038 w
10 S f
( b)1 71(= -)1 167 2 1313 6038 t
10 R f
(\()1559 6038 w
10 I f
(v)1600 6038 w
10 R f
(,)1652 6038 w
10 I f
(f)1726 6038 w
7 R f
(0)1765 6058 w
10 R f
(\) for all intermediate vertices)4 1214 1 1816 6038 t
10 I f
(v)3069 6038 w
10 R f
(in)3152 6038 w
10 I f
(V)3269 6038 w
10 R f
( not)1 168( that a balancing preflow may)5 1259(. Note)1 283 3 3330 6038 t
(exist because of the capacity bounds on the edges.)8 2004 1 720 6158 t
10 B f
(Theorem 3.1.)1 565 1 720 6314 t
10 R f
(\(Decomposition of a feasible flow\).)4 1430 1 1312 6314 t
10 I f
(f)2794 6314 w
10 R f
( on)1 128(is a flow)2 348 2 2849 6314 t
10 I f
(G)3353 6314 w
10 R f
(if and only if)3 528 1 3453 6314 t
10 I f
(f)4009 6314 w
10 S f
(=)4102 6314 w
10 I f
(f)4214 6314 w
7 R f
(0)4253 6334 w
10 S f
(+)4345 6314 w
10 I f
(f)4457 6314 w
7 I f
(b)4496 6334 w
10 R f
(, where)1 296 1 4539 6314 t
10 I f
(f)4863 6314 w
7 R f
(0)4902 6334 w
10 R f
(is)4973 6314 w
(the basic preflow and)3 856 1 720 6434 t
10 I f
(f)1601 6434 w
7 I f
(b)1640 6454 w
10 R f
(is a balancing preflow on)4 1009 1 1708 6434 t
10 I f
(G)2742 6434 w
10 R f
(.)2814 6434 w
10 B f
(Proof.)720 6614 w
10 R f
(Let)1046 6614 w
10 I f
(f)1242 6614 w
7 I f
(b)1281 6634 w
10 R f
( Since)1 310(be a balancing preflow.)3 1050 2 1387 6614 t
10 I f
(l)2810 6614 w
10 R f
(\()2846 6614 w
10 I f
(e)2887 6614 w
10 R f
(\))2939 6614 w
10 S f
(\243)3021 6614 w
10 I f
(f)3125 6614 w
7 R f
(0)3164 6634 w
10 R f
(\()3215 6614 w
10 I f
(e)3256 6614 w
10 R f
(\))3308 6614 w
10 S f
(+)3398 6614 w
10 I f
(f)3510 6614 w
7 I f
(b)3549 6634 w
10 R f
(\()3600 6614 w
10 I f
(e)3641 6614 w
10 R f
(\))3693 6614 w
10 S f
(\243)3789 6614 w
10 I f
(u)3907 6614 w
10 R f
(\()3965 6614 w
10 I f
(e)4006 6614 w
10 R f
(\) for all)2 375 1 4058 6614 t
10 I f
(e)4496 6614 w
10 R f
(in)4604 6614 w
10 I f
(E)4746 6614 w
10 R f
(, and)1 233 1 4807 6614 t
10 S f
(b)720 6734 w
10 R f
(\()783 6734 w
10 I f
(v)824 6734 w
10 R f
(,)876 6734 w
10 I f
(f)950 6734 w
7 R f
(0)989 6754 w
10 S f
(+)1081 6734 w
10 I f
(f)1193 6734 w
7 I f
(b)1232 6754 w
10 R f
(\) =)1 114 1 1283 6734 t
10 S f
(b)1422 6734 w
10 R f
(\()1485 6734 w
10 I f
(v)1526 6734 w
10 R f
(,)1578 6734 w
10 I f
(f)1652 6734 w
7 R f
(0)1691 6754 w
10 R f
(\) +)1 114 1 1742 6734 t
10 S f
(b)1881 6734 w
10 R f
(\()1944 6734 w
10 I f
(v)1985 6734 w
10 R f
(,)2037 6734 w
10 I f
(f)2111 6734 w
7 I f
(b)2150 6754 w
10 R f
(\) = 0 for all intermediate vertices)6 1314 1 2201 6734 t
10 I f
(v)3540 6734 w
10 R f
(,)3584 6734 w
10 I f
(f)3634 6734 w
7 R f
(0)3673 6754 w
10 S f
(+)3765 6734 w
10 I f
(f)3877 6734 w
7 I f
(b)3916 6754 w
10 R f
(is a feasible flow.)3 704 1 3984 6734 t
( hand, given a flow)4 840(On the other)2 533 2 970 6890 t
10 I f
(f)2386 6890 w
10 R f
(on)2457 6890 w
10 I f
(G)2600 6890 w
10 R f
(, let)1 168 1 2672 6890 t
10 I f
(f)2883 6890 w
7 I f
(b)2922 6910 w
10 S f
(=)3014 6890 w
10 I f
(f)3126 6890 w
10 S f
(-)3219 6890 w
10 I f
(f)3331 6890 w
7 R f
(0)3370 6910 w
10 R f
(, where)1 311 1 3413 6890 t
10 I f
(f)3767 6890 w
7 R f
(0)3806 6910 w
10 R f
( Since)1 290(is the basic preflow.)3 858 2 3892 6890 t
10 I f
(l)720 7010 w
10 R f
(\()756 7010 w
10 I f
(e)797 7010 w
10 R f
(\))849 7010 w
10 S f
(\243)931 7010 w
10 I f
(f)1035 7010 w
10 R f
(\()1079 7010 w
10 I f
(e)1120 7010 w
10 R f
(\))1172 7010 w
10 S f
(\243)1254 7010 w
10 I f
(u)1350 7010 w
10 R f
(\()1408 7010 w
10 I f
(e)1449 7010 w
10 R f
(\) for all)2 375 1 1501 7010 t
10 I f
(e)1939 7010 w
10 R f
(in)2046 7010 w
10 I f
(E)2187 7010 w
10 R f
(,)2248 7010 w
10 I f
(f)2336 7010 w
7 I f
(b)2375 7030 w
10 R f
( intermediate vertices)2 933( all)1 163( For)1 227(is a preflow.)2 572 4 2481 7010 t
10 I f
(v)4438 7010 w
10 R f
(,)4482 7010 w
10 S f
(b)4569 7010 w
10 R f
(\()4632 7010 w
10 I f
(v)4673 7010 w
10 R f
(,)4725 7010 w
10 I f
(f)4799 7010 w
7 I f
(b)4838 7030 w
10 R f
(\) =)1 151 1 4889 7010 t
10 S f
(b)720 7130 w
10 R f
(\()783 7130 w
10 I f
(v)824 7130 w
10 R f
(,)876 7130 w
10 I f
(f)950 7130 w
10 R f
(\))1002 7130 w
10 S f
(- b)1 159 1 1092 7130 t
10 R f
(\()1259 7130 w
10 I f
(v)1300 7130 w
10 R f
(,)1352 7130 w
10 I f
(f)1426 7130 w
7 R f
(0)1465 7150 w
10 R f
(\) = 0)2 189 1 1516 7130 t
10 S f
(- b)1 159 1 1754 7130 t
10 R f
(\()1921 7130 w
10 I f
(v)1962 7130 w
10 R f
(,)2014 7130 w
10 I f
(f)2088 7130 w
7 R f
(0)2127 7150 w
10 R f
(\) and therefore)2 586 1 2178 7130 t
10 I f
(f)2789 7130 w
7 I f
(b)2828 7150 w
10 R f
(is balancing.)1 505 1 2896 7130 t
10 S1 f
()3451 7130 w
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10 S1 f
3451 7130 m
50 build_sq
3501 7130 m
10 R f
(Since the basic preflow depends only on)6 1614 1 970 7286 t
10 I f
(G)2609 7286 w
10 R f
(and has fixed cost and flow value, we have:)8 1743 1 2706 7286 t
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4 pagesetup
10 R f
(- 4 -)2 166 1 2797 480 t
10 B f
(Theorem 3.2.)1 574 1 720 840 t
10 R f
(The minimum-flow problems)2 1205 1 1330 840 t
10 I f
(F)2571 840 w
7 I f
(i)2637 859 w
7 R f
(\( 0 \))2 91 1 2637 800 t
10 R f
(, for)1 178 1 2736 840 t
10 I f
(i)2951 840 w
10 S f
(=)3028 840 w
10 R f
( are equivalent to the following corre-)6 1585( 3,)1 116( ,)1 33( 2)1 91(1 ,)1 83 5 3132 840 t
(sponding balancing-preflow problems:)2 1548 1 720 960 t
10 B f
(Problem)720 1176 w
10 I f
(BF)1111 1176 w
7 R f
(1)1244 1196 w
10 R f
(.)1287 1176 w
10 I f
(Minimum balancing preflow)2 1139 1 1362 1176 t
10 R f
(.)2501 1176 w
(Find a balancing preflow for)4 1142 1 970 1332 t
10 I f
(G)2137 1332 w
10 R f
(with minimum value.)2 859 1 2234 1332 t
10 B f
(Problem)720 1548 w
10 I f
(BF)1111 1548 w
7 R f
(2)1244 1568 w
10 R f
(.)1287 1548 w
10 I f
(Minimum-cost minimum balancing preflow)3 1730 1 1362 1548 t
10 R f
(.)3092 1548 w
(Among the minimum balancing preflows, find one with minimum cost.)9 2858 1 970 1704 t
10 B f
(Problem)720 1920 w
10 I f
(BF)1111 1920 w
7 R f
(3)1244 1940 w
10 R f
(.)1287 1920 w
10 I f
(Minimum-cost balancing preflow)2 1333 1 1362 1920 t
10 R f
(.)2695 1920 w
(Find a balancing preflow for)4 1142 1 970 2076 t
10 I f
(G)2137 2076 w
10 R f
(with minimum cost.)2 804 1 2234 2076 t
10 B f
( Graphs)1 348(3.2. Balancing)1 629 2 720 2316 t
10 R f
(From the directed graph)3 983 1 970 2477 t
10 I f
(G)1985 2477 w
10 R f
(, we construct a)3 647 1 2057 2477 t
10 I f
(balancing graph G)2 777 1 2737 2477 t
10 S1 f
(_ _)1 62 1 3452 2384 t
10 R f
(, and the balancing-preflow problems)4 1526 1 3514 2477 t
(on)720 2602 w
10 I f
(G)846 2602 w
10 R f
( problems on)2 522(can be reduced to corresponding maximum-flow)5 1948 2 944 2602 t
10 I f
(G)3439 2602 w
10 S1 f
(_ _)1 62 1 3449 2509 t
10 R f
(. To construct)2 552 1 3511 2602 t
10 I f
(G)4088 2602 w
10 S1 f
(_ _)1 62 1 4098 2509 t
10 R f
(, we add a new source)5 880 1 4160 2602 t
(vertex)720 2722 w
10 I f
(s)994 2722 w
10 S f
(\242)1041 2722 w
10 R f
(and a new sink vertex)4 870 1 1091 2722 t
10 I f
(t)1986 2722 w
10 S f
(\242)2022 2722 w
10 R f
(to)2072 2722 w
10 I f
(G)2175 2722 w
10 R f
( following edges are also added to)6 1364(. The)1 230 2 2247 2722 t
10 I f
(G)3866 2722 w
10 R f
(:)3938 2722 w
( the edges \()3 457(i\) Add)1 422 2 720 2878 t
10 I f
(s)1607 2878 w
10 S f
(\242)1654 2878 w
10 R f
(,)1687 2878 w
10 I f
(s)1753 2878 w
10 R f
(\) and \()2 260 1 1800 2878 t
10 I f
(t)2068 2878 w
10 R f
(,)2104 2878 w
10 I f
(t)2170 2878 w
10 S f
(\242)2206 2878 w
10 R f
(\).)2239 2878 w
( each)1 207(ii\) For)1 389 2 720 3039 t
10 I f
(v)1341 3039 w
10 R f
(in)1410 3039 w
10 I f
(S)1513 3039 w
10 R f
(, add the edge \()4 612 1 1563 3039 t
10 I f
(s)2183 3039 w
10 S f
(\242)2230 3039 w
10 R f
(,)2263 3039 w
10 I f
(v)2329 3039 w
10 R f
(\) with capacity upper bound)4 1120 1 2381 3039 t
10 I f
(u)3526 3039 w
10 S1 f
(_)3528 2971 w
10 R f
(\()3584 3039 w
10 I f
(s)3625 3039 w
10 S f
(\242)3672 3039 w
10 R f
(,)3705 3039 w
10 I f
(v)3771 3039 w
10 R f
(\))3823 3039 w
10 S f
( b)1 71(= -)1 167 2 3913 3039 t
10 R f
(\()4159 3039 w
10 I f
(v)4200 3039 w
10 R f
(,)4252 3039 w
10 I f
(f)4326 3039 w
7 R f
(0)4365 3059 w
10 R f
(\).)4416 3039 w
( each)1 207(iii\) For)1 389 2 720 3200 t
10 I f
(w)1341 3200 w
10 R f
(in)1433 3200 w
10 I f
(D)1536 3200 w
10 R f
(, add the edge \()4 612 1 1608 3200 t
10 I f
(w)2228 3200 w
10 R f
(,)2303 3200 w
10 I f
(t)2369 3200 w
10 S f
(\242)2405 3200 w
10 R f
(\) with capacity upper bound)4 1120 1 2438 3200 t
10 I f
(u)3583 3200 w
10 S1 f
(_)3585 3132 w
10 R f
(\()3641 3200 w
10 I f
(w)3682 3200 w
10 R f
(,)3757 3200 w
10 I f
(t)3823 3200 w
10 S f
(\242)3859 3200 w
10 R f
(\))3892 3200 w
10 S f
(= b)1 159 1 3982 3200 t
10 R f
(\()4149 3200 w
10 I f
(w)4190 3200 w
10 R f
(,)4265 3200 w
10 I f
(f)4339 3200 w
7 R f
(0)4378 3220 w
10 R f
(\).)4429 3200 w
(The cost of each edge in)5 1027 1 720 3356 t
10 I f
(G)1783 3356 w
10 R f
( capacity)1 369( The)1 217( added edge is zero.)4 837(remains what it was, and the cost of each)8 1726 4 1891 3356 t
( in)1 121(lower bound of all the edges)5 1229 2 720 3481 t
10 I f
(G)2113 3481 w
10 S1 f
(_ _)1 62 1 2123 3388 t
10 R f
( each edge)2 456( For)1 207(is zero.)1 306 3 2228 3481 t
10 I f
(e)3240 3481 w
10 R f
(in)3327 3481 w
10 I f
(G)3448 3481 w
10 R f
(, the capacity upper bound becomes)5 1520 1 3520 3481 t
10 I f
(u)720 3606 w
10 S1 f
(_)722 3538 w
10 R f
(\()778 3606 w
10 I f
(e)819 3606 w
10 R f
(\))871 3606 w
10 S f
(=)961 3606 w
10 I f
(u)1065 3606 w
10 R f
(\()1123 3606 w
10 I f
(e)1164 3606 w
10 R f
(\))1216 3606 w
10 S f
(-)1306 3606 w
10 I f
(l)1410 3606 w
10 R f
(\()1446 3606 w
10 I f
(e)1487 3606 w
10 R f
( capacity upper bounds of \()5 1114(\). The)1 268 2 1539 3606 t
10 I f
(s)2929 3606 w
10 S f
(\242)2976 3606 w
10 R f
(,)3009 3606 w
10 I f
(s)3075 3606 w
10 R f
(\) and \()2 272 1 3122 3606 t
10 I f
(t)3402 3606 w
10 R f
(,)3438 3606 w
10 I f
(t)3504 3606 w
10 S f
(\242)3540 3606 w
10 R f
( Obviously,)1 498(\) will be specified later.)4 969 2 3573 3606 t
10 I f
(G)720 3731 w
10 S1 f
(_ _)1 62 1 730 3638 t
10 R f
(has)817 3731 w
10 I f
(O)975 3731 w
10 R f
(\()1055 3731 w
10 I f
(m)1096 3731 w
10 R f
(\) edges and)2 454 1 1176 3731 t
10 I f
(O)1655 3731 w
10 R f
(\()1735 3731 w
10 I f
(n)1776 3731 w
10 R f
(\) vertices.)1 393 1 1834 3731 t
(Given a flow)2 529 1 970 3892 t
10 I f
(F)1528 3892 w
10 R f
(on)1618 3892 w
10 I f
(G)1747 3892 w
10 S1 f
(_ _)1 62 1 1757 3799 t
10 R f
(, the)1 176 1 1819 3892 t
10 I f
(induced)2024 3892 w
10 R f
(preflow)2369 3892 w
10 I f
(f)2708 3892 w
10 R f
(on)2765 3892 w
10 I f
(G)2894 3892 w
10 R f
(is defined as)2 507 1 2995 3892 t
10 I f
(f)3531 3892 w
10 R f
(\()3575 3892 w
10 I f
(e)3616 3892 w
10 R f
(\))3668 3892 w
10 S f
(=)3758 3892 w
10 I f
(F)3862 3892 w
10 R f
(\()3931 3892 w
10 I f
(e)3972 3892 w
10 R f
( all)1 130(\) for)1 178 2 4024 3892 t
10 I f
(e)4362 3892 w
10 S f
(\316)4447 3892 w
10 I f
(E)4559 3892 w
10 R f
( now)1 202(. We)1 218 2 4620 3892 t
(reduce the balancing-preflow problems to maximum-flow problems on balancing graphs.)9 3566 1 720 4012 t
10 B f
( flow)1 208(3.3. Minimum)1 628 2 720 4252 t
10 R f
(To present the algorithms, we use the concept of a)9 2079 1 970 4408 t
10 I f
(residual graph)1 608 1 3082 4408 t
10 R f
( Let)1 191(of a flow.)2 401 2 3723 4408 t
10 I f
(f)4349 4408 w
10 R f
(be a flow on)3 523 1 4411 4408 t
10 I f
(G)4968 4408 w
10 R f
( and upper bound)3 705(with capacity lower bound zero)4 1274 2 720 4528 t
10 I f
(u)2727 4528 w
10 R f
(. The)1 233 1 2777 4528 t
10 I f
(residual capacity)1 707 1 3038 4528 t
10 R f
(\(upper bound\) of \()3 743 1 3773 4528 t
10 I f
(v)4524 4528 w
10 R f
(,)4576 4528 w
10 I f
(w)4642 4528 w
10 R f
(\) in)1 139 1 4717 4528 t
10 I f
(E)4884 4528 w
10 R f
(is)4973 4528 w
10 I f
(u)720 4653 w
10 S1 f
(_)722 4585 w
10 R f
(\()778 4653 w
10 I f
(v)819 4653 w
10 R f
(,)871 4653 w
10 I f
(w)937 4653 w
10 R f
(\))1012 4653 w
10 S f
(=)1083 4653 w
10 I f
(u)1187 4653 w
10 R f
(\()1245 4653 w
10 I f
(v)1286 4653 w
10 R f
(,)1338 4653 w
10 I f
(w)1404 4653 w
10 R f
(\))1479 4653 w
10 S f
(-)1569 4653 w
10 I f
(f)1681 4653 w
10 R f
(\()1725 4653 w
10 I f
(v)1766 4653 w
10 R f
(,)1818 4653 w
10 I f
(w)1884 4653 w
10 R f
( \()1 71(\). If)1 187 2 1959 4653 t
10 I f
(v)2225 4653 w
10 R f
(,)2277 4653 w
10 I f
(w)2343 4653 w
10 R f
(\) is in)2 254 1 2418 4653 t
10 I f
(E)2710 4653 w
10 R f
(but \()1 199 1 2809 4653 t
10 I f
(w)3016 4653 w
10 R f
(,)3091 4653 w
10 I f
(v)3157 4653 w
10 R f
( edge \()2 299(\) is not, then add an)5 853 2 3209 4653 t
10 I f
(w)4369 4653 w
10 R f
(,)4444 4653 w
10 I f
(v)4510 4653 w
10 R f
(\) to)1 150 1 4562 4653 t
10 I f
(G)4751 4653 w
10 R f
(with)4862 4653 w
(capacity upper bound)2 867 1 720 4778 t
10 I f
(u)1616 4778 w
10 S1 f
(_)1618 4710 w
10 R f
(\()1674 4778 w
10 I f
(w)1715 4778 w
10 R f
(,)1790 4778 w
10 I f
(v)1856 4778 w
10 R f
(\))1908 4778 w
10 S f
(=)1998 4778 w
10 I f
(f)2110 4778 w
10 R f
(\()2154 4778 w
10 I f
(v)2195 4778 w
10 R f
(,)2247 4778 w
10 I f
(w)2313 4778 w
10 R f
( residual graph)2 601(\). The)1 267 2 2388 4778 t
10 I f
(R)3285 4778 w
10 R f
(\()3354 4778 w
10 I f
(f)3411 4778 w
10 R f
(\) from a flow)3 538 1 3463 4778 t
10 I f
(f)4029 4778 w
10 R f
(is the graph with vertex)4 955 1 4085 4778 t
(set)720 4903 w
10 I f
(V)857 4903 w
10 R f
(, source)1 311 1 918 4903 t
10 I f
(s)1255 4903 w
10 R f
(, sink)1 218 1 1294 4903 t
10 I f
(t)1538 4903 w
10 R f
( \()1 60(, and an edge)3 529 2 1566 4903 t
10 I f
(v)2163 4903 w
10 R f
(,)2215 4903 w
10 I f
(w)2281 4903 w
10 R f
(\) of capacity)2 502 1 2356 4903 t
10 I f
(u)2885 4903 w
10 S1 f
(_)2887 4835 w
10 R f
(\()2943 4903 w
10 I f
(v)2984 4903 w
10 R f
(,)3036 4903 w
10 I f
(w)3102 4903 w
10 R f
(\), where)1 328 1 3177 4903 t
10 I f
(u)3532 4903 w
10 S1 f
(_)3534 4835 w
10 R f
(\()3590 4903 w
10 I f
(v)3631 4903 w
10 R f
(,)3683 4903 w
10 I f
(w)3749 4903 w
10 R f
(\))3824 4903 w
10 S f
(>)3914 4903 w
10 R f
( costs of the edges)4 740(0. The)1 282 2 4018 4903 t
(in)720 5023 w
10 I f
(R)825 5023 w
10 R f
(\()894 5023 w
10 I f
(f)951 5023 w
10 R f
( as)1 109(\) are defined)2 507 2 1003 5023 t
10 S f
(D)1645 5023 w
10 R f
(\()1714 5023 w
10 I f
(v)1755 5023 w
10 R f
(,)1807 5023 w
10 I f
(w)1873 5023 w
10 R f
(\))1948 5023 w
10 S f
(=)2038 5023 w
10 I f
(cost)2142 5023 w
10 R f
(\()2311 5023 w
10 I f
(v)2352 5023 w
10 R f
(,)2404 5023 w
10 I f
(w)2470 5023 w
10 R f
(\) if \()2 179 1 2545 5023 t
10 I f
(v)2732 5023 w
10 R f
(,)2784 5023 w
10 I f
(w)2850 5023 w
10 R f
(\))2925 5023 w
10 S f
(\316)3007 5023 w
10 I f
(E)3119 5023 w
10 R f
(, and)1 195 1 3180 5023 t
10 S f
(D)3401 5023 w
10 R f
(\()3470 5023 w
10 I f
(v)3511 5023 w
10 R f
(,)3563 5023 w
10 I f
(w)3629 5023 w
10 R f
(\))3704 5023 w
10 S f
(= -)1 167 1 3794 5023 t
10 I f
(cost)3977 5023 w
10 R f
(\()4146 5023 w
10 I f
(w)4187 5023 w
10 R f
(,)4262 5023 w
10 I f
(v)4328 5023 w
10 R f
(\) if \()2 179 1 4380 5023 t
10 I f
(w)4567 5023 w
10 R f
(,)4642 5023 w
10 I f
(v)4708 5023 w
10 R f
(\))4760 5023 w
10 S f
(\316)4842 5023 w
10 I f
(E)4954 5023 w
10 R f
(.)5015 5023 w
(We need the following well-known lemma, see, for example, [Tarjan, 1983].)10 3068 1 720 5143 t
10 B f
(Lemma 3.1.)1 504 1 720 5299 t
10 R f
(Let)1251 5299 w
10 I f
(f)1411 5299 w
10 R f
(be any flow and)3 646 1 1466 5299 t
10 I f
(f *)1 102 1 2139 5299 t
10 R f
(a maximum flow on)3 817 1 2268 5299 t
10 I f
(G)3113 5299 w
10 R f
(. If)1 119 1 3185 5299 t
10 I f
(R)3332 5299 w
10 R f
(\()3401 5299 w
10 I f
(f)3458 5299 w
10 R f
(\) is the residual graph from)5 1099 1 3510 5299 t
10 I f
(f)4637 5299 w
10 R f
(, then the)2 375 1 4665 5299 t
(value of a maximum flow on)5 1157 1 720 5419 t
10 I f
(R)1902 5419 w
10 R f
(\()1971 5419 w
10 I f
(f)2028 5419 w
10 R f
(\) is)1 125 1 2080 5419 t
10 I f
(value)2230 5419 w
10 R f
(\()2454 5419 w
10 I f
(f *)1 102 1 2511 5419 t
10 R f
(\))2621 5419 w
10 S f
(-)2711 5419 w
10 I f
(value)2815 5419 w
10 R f
(\()3039 5419 w
10 I f
(f)3096 5419 w
10 R f
(\).)3148 5419 w
10 S1 f
()3256 5419 w
3256 5419 m
50 build_sq
3306 5419 m
10 R f
( a minimum flow, we use the following step-wise balancing on)10 2597(To find)1 304 2 970 5580 t
10 I f
(G)3904 5580 w
10 S1 f
(_ _)1 62 1 3914 5487 t
10 R f
( finding the maxi-)3 743(. When)1 321 2 3976 5580 t
( induced preflow on)3 816( The)1 210(mum flows, we use the Sleator-Tarjan algorithm [Tarjan, 1983].)8 2605 3 720 5700 t
10 I f
(G)4380 5700 w
10 R f
(is a minimum)2 559 1 4481 5700 t
(balancing preflow.)1 748 1 720 5820 t
10 B f
(Algorithm 1.)1 544 1 720 5976 t
10 R f
(Minimum flow.)1 634 1 1289 5976 t
( the balancing graph)3 812(1. Construct)1 639 2 720 6137 t
10 I f
(G)2196 6137 w
10 S1 f
(_ _)1 62 1 2206 6044 t
10 R f
(and set)1 280 1 2293 6137 t
10 I f
(u)2598 6137 w
10 S1 f
(_)2600 6069 w
10 R f
(\()2656 6137 w
10 I f
(s)2697 6137 w
10 S f
(\242)2744 6137 w
10 R f
(,)2777 6137 w
10 I f
(s)2843 6137 w
10 R f
(\) =)1 114 1 2890 6137 t
10 I f
(u)3029 6137 w
10 S1 f
(_)3031 6069 w
10 R f
(\()3087 6137 w
10 I f
(t)3128 6137 w
10 R f
(,)3164 6137 w
10 I f
(t)3230 6137 w
10 S f
(\242)3266 6137 w
10 R f
(\) = 0.)2 214 1 3299 6137 t
( a maximum flow)3 708(2. Find)1 434 2 720 6298 t
10 I f
(F)1887 6298 w
7 R f
(1)1959 6318 w
10 R f
(on)2027 6298 w
10 I f
(G)2152 6298 w
10 S1 f
(_ _)1 62 1 2162 6205 t
10 R f
(, and let)2 319 1 2224 6298 t
10 I f
(R)2568 6298 w
7 R f
(1)2640 6318 w
10 R f
(be the residual graph.)3 859 1 2708 6298 t
(3. Set)1 378 1 720 6459 t
10 I f
(u)1123 6459 w
10 S1 f
(_)1125 6391 w
10 R f
(\()1181 6459 w
10 I f
(s)1222 6459 w
10 S f
(\242)1269 6459 w
10 R f
(,)1302 6459 w
10 I f
(s)1368 6459 w
10 R f
(\))1415 6459 w
10 S f
(= \245)1 177 1 1505 6459 t
10 R f
(and)1707 6459 w
10 I f
(u)1876 6459 w
10 S1 f
(_)1878 6391 w
10 R f
(\()1934 6459 w
10 I f
(v)1975 6459 w
10 R f
(,)2027 6459 w
10 I f
(s)2093 6459 w
10 S f
(\242)2140 6459 w
10 R f
(\))2173 6459 w
10 S f
(=)2263 6459 w
10 R f
(0,)2367 6459 w
10 I f
(v)2467 6459 w
10 S f
(\316)2552 6459 w
10 I f
(S)2664 6459 w
10 R f
(, in)1 128 1 2714 6459 t
10 I f
(R)2867 6459 w
7 R f
(1)2939 6479 w
10 R f
(, and let)2 319 1 2982 6459 t
10 I f
(R)3326 6459 w
10 S1 f
(_ _)1 51 1 3336 6366 t
7 R f
(1)3398 6479 w
10 R f
(be the modified graph.)3 904 1 3466 6459 t
( a maximum flow)3 708(4. Find)1 434 2 720 6620 t
10 I f
(f)1887 6620 w
7 R f
(2)1926 6640 w
10 R f
(on)1994 6620 w
10 I f
(R)2119 6620 w
10 S1 f
(_ _)1 51 1 2129 6527 t
7 R f
(1)2191 6640 w
10 R f
(. Let)1 208 1 2234 6620 t
10 I f
(R)2467 6620 w
7 R f
(2)2539 6640 w
10 R f
(be the residual graph, and let)5 1153 1 2607 6620 t
10 I f
(F)3785 6620 w
7 R f
(2)3857 6640 w
10 S f
(=)3949 6620 w
10 I f
(F)4053 6620 w
7 R f
(1)4125 6640 w
10 S f
(+)4217 6620 w
10 I f
(f)4329 6620 w
7 R f
(2)4368 6640 w
10 R f
(.)4411 6620 w
(5. If)1 316 1 720 6776 t
10 I f
(value)1061 6776 w
10 R f
(\()1285 6776 w
10 I f
(F)1326 6776 w
7 R f
(2)1398 6796 w
10 R f
(\))1449 6776 w
10 S f
(<)1539 6776 w
7 I f
(w)1643 6876 w
7 S f
(\316)1718 6876 w
7 I f
(D)1796 6876 w
15 S f
(S)1700 6806 w
10 S f
(b)1854 6776 w
10 R f
(\()1917 6776 w
10 I f
(w)1958 6776 w
10 R f
(,)2033 6776 w
10 I f
(f)2107 6776 w
7 R f
(0)2146 6796 w
10 R f
( is no feasible flow on)5 885( There)1 282(\), then abort.)2 510 3 2197 6776 t
10 I f
(G)3899 6776 w
10 R f
(.)3971 6776 w
(6. Set)1 378 1 720 7017 t
10 S f
(m =)1 162 1 1123 7017 t
10 I f
(u)1334 7017 w
10 S1 f
(_)1336 6949 w
10 R f
(\()1392 7017 w
10 I f
(s)1433 7017 w
10 S f
(\242)1480 7017 w
10 R f
(,)1513 7017 w
10 I f
(s)1579 7017 w
10 R f
(\))1626 7017 w
10 S f
(=)1716 7017 w
10 I f
(F)1820 7017 w
7 R f
(2)1892 7037 w
10 R f
(\()1943 7017 w
10 I f
(s)1984 7017 w
10 S f
(\242)2031 7017 w
10 R f
(,)2064 7017 w
10 I f
(s)2130 7017 w
10 R f
(\) and)1 202 1 2177 7017 t
10 I f
(u)2404 7017 w
10 S1 f
(_)2406 6949 w
10 R f
(\()2462 7017 w
10 I f
(t)2503 7017 w
10 R f
(,)2539 7017 w
10 I f
(t)2605 7017 w
10 S f
(\242)2641 7017 w
10 R f
(\))2674 7017 w
10 S f
(= \245)1 177 1 2764 7017 t
10 R f
(in)2966 7017 w
10 I f
(R)3069 7017 w
7 R f
(2)3141 7037 w
10 R f
(, and let)2 319 1 3184 7017 t
10 I f
(R)3528 7017 w
10 S1 f
(_ _)1 51 1 3538 6924 t
7 R f
(2)3600 7037 w
10 R f
(be the modified graph.)3 904 1 3668 7017 t
( a maximum flow)3 708(7. Find)1 434 2 720 7178 t
10 I f
(f)1887 7178 w
7 R f
(3)1926 7198 w
10 R f
(on)1994 7178 w
10 I f
(R)2119 7178 w
10 S1 f
(_ _)1 51 1 2129 7085 t
7 R f
(2)2191 7198 w
10 R f
(, and let)2 319 1 2234 7178 t
10 I f
(F)2578 7178 w
7 R f
(3)2650 7198 w
10 S f
(=)2742 7178 w
10 I f
(F)2846 7178 w
7 R f
(2)2918 7198 w
10 S f
(+)3010 7178 w
10 I f
(f)3122 7178 w
7 R f
(3)3161 7198 w
10 R f
(.)3204 7178 w
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5 pagesetup
10 R f
(- 5 -)2 166 1 2797 480 t
(8. If)1 316 1 720 840 t
10 I f
(value)1061 840 w
10 R f
(\()1285 840 w
10 I f
(F)1326 840 w
7 R f
(3)1398 860 w
10 R f
(\))1449 840 w
10 S f
(< m +)2 266 1 1539 840 t
7 I f
(v)1854 940 w
7 S f
(\316)1913 940 w
7 I f
(S)1991 940 w
15 S f
(S)1895 870 w
10 R f
([)2033 840 w
10 S f
(- b)1 126 1 2082 840 t
10 R f
(\()2216 840 w
10 I f
(v)2257 840 w
10 R f
(,)2309 840 w
10 I f
(f)2383 840 w
7 R f
(0)2422 860 w
10 R f
( is no feasible flow on)5 885( There)1 282( then abort.)2 452(\) ],)1 99 4 2473 840 t
10 I f
(G)4216 840 w
10 R f
(.)4288 840 w
(9. Let)1 383 1 720 1076 t
10 I f
(f)1132 1076 w
7 I f
(b)1171 1096 w
10 R f
(be the induced preflow of)4 1041 1 1243 1076 t
10 I f
(F)2313 1076 w
7 R f
(3)2385 1096 w
10 R f
(on)2458 1076 w
10 I f
(G)2588 1076 w
10 R f
(, and let)2 329 1 2660 1076 t
10 I f
(f)3019 1076 w
10 S f
(=)3112 1076 w
10 I f
(f)3224 1076 w
7 I f
(b)3263 1096 w
10 S f
(+)3355 1076 w
10 I f
(f)3467 1076 w
7 R f
(0)3506 1096 w
10 R f
(, where)1 298 1 3549 1076 t
10 I f
(f)3877 1076 w
7 R f
(0)3916 1096 w
10 R f
(is the basic preflow;)3 822 1 3989 1076 t
10 I f
(f)4841 1076 w
10 R f
(is a)1 141 1 4899 1076 t
(minimum flow on)2 723 1 970 1196 t
10 I f
(G)1718 1196 w
10 R f
(.)1790 1196 w
10 S1 f
()1865 1196 w
1865 1196 m
50 build_sq
1915 1196 m
10 R f
( 1 and 2)3 328( Steps)1 270( preflow is constructed from three-stage balancing.)6 2049(Intuitively, the minimum balancing)3 1423 4 970 1352 t
(balance vertices in)2 758 1 720 1472 t
10 I f
(S)1511 1472 w
10 R f
(and)1594 1472 w
10 I f
(D)1771 1472 w
10 R f
(as much as possible without using flow from)7 1847 1 1876 1472 t
10 I f
(s)3756 1472 w
10 R f
(or)3828 1472 w
10 I f
(t)3944 1472 w
10 R f
( the)1 154( 3 to 5 balance)4 614(. Steps)1 300 3 3972 1472 t
(remaining vertices in)2 851 1 720 1592 t
10 I f
(D)1600 1592 w
10 R f
(using flow from)2 652 1 1701 1592 t
10 I f
(s)2383 1592 w
10 R f
(only, and Steps 6 to 8 balance the remaining vertices in)10 2261 1 2452 1592 t
10 I f
(S)4743 1592 w
10 R f
(using)4823 1592 w
(flow from)1 402 1 720 1712 t
10 I f
(t)1147 1712 w
10 R f
( analyze Algorithm 1, we need the following lemma.)8 2108(only. To)1 364 2 1200 1712 t
10 B f
(Lemma 3.2.)1 502 1 720 1868 t
10 R f
(Let)1247 1868 w
10 I f
(f)1405 1868 w
7 I f
(b)1444 1888 w
10 I f
(*)1495 1868 w
10 R f
(be a minimum balancing preflow on)5 1451 1 1570 1868 t
10 I f
(G)3046 1868 w
10 R f
(with value)1 419 1 3143 1868 t
10 S f
(m)3587 1868 w
10 I f
(*)3653 1868 w
10 R f
(. Then)1 280 1 3703 1868 t
( exists a flow)3 551(i\) there)1 449 2 720 2029 t
10 I f
(F)1752 2029 w
7 R f
(1)1824 2049 w
10 I f
(*)1875 2029 w
10 R f
(on)1957 2029 w
10 I f
(G)2089 2029 w
10 S1 f
(_ _)1 62 1 2099 1936 t
10 R f
(with)2194 2029 w
10 I f
(u)2405 2029 w
10 S1 f
(_)2407 1961 w
10 R f
(\()2463 2029 w
10 I f
(s)2504 2029 w
10 S f
(\242)2551 2029 w
10 R f
(,)2584 2029 w
10 I f
(s)2650 2029 w
10 R f
(\))2697 2029 w
10 S f
(=)2787 2029 w
10 I f
(u)2891 2029 w
10 S1 f
(_)2893 1961 w
10 R f
(\()2949 2029 w
10 I f
(t)2990 2029 w
10 R f
(,)3026 2029 w
10 I f
(t)3092 2029 w
10 S f
(\242)3128 2029 w
10 R f
(\))3161 2029 w
10 S f
(=)3251 2029 w
10 R f
(0 such that)2 449 1 3355 2029 t
10 I f
(value)3837 2029 w
10 R f
(\()4061 2029 w
10 I f
(F)4102 2029 w
7 R f
(1)4174 2049 w
10 I f
(*)4225 2029 w
10 R f
(\))4283 2029 w
10 S f
(=)4349 2029 w
7 I f
(w)4453 2129 w
7 S f
(\316)4528 2129 w
7 I f
(D)4606 2129 w
15 S f
(S)4510 2059 w
10 S f
(b)4664 2029 w
10 R f
(\()4727 2029 w
10 I f
(w)4768 2029 w
10 R f
(,)4843 2029 w
10 I f
(f)4917 2029 w
7 R f
(0)4956 2049 w
10 R f
(\))5007 2029 w
10 S f
(- m)1 162 1 970 2229 t
10 I f
(*)1140 2229 w
10 R f
(;)1190 2229 w
( exists a flow)3 647(ii\) there)1 449 2 720 2390 t
10 I f
(F)1880 2390 w
7 R f
(2)1952 2410 w
10 I f
(*)2003 2390 w
10 R f
(on)2117 2390 w
10 I f
(G)2281 2390 w
10 S1 f
(_ _)1 62 1 2291 2297 t
10 R f
(with)2418 2390 w
10 I f
(u)2661 2390 w
10 S1 f
(_)2663 2322 w
10 R f
(\()2719 2390 w
10 I f
(s)2760 2390 w
10 S f
(\242)2807 2390 w
10 R f
(,)2840 2390 w
10 I f
(s)2906 2390 w
10 R f
(\))2953 2390 w
10 S f
(= \245)1 177 1 3043 2390 t
10 R f
(and)3285 2390 w
10 I f
(u)3494 2390 w
10 S1 f
(_)3496 2322 w
10 R f
(\()3552 2390 w
10 I f
(t)3593 2390 w
10 R f
(,)3629 2390 w
10 I f
(t)3695 2390 w
10 S f
(\242)3731 2390 w
10 R f
(\))3764 2390 w
10 S f
(=)3854 2390 w
10 R f
(0, such that)2 538 1 3958 2390 t
10 I f
(value)4561 2390 w
10 R f
(\()4785 2390 w
10 I f
(F)4826 2390 w
7 R f
(2)4898 2410 w
10 I f
(*)4949 2390 w
10 R f
(\))5007 2390 w
10 S f
(=)970 2510 w
7 I f
(w)1074 2610 w
7 S f
(\316)1149 2610 w
7 I f
(D)1227 2610 w
15 S f
(S)1131 2540 w
10 S f
(b)1285 2510 w
10 R f
(\()1348 2510 w
10 I f
(w)1389 2510 w
10 R f
(,)1464 2510 w
10 I f
(f)1538 2510 w
7 R f
(0)1577 2530 w
10 R f
(\); and)1 230 1 1628 2510 t
( exists a flow)3 578(iii\) there)1 449 2 720 2751 t
10 I f
(F)1788 2751 w
7 R f
(3)1860 2771 w
10 I f
(*)1911 2751 w
10 R f
(on)2003 2751 w
10 I f
(G)2145 2751 w
10 S1 f
(_ _)1 62 1 2155 2658 t
10 R f
(with)2259 2751 w
10 I f
(u)2479 2751 w
10 S1 f
(_)2481 2683 w
10 R f
(\()2537 2751 w
10 I f
(s)2578 2751 w
10 S f
(\242)2625 2751 w
10 R f
(,)2658 2751 w
10 I f
(s)2724 2751 w
10 R f
(\))2771 2751 w
10 S f
(= m)1 162 1 2861 2751 t
10 I f
(*)3031 2751 w
10 R f
(and)3123 2751 w
10 I f
(u)3309 2751 w
10 S1 f
(_)3311 2683 w
10 R f
(\()3367 2751 w
10 I f
(t)3408 2751 w
10 R f
(,)3444 2751 w
10 I f
(t)3510 2751 w
10 S f
(\242)3546 2751 w
10 R f
(\))3579 2751 w
10 S f
(= \245)1 177 1 3669 2751 t
10 R f
(such that)1 375 1 3888 2751 t
10 I f
(value)4305 2751 w
10 R f
(\()4529 2751 w
10 I f
(F)4570 2751 w
7 R f
(3)4642 2771 w
10 I f
(*)4693 2751 w
10 R f
(\) =)1 131 1 4751 2751 t
10 S f
(m)4924 2751 w
10 I f
(*)4990 2751 w
10 S f
(+)970 2871 w
7 I f
(v)1074 2971 w
7 S f
(\316)1133 2971 w
7 I f
(S)1211 2971 w
15 S f
(S)1115 2901 w
10 R f
([)1253 2871 w
10 S f
(- b)1 126 1 1302 2871 t
10 R f
(\()1436 2871 w
10 I f
(v)1477 2871 w
10 R f
(,)1529 2871 w
10 I f
(f)1603 2871 w
7 R f
(0)1642 2891 w
10 R f
(\) ].)1 99 1 1693 2871 t
10 B f
(Proof.)720 3112 w
10 R f
(We define a flow)3 767 1 1058 3112 t
10 I f
(F)1876 3112 w
7 R f
(3)1948 3132 w
10 I f
(*)1999 3112 w
10 R f
(on)2100 3112 w
10 I f
(G)2251 3112 w
10 S1 f
(_ _)1 62 1 2261 3019 t
10 R f
(with)2374 3112 w
10 I f
(u)2603 3112 w
10 S1 f
(_)2605 3044 w
10 R f
(\()2661 3112 w
10 I f
(s)2702 3112 w
10 S f
(\242)2749 3112 w
10 R f
(,)2782 3112 w
10 I f
(s)2848 3112 w
10 R f
(\))2895 3112 w
10 S f
(= m)1 162 1 2985 3112 t
10 I f
(*)3155 3112 w
10 R f
(and)3256 3112 w
10 I f
(u)3451 3112 w
10 S1 f
(_)3453 3044 w
10 R f
(\()3509 3112 w
10 I f
(t)3550 3112 w
10 R f
(,)3586 3112 w
10 I f
(t)3652 3112 w
10 S f
(\242)3688 3112 w
10 R f
(\))3721 3112 w
10 S f
(= \245)1 177 1 3811 3112 t
10 R f
(as follows. Let)2 643 1 4039 3112 t
10 I f
(F)4733 3112 w
7 R f
(3)4805 3132 w
10 I f
(*)4856 3112 w
10 R f
(\()4914 3112 w
10 I f
(e)4955 3112 w
10 R f
(\))5007 3112 w
10 S f
(=)720 3232 w
10 I f
(f)832 3232 w
7 I f
(b)871 3252 w
10 I f
(*)922 3232 w
10 R f
(\()980 3232 w
10 I f
(e)1021 3232 w
10 R f
(\) for)1 178 1 1073 3232 t
10 I f
(e)1280 3232 w
10 S f
(\316)1365 3232 w
10 I f
(E)1477 3232 w
10 R f
(, and let)2 327 1 1538 3232 t
10 I f
(F)1894 3232 w
7 R f
(3)1966 3252 w
10 I f
(*)2017 3232 w
10 R f
(\()2075 3232 w
10 I f
(s)2116 3232 w
10 S f
(\242)2163 3232 w
10 R f
(,)2196 3232 w
10 I f
(s)2262 3232 w
10 R f
(\) =)1 118 1 2309 3232 t
10 S f
(m)2456 3232 w
10 I f
(*)2522 3232 w
10 R f
(,)2572 3232 w
10 I f
(F)2625 3232 w
7 R f
(3)2697 3252 w
10 I f
(*)2748 3232 w
10 R f
(\()2806 3232 w
10 I f
(s)2847 3232 w
10 S f
(\242)2894 3232 w
10 R f
(,)2927 3232 w
10 I f
(v)2993 3232 w
10 R f
(\) =)1 117 1 3045 3232 t
10 S f
(b)3190 3232 w
10 R f
(\()3253 3232 w
10 I f
(v)3294 3232 w
10 R f
(,)3346 3232 w
10 I f
(f)3420 3232 w
7 I f
(b)3459 3252 w
10 I f
(*)3510 3232 w
10 R f
(\),)3568 3232 w
10 I f
(v)3654 3232 w
10 S f
(\316)3739 3232 w
10 I f
(S)3851 3232 w
10 R f
(,)3901 3232 w
10 I f
(F)3954 3232 w
7 R f
(3)4026 3252 w
10 I f
(*)4077 3232 w
10 R f
(\()4135 3232 w
10 I f
(t)4176 3232 w
10 R f
(,)4212 3232 w
10 I f
(t)4278 3232 w
10 S f
(\242)4314 3232 w
10 R f
(\))4347 3232 w
10 S f
( b)1 71(= -)1 167 2 4437 3232 t
10 R f
(\()4683 3232 w
10 I f
(t)4724 3232 w
10 R f
(,)4760 3232 w
10 I f
(f)4834 3232 w
7 I f
(b)4873 3252 w
10 I f
(*)4924 3232 w
10 R f
(\),)4982 3232 w
(and)720 3357 w
10 I f
(F)894 3357 w
7 R f
(3)966 3377 w
10 I f
(*)1017 3357 w
10 R f
(\()1075 3357 w
10 I f
(w)1116 3357 w
10 R f
(,)1191 3357 w
10 I f
(t)1257 3357 w
10 S f
(\242)1293 3357 w
10 R f
(\) =)1 119 1 1326 3357 t
10 S f
(- b)1 126 1 1475 3357 t
10 R f
(\()1609 3357 w
10 I f
(w)1650 3357 w
10 R f
(,)1725 3357 w
10 I f
(f)1799 3357 w
7 I f
(b)1838 3377 w
10 I f
(*)1889 3357 w
10 R f
(\),)1947 3357 w
10 I f
(w)2035 3357 w
10 S f
(\316)2143 3357 w
10 I f
(D)2255 3357 w
10 R f
(. Then)1 285 1 2327 3357 t
10 I f
(F)2642 3357 w
7 R f
(3)2714 3377 w
10 I f
(*)2765 3357 w
10 R f
(is a flow on)3 484 1 2845 3357 t
10 I f
(G)3359 3357 w
10 S1 f
(_ _)1 62 1 3369 3264 t
10 R f
(with)3461 3357 w
10 I f
(u)3670 3357 w
10 S1 f
(_)3672 3289 w
10 R f
(\()3728 3357 w
10 I f
(s)3769 3357 w
10 S f
(\242)3816 3357 w
10 R f
(,)3849 3357 w
10 I f
(s)3915 3357 w
10 R f
(\))3962 3357 w
10 S f
(= m)1 162 1 4052 3357 t
10 I f
(*)4222 3357 w
10 R f
(and)4303 3357 w
10 I f
(u)4478 3357 w
10 S1 f
(_)4480 3289 w
10 R f
(\()4536 3357 w
10 I f
(t)4577 3357 w
10 R f
(,)4613 3357 w
10 I f
(t)4679 3357 w
10 S f
(\242)4715 3357 w
10 R f
(\))4748 3357 w
10 S f
(= \245)1 177 1 4838 3357 t
10 R f
(.)5015 3357 w
(We call)1 344 1 720 3482 t
10 I f
(F)1126 3482 w
10 R f
(the)1249 3482 w
10 I f
(derived)1433 3482 w
10 R f
(flow on)1 345 1 1794 3482 t
10 I f
(G)2201 3482 w
10 S1 f
(_ _)1 62 1 2211 3389 t
10 R f
(from)2335 3482 w
10 I f
(f)2591 3482 w
7 I f
(b)2630 3502 w
10 I f
(*)2681 3482 w
10 R f
(. Since)1 334 1 2731 3482 t
10 I f
(f)3127 3482 w
7 I f
(b)3166 3502 w
10 I f
(*)3217 3482 w
10 R f
(is balancing,)1 542 1 3329 3482 t
10 I f
(value)3933 3482 w
10 R f
(\()4157 3482 w
10 I f
(F)4198 3482 w
7 R f
(3)4270 3502 w
10 I f
(*)4321 3482 w
10 R f
(\) =)1 151 1 4379 3482 t
10 I f
(F)4592 3482 w
7 R f
(3)4664 3502 w
10 I f
(*)4715 3482 w
10 R f
(\()4773 3482 w
10 I f
(s)4814 3482 w
10 S f
(\242)4861 3482 w
10 R f
(,)4894 3482 w
10 I f
(s)4960 3482 w
10 R f
(\))5007 3482 w
10 S f
(+)720 3602 w
7 I f
(v)824 3702 w
7 S f
(\316)883 3702 w
7 I f
(S)961 3702 w
15 S f
(S)865 3632 w
10 I f
(F)995 3602 w
7 R f
(3)1067 3622 w
10 I f
(*)1118 3602 w
10 R f
(\()1176 3602 w
10 I f
(s)1217 3602 w
10 S f
(\242)1264 3602 w
10 R f
(,)1297 3602 w
10 I f
(v)1363 3602 w
10 R f
(\) =)1 114 1 1415 3602 t
10 S f
(m)1554 3602 w
10 I f
(*)1620 3602 w
10 S f
(+)1695 3602 w
7 I f
(v)1799 3702 w
7 S f
(\316)1858 3702 w
7 I f
(S)1936 3702 w
15 S f
(S)1840 3632 w
10 S f
(b)1978 3602 w
10 R f
(\()2041 3602 w
10 I f
(v)2082 3602 w
10 R f
(,)2134 3602 w
10 I f
(f)2208 3602 w
7 I f
(b)2247 3622 w
10 I f
(*)2298 3602 w
10 R f
(\) =)1 114 1 2356 3602 t
10 S f
(m)2495 3602 w
10 I f
(*)2561 3602 w
10 S f
(+)2636 3602 w
7 I f
(v)2740 3702 w
7 S f
(\316)2799 3702 w
7 I f
(S)2877 3702 w
15 S f
(S)2781 3632 w
10 R f
([)2919 3602 w
10 S f
(- b)1 126 1 2968 3602 t
10 R f
(\()3102 3602 w
10 I f
(v)3143 3602 w
10 R f
(,)3195 3602 w
10 I f
(f)3269 3602 w
7 R f
(0)3308 3622 w
10 R f
( \(iii\) has now been proved.)5 1070( Part)1 211(\) ].)1 99 3 3359 3602 t
(We successively reduce the minimum balancing preflow)6 2340 1 970 3838 t
10 I f
(f)3348 3838 w
7 I f
(b)3387 3858 w
10 I f
(*)3438 3838 w
10 R f
(along)3526 3838 w
10 I f
(v)3786 3838 w
10 S f
(-)3854 3838 w
10 I f
(t)3925 3838 w
10 R f
(paths, where)1 517 1 3991 3838 t
10 I f
(v)4546 3838 w
10 S f
(\316)4631 3838 w
10 I f
(S)4743 3838 w
10 R f
(, such)1 247 1 4793 3838 t
(that the reduced preflow)3 981 1 720 3963 t
10 I f
(f)1729 3963 w
10 S1 f
(_)1723 3870 w
7 I f
(b)1768 3983 w
10 I f
(*)1819 3963 w
10 R f
(has)1897 3963 w
10 S f
(b)2058 3963 w
10 R f
(\()2121 3963 w
10 I f
(t)2162 3963 w
10 R f
(,)2198 3963 w
10 I f
(f)2272 3963 w
10 S1 f
(_)2266 3870 w
7 I f
(b)2311 3983 w
10 I f
(*)2362 3963 w
10 R f
( let)1 127( Similarly,)1 450(\) = 0.)2 218 3 2420 3963 t
10 I f
(F)3242 3963 w
7 R f
(2)3314 3983 w
10 I f
(*)3365 3963 w
10 R f
(be the derived flow from)4 1000 1 3442 3963 t
10 I f
(f)4469 3963 w
10 S1 f
(_)4463 3870 w
7 I f
(b)4508 3983 w
10 I f
(*)4559 3963 w
10 R f
(on)4636 3963 w
10 I f
(G)4763 3963 w
10 S1 f
(_ _)1 62 1 4773 3870 t
10 R f
(with)4862 3963 w
10 I f
(u)720 4088 w
10 S1 f
(_)722 4020 w
10 R f
(\()778 4088 w
10 I f
(s)819 4088 w
10 S f
(\242)866 4088 w
10 R f
(,)899 4088 w
10 I f
(s)965 4088 w
10 R f
(\))1012 4088 w
10 S f
(= \245)1 177 1 1102 4088 t
10 R f
(and)1319 4088 w
10 I f
(u)1503 4088 w
10 S1 f
(_)1505 4020 w
10 R f
(\()1561 4088 w
10 I f
(t)1602 4088 w
10 R f
(,)1638 4088 w
10 I f
(t)1704 4088 w
10 S f
(\242)1740 4088 w
10 R f
(\))1773 4088 w
10 S f
(=)1863 4088 w
10 R f
(0. Since)1 362 1 1967 4088 t
10 I f
(f)2369 4088 w
7 I f
(b)2408 4108 w
10 I f
(*)2459 4088 w
10 R f
( the balancing indices of)4 1040(is balancing, and only)3 922 2 2549 4088 t
10 I f
(v)4552 4088 w
10 R f
(in)4637 4088 w
10 I f
(S)4756 4088 w
10 R f
(were)4847 4088 w
(changed during the reduction of)4 1283 1 720 4213 t
10 I f
(f)2030 4213 w
7 I f
(b)2069 4233 w
10 I f
(*)2120 4213 w
10 R f
(,)2170 4213 w
10 I f
(F)2222 4213 w
7 R f
(2)2294 4233 w
10 I f
(*)2345 4213 w
10 R f
(\()2403 4213 w
10 I f
(w)2444 4213 w
10 R f
(,)2519 4213 w
10 I f
(t)2585 4213 w
10 S f
(\242)2621 4213 w
10 R f
(\) =)1 116 1 2654 4213 t
10 S f
(- b)1 126 1 2797 4213 t
10 R f
(\()2931 4213 w
10 I f
(w)2972 4213 w
10 R f
(,)3047 4213 w
10 I f
(f)3121 4213 w
10 S1 f
(_)3115 4120 w
7 I f
(b)3160 4233 w
10 I f
(*)3211 4213 w
10 R f
(\) =)1 116 1 3269 4213 t
10 S f
(- b)1 126 1 3412 4213 t
10 R f
(\()3546 4213 w
10 I f
(w)3587 4213 w
10 R f
(,)3662 4213 w
10 I f
(f)3736 4213 w
7 I f
(b)3775 4233 w
10 I f
(*)3826 4213 w
10 R f
(\) =)1 116 1 3884 4213 t
10 S f
(b)4026 4213 w
10 R f
(\()4089 4213 w
10 I f
(w)4130 4213 w
10 R f
(,)4205 4213 w
10 I f
(f)4279 4213 w
7 R f
(0)4318 4233 w
10 R f
(\),)4369 4213 w
10 I f
(w)4453 4213 w
10 S f
(\316)4561 4213 w
10 I f
(D)4673 4213 w
10 R f
(. Now,)1 295 1 4745 4213 t
10 I f
(value)720 4333 w
10 R f
(\()944 4333 w
10 I f
(F)985 4333 w
7 R f
(2)1057 4353 w
10 I f
(*)1108 4333 w
10 R f
(\) =)1 114 1 1166 4333 t
7 I f
(w)1305 4433 w
7 S f
(\316)1380 4433 w
7 I f
(D)1458 4433 w
15 S f
(S)1362 4363 w
10 I f
(F)1508 4333 w
7 R f
(2)1580 4353 w
10 I f
(*)1631 4333 w
10 R f
(\()1689 4333 w
10 I f
(w)1730 4333 w
10 R f
(,)1805 4333 w
10 I f
(t)1871 4333 w
10 S f
(\242)1907 4333 w
10 R f
(\) =)1 114 1 1940 4333 t
7 I f
(w)2079 4433 w
7 S f
(\316)2154 4433 w
7 I f
(D)2232 4433 w
15 S f
(S)2136 4363 w
10 S f
(b)2290 4333 w
10 R f
(\()2353 4333 w
10 I f
(w)2394 4333 w
10 R f
(,)2469 4333 w
10 I f
(f)2543 4333 w
7 R f
(0)2582 4353 w
10 R f
(\), and Part \(ii\) has been established.)6 1425 1 2633 4333 t
(We further reduce)2 764 1 970 4599 t
10 I f
(f)1779 4599 w
10 S1 f
(_)1773 4506 w
7 I f
(b)1818 4619 w
10 I f
(*)1869 4599 w
10 R f
(along)1964 4599 w
10 I f
(s)2231 4599 w
10 S f
(-)2294 4599 w
10 I f
(w)2365 4599 w
10 R f
(paths, where)1 525 1 2477 4599 t
10 I f
(w)3048 4599 w
10 S f
(\316)3156 4599 w
10 I f
(D)3268 4599 w
10 R f
(, such that the reduced preflow)5 1335 1 3340 4599 t
10 I f
(f)4721 4599 w
10 S1 f
(_)4715 4506 w
(_)4715 4481 w
7 I f
(b)4760 4619 w
10 I f
(*)4811 4599 w
10 R f
(has)4907 4599 w
10 S f
(b)720 4749 w
10 R f
(\()783 4749 w
10 I f
(s)824 4749 w
10 R f
(,)871 4749 w
10 I f
(f)945 4749 w
10 S1 f
(_)939 4656 w
(_)939 4631 w
7 I f
(b)984 4769 w
10 I f
(*)1035 4749 w
10 R f
(\))1093 4749 w
10 S f
(=)1183 4749 w
10 R f
(0. Let)1 262 1 1287 4749 t
10 I f
(F)1578 4749 w
7 R f
(1)1650 4769 w
10 I f
(*)1701 4749 w
10 R f
(be the derived flow from)4 1008 1 1780 4749 t
10 I f
(f)2817 4749 w
10 S1 f
(_)2811 4656 w
(_)2811 4631 w
7 I f
(b)2856 4769 w
10 I f
(*)2907 4749 w
10 R f
(on)2986 4749 w
10 I f
(G)3114 4749 w
10 S1 f
(_ _)1 62 1 3124 4656 t
10 R f
(with)3214 4749 w
10 I f
(u)3420 4749 w
10 S1 f
(_)3422 4681 w
10 R f
(\()3478 4749 w
10 I f
(s)3519 4749 w
10 S f
(\242)3566 4749 w
10 R f
(,)3599 4749 w
10 I f
(s)3665 4749 w
10 R f
(\))3712 4749 w
10 S f
(=)3802 4749 w
10 I f
(u)3906 4749 w
10 S1 f
(_)3908 4681 w
10 R f
(\()3964 4749 w
10 I f
(t)4005 4749 w
10 R f
(,)4041 4749 w
10 I f
(t)4107 4749 w
10 S f
(\242)4143 4749 w
10 R f
(\))4176 4749 w
10 S f
(=)4266 4749 w
10 R f
( we)1 144(0. Similarly,)1 526 2 4370 4749 t
(have)720 4869 w
10 I f
(value)933 4869 w
10 R f
(\()1157 4869 w
10 I f
(F)1198 4869 w
7 R f
(1)1270 4889 w
10 I f
(*)1321 4869 w
10 R f
(\) =)1 114 1 1379 4869 t
7 I f
(w)1518 4969 w
7 S f
(\316)1593 4969 w
7 I f
(D)1671 4969 w
15 S f
(S)1575 4899 w
10 S f
(b)1729 4869 w
10 R f
(\()1792 4869 w
10 I f
(w)1833 4869 w
10 R f
(,)1908 4869 w
10 I f
(f)1982 4869 w
7 R f
(0)2021 4889 w
10 R f
(\))2072 4869 w
10 S f
(- m)1 162 1 2130 4869 t
10 I f
(*)2300 4869 w
10 R f
(, and Part \(i\) has been proved.)6 1197 1 2350 4869 t
10 S1 f
()3597 4869 w
3597 4869 m
50 build_sq
3647 4869 m
10 B f
(Theorem 3.3.)1 573 1 720 5105 t
10 R f
(Algorithm 1 determines the existence of a flow on)8 2088 1 1328 5105 t
10 I f
(G)3451 5105 w
10 R f
(and constructs a minimum flow)4 1306 1 3558 5105 t
10 I f
(f)4899 5105 w
10 R f
(in)4962 5105 w
(time)720 5225 w
10 I f
(O)923 5225 w
10 R f
(\()1003 5225 w
10 I f
(mn)1044 5225 w
10 R f
(log)1207 5225 w
10 I f
(n)1376 5225 w
10 R f
(\).)1434 5225 w
10 B f
(Proof.)720 5381 w
10 R f
( is no feasible flow on)5 915(We first show that if the algorithm aborts at Steps 5 or 8, there)13 2562 2 1038 5381 t
10 I f
(G)4546 5381 w
10 R f
( then)1 203(. We)1 219 2 4618 5381 t
( the preflow)2 492(show that)1 392 2 720 5501 t
10 I f
(f)1634 5501 w
7 I f
(b)1673 5521 w
10 R f
(constructed in Step 9 is a minimum balancing preflow and, therefore,)10 2818 1 1746 5501 t
10 I f
(f)4594 5501 w
10 R f
(is a mini-)2 388 1 4652 5501 t
(mum flow on)2 539 1 720 5621 t
10 I f
(G)1284 5621 w
10 R f
(.)1356 5621 w
(In Step 4, we construct a maximum flow on graph)9 2066 1 970 5782 t
10 I f
(R)3068 5782 w
10 S1 f
(_ _)1 51 1 3078 5689 t
7 R f
(1)3140 5802 w
10 R f
(after setting)1 481 1 3215 5782 t
10 I f
(u)3728 5782 w
10 S1 f
(_)3730 5714 w
10 R f
(\()3786 5782 w
10 I f
(v)3827 5782 w
10 R f
(,)3879 5782 w
10 I f
(s)3945 5782 w
10 S f
(\242)3992 5782 w
10 R f
( for each)2 364(\) to zero)2 346 2 4025 5782 t
10 I f
(v)4768 5782 w
10 S f
(\316)4853 5782 w
10 I f
(S)4965 5782 w
10 R f
(.)5015 5782 w
(Since)720 5907 w
10 I f
(t)974 5907 w
10 S f
(\242)1010 5907 w
10 R f
(is not reachable from)3 866 1 1067 5907 t
10 I f
(s)1965 5907 w
10 S f
(\242)2012 5907 w
10 R f
(in)2068 5907 w
10 I f
(R)2177 5907 w
7 R f
(2)2249 5927 w
10 R f
(, and setting)2 498 1 2292 5907 t
10 I f
(u)2821 5907 w
10 S1 f
(_)2823 5839 w
10 R f
(\()2879 5907 w
10 I f
(v)2920 5907 w
10 R f
(,)2972 5907 w
10 I f
(s)3038 5907 w
10 S f
(\242)3085 5907 w
10 R f
(\) back to)2 361 1 3118 5907 t
10 I f
(F)3510 5907 w
7 R f
(1)3582 5927 w
10 R f
(\()3633 5907 w
10 I f
(s)3674 5907 w
10 S f
(\242)3721 5907 w
10 R f
(,)3754 5907 w
10 I f
(v)3820 5907 w
10 R f
(\) does not provide any)4 917 1 3872 5907 t
10 I f
(s)4820 5907 w
10 S f
(\242 -)1 96 1 4867 5907 t
10 I f
(t)4979 5907 w
10 S f
(\242)5015 5907 w
10 R f
(paths in)1 314 1 720 6032 t
10 I f
(R)1059 6032 w
7 R f
(2)1131 6052 w
10 R f
(,)1174 6032 w
10 I f
(F)1224 6032 w
7 R f
(2)1296 6052 w
10 R f
(is a maximum flow on)4 900 1 1364 6032 t
10 I f
(G)2289 6032 w
10 S1 f
(_ _)1 62 1 2299 5939 t
10 R f
(with)2386 6032 w
10 I f
(u)2589 6032 w
10 S1 f
(_)2591 5964 w
10 R f
(\()2647 6032 w
10 I f
(s)2688 6032 w
10 S f
(\242)2735 6032 w
10 R f
(,)2768 6032 w
10 I f
(s)2834 6032 w
10 R f
(\))2881 6032 w
10 S f
(= \245)1 177 1 2971 6032 t
10 R f
(and)3173 6032 w
10 I f
(u)3342 6032 w
10 S1 f
(_)3344 5964 w
10 R f
(\()3400 6032 w
10 I f
(t)3441 6032 w
10 R f
(,)3477 6032 w
10 I f
(t)3543 6032 w
10 S f
(\242)3579 6032 w
10 R f
(\))3612 6032 w
10 S f
(=)3702 6032 w
10 R f
(0.)3806 6032 w
( exist balancing preflows on)4 1146(Assume that Algorithm 1 aborts at Step 5 and that there)10 2266 2 970 6188 t
10 I f
(G)4412 6188 w
10 R f
(. Let)1 188 1 4484 6188 t
10 I f
(f)4702 6188 w
7 I f
(b)4741 6208 w
10 I f
(*)4792 6188 w
10 R f
(be a)1 168 1 4872 6188 t
(minimum balancing preflow with value)4 1594 1 720 6313 t
10 S f
(m)2342 6313 w
10 I f
(*)2408 6313 w
10 R f
( by \(ii\) of Lemma 3.2, there exists a flow)9 1666(. Then)1 283 2 2458 6313 t
10 I f
(F)4435 6313 w
7 R f
(2)4507 6333 w
10 I f
(*)4558 6313 w
10 R f
(on)4636 6313 w
10 I f
(G)4763 6313 w
10 S1 f
(_ _)1 62 1 4773 6220 t
10 R f
(with)4862 6313 w
10 I f
(u)720 6438 w
10 S1 f
(_)722 6370 w
10 R f
(\()778 6438 w
10 I f
(s)819 6438 w
10 S f
(\242)866 6438 w
10 R f
(,)899 6438 w
10 I f
(s)965 6438 w
10 R f
(\))1012 6438 w
10 S f
(= \245)1 177 1 1102 6438 t
10 R f
(and)1305 6438 w
10 I f
(u)1475 6438 w
10 S1 f
(_)1477 6370 w
10 R f
(\()1533 6438 w
10 I f
(t)1574 6438 w
10 R f
(,)1610 6438 w
10 I f
(t)1676 6438 w
10 S f
(\242)1712 6438 w
10 R f
(\))1745 6438 w
10 S f
(=)1835 6438 w
10 R f
(0, with)1 280 1 1939 6438 t
10 I f
(value)2246 6438 w
10 R f
(\()2470 6438 w
10 I f
(F)2511 6438 w
7 R f
(2)2583 6458 w
10 I f
(*)2634 6438 w
10 R f
(\) =)1 116 1 2692 6438 t
7 I f
(w)2835 6538 w
7 S f
(\316)2910 6538 w
7 I f
(D)2988 6538 w
15 S f
(S)2892 6468 w
10 S f
(b)3046 6438 w
10 R f
(\()3109 6438 w
10 I f
(w)3150 6438 w
10 R f
(,)3225 6438 w
10 I f
(f)3299 6438 w
7 R f
(0)3338 6458 w
10 R f
( the other hand, the maximum flow)6 1419(\). On)1 232 2 3389 6438 t
10 I f
(F)720 6643 w
7 R f
(2)792 6663 w
10 R f
(on)873 6643 w
10 I f
(G)1011 6643 w
10 S1 f
(_ _)1 62 1 1021 6550 t
10 R f
(with)1121 6643 w
10 I f
(u)1337 6643 w
10 S1 f
(_)1339 6575 w
10 R f
(\()1395 6643 w
10 I f
(s)1436 6643 w
10 S f
(\242)1483 6643 w
10 R f
(,)1516 6643 w
10 I f
(s)1582 6643 w
10 R f
(\) =)1 127 1 1629 6643 t
10 S f
(\245)1794 6643 w
10 R f
(and)1905 6643 w
10 I f
(u)2087 6643 w
10 S1 f
(_)2089 6575 w
10 R f
(\()2145 6643 w
10 I f
(t)2186 6643 w
10 R f
(,)2222 6643 w
10 I f
(t)2288 6643 w
10 S f
(\242)2324 6643 w
10 R f
(\))2357 6643 w
10 S f
(=)2447 6643 w
10 R f
(0 has)1 221 1 2551 6643 t
10 I f
(value)2810 6643 w
10 R f
(\()3034 6643 w
10 I f
(F)3075 6643 w
7 R f
(2)3147 6663 w
10 R f
(\))3198 6643 w
10 S f
(<)3269 6643 w
7 I f
(w)3362 6743 w
7 S f
(\316)3437 6743 w
7 I f
(D)3515 6743 w
15 S f
(S)3419 6673 w
10 S f
(b)3573 6643 w
10 R f
(\()3636 6643 w
10 I f
(w)3677 6643 w
10 R f
(,)3752 6643 w
10 I f
(f)3826 6643 w
7 R f
(0)3865 6663 w
10 R f
(\))3916 6643 w
10 S f
(=)4006 6643 w
10 I f
(value)4110 6643 w
10 R f
(\()4334 6643 w
10 I f
(F)4375 6643 w
7 R f
(2)4447 6663 w
10 I f
(*)4498 6643 w
10 R f
( is a)2 185(\). This)1 299 2 4556 6643 t
( if Algorithm 1 aborts at Step 5, there is no balancing preflow on)13 2612(contradiction. Therefore,)1 1021 2 720 6843 t
10 I f
(G)4380 6843 w
10 R f
(, and by Theo-)3 588 1 4452 6843 t
(rem 3.2, there is no feasible flow on)7 1439 1 720 6963 t
10 I f
(G)2184 6963 w
10 R f
(.)2256 6963 w
( exist balancing preflows on)4 1146(Assume that Algorithm 1 aborts at Step 8 and that there)10 2266 2 970 7119 t
10 I f
(G)4412 7119 w
10 R f
(. Let)1 188 1 4484 7119 t
10 I f
(f)4702 7119 w
7 I f
(b)4741 7139 w
10 I f
(*)4792 7119 w
10 R f
(be a)1 168 1 4872 7119 t
(minimum balancing preflow with value)4 1602 1 720 7244 t
10 S f
(m)2352 7244 w
10 I f
(*)2418 7244 w
10 R f
( \(i\) of Lemma 3.2, there exists a flow)8 1518( by)1 130(. Then)1 285 3 2468 7244 t
10 I f
(F)4430 7244 w
7 R f
(1)4502 7264 w
10 I f
(*)4553 7244 w
10 R f
(on)4632 7244 w
10 I f
(G)4761 7244 w
10 S1 f
(_ _)1 62 1 4771 7151 t
10 R f
(with)4862 7244 w
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6 pagesetup
10 R f
(- 6 -)2 166 1 2797 480 t
10 I f
(u)720 845 w
10 S1 f
(_)722 777 w
10 R f
(\()778 845 w
10 I f
(s)819 845 w
10 S f
(\242)866 845 w
10 R f
(,)899 845 w
10 I f
(s)965 845 w
10 R f
(\))1012 845 w
10 S f
(=)1102 845 w
10 I f
(u)1206 845 w
10 S1 f
(_)1208 777 w
10 R f
(\()1264 845 w
10 I f
(t)1305 845 w
10 R f
(,)1341 845 w
10 I f
(t)1407 845 w
10 S f
(\242)1443 845 w
10 R f
(\))1476 845 w
10 S f
(=)1566 845 w
10 R f
(0, with)1 293 1 1670 845 t
10 I f
(value)2003 845 w
10 R f
(\()2227 845 w
10 I f
(F)2268 845 w
7 R f
(1)2340 865 w
10 I f
(*)2391 845 w
10 R f
(\) =)1 129 1 2449 845 t
7 I f
(w)2618 945 w
7 S f
(\316)2693 945 w
7 I f
(D)2771 945 w
15 S f
(S)2675 875 w
10 S f
(b)2829 845 w
10 R f
(\()2892 845 w
10 I f
(w)2933 845 w
10 R f
(,)3008 845 w
10 I f
(f)3082 845 w
7 R f
(0)3121 865 w
10 R f
(\))3172 845 w
10 S f
(- m)1 162 1 3262 845 t
10 I f
(*)3432 845 w
10 R f
( Algorithm 1 did not abort at)6 1246(. Since)1 312 2 3482 845 t
(Step 5,)1 295 1 720 1050 t
10 I f
(value)1057 1050 w
10 R f
(\()1281 1050 w
10 I f
(F)1322 1050 w
7 R f
(2)1394 1070 w
10 R f
(\) =)1 131 1 1445 1050 t
7 I f
(w)1618 1150 w
7 S f
(\316)1693 1150 w
7 I f
(D)1771 1150 w
15 S f
(S)1675 1080 w
10 S f
(b)1829 1050 w
10 R f
(\()1892 1050 w
10 I f
(w)1933 1050 w
10 R f
(,)2008 1050 w
10 I f
(f)2082 1050 w
7 R f
(0)2121 1070 w
10 R f
(\). Since)1 347 1 2172 1050 t
10 I f
(F)2561 1050 w
7 R f
(1)2633 1070 w
10 R f
(is a maximum flow on)4 968 1 2718 1050 t
10 I f
(G)3728 1050 w
10 S1 f
(_ _)1 62 1 3738 957 t
10 R f
(with)3842 1050 w
10 I f
(u)4061 1050 w
10 S1 f
(_)4063 982 w
10 R f
(\()4119 1050 w
10 I f
(s)4160 1050 w
10 S f
(\242)4207 1050 w
10 R f
(,)4240 1050 w
10 I f
(s)4306 1050 w
10 R f
(\) =)1 130 1 4353 1050 t
10 I f
(u)4524 1050 w
10 S1 f
(_)4526 982 w
10 R f
(\()4582 1050 w
10 I f
(t)4623 1050 w
10 R f
(,)4659 1050 w
10 I f
(t)4725 1050 w
10 S f
(\242)4761 1050 w
10 R f
(\) = 0,)2 246 1 4794 1050 t
10 I f
(value)720 1250 w
10 R f
(\()944 1250 w
10 I f
(F)985 1250 w
7 R f
(1)1057 1270 w
10 R f
(\) =)1 114 1 1108 1250 t
10 I f
(value)1247 1250 w
10 R f
(\()1471 1250 w
10 I f
(F)1512 1250 w
7 R f
(2)1584 1270 w
10 R f
(\))1635 1250 w
10 S f
( =)1 80(- m)1 162 2 1693 1250 t
7 I f
(w)1984 1350 w
7 S f
(\316)2059 1350 w
7 I f
(D)2137 1350 w
15 S f
(S)2041 1280 w
10 S f
(b)2195 1250 w
10 R f
(\()2258 1250 w
10 I f
(w)2299 1250 w
10 R f
(,)2374 1250 w
10 I f
(f)2448 1250 w
7 R f
(0)2487 1270 w
10 R f
(\))2538 1250 w
10 S f
( \263)1 80(- m)1 162 2 2596 1250 t
10 I f
(value)2879 1250 w
10 R f
(\()3103 1250 w
10 I f
(F)3144 1250 w
7 R f
(1)3216 1270 w
10 I f
(*)3267 1250 w
10 R f
(\) =)1 114 1 3325 1250 t
7 I f
(w)3464 1350 w
7 S f
(\316)3539 1350 w
7 I f
(D)3617 1350 w
15 S f
(S)3521 1280 w
10 S f
(b)3675 1250 w
10 R f
(\()3738 1250 w
10 I f
(w)3779 1250 w
10 R f
(,)3854 1250 w
10 I f
(f)3928 1250 w
7 R f
(0)3967 1270 w
10 R f
(\))4018 1250 w
10 S f
(- m)1 129 1 4101 1250 t
10 I f
(*)4238 1250 w
10 R f
(. Thus)1 275 1 4288 1250 t
10 S f
(m \243 m)2 253 1 4588 1250 t
10 I f
(*)4849 1250 w
10 R f
(.)4899 1250 w
(Since)970 1491 w
10 I f
(F)1218 1491 w
7 R f
(3)1290 1511 w
10 R f
(is a maximum flow on)4 904 1 1359 1491 t
10 I f
(G)2289 1491 w
10 S1 f
(_ _)1 62 1 2299 1398 t
10 R f
(with)2387 1491 w
10 I f
(u)2591 1491 w
10 S1 f
(_)2593 1423 w
10 R f
(\()2649 1491 w
10 I f
(s)2690 1491 w
10 S f
(\242)2737 1491 w
10 R f
(,)2770 1491 w
10 I f
(s)2836 1491 w
10 R f
(\))2883 1491 w
10 S f
(= m)1 162 1 2973 1491 t
10 R f
(and)3162 1491 w
10 I f
(u)3333 1491 w
10 S1 f
(_)3335 1423 w
10 R f
(\()3391 1491 w
10 I f
(t)3432 1491 w
10 R f
(,)3468 1491 w
10 I f
(t)3534 1491 w
10 S f
(\242)3570 1491 w
10 R f
(\))3603 1491 w
10 S f
(= \245)1 177 1 3693 1491 t
10 R f
(, a maximum flow on)4 866 1 3870 1491 t
10 I f
(G)4763 1491 w
10 S1 f
(_ _)1 62 1 4773 1398 t
10 R f
(with)4862 1491 w
10 I f
(u)720 1616 w
10 S1 f
(_)722 1548 w
10 R f
(\()778 1616 w
10 I f
(s)819 1616 w
10 S f
(\242)866 1616 w
10 R f
(,)899 1616 w
10 I f
(s)965 1616 w
10 R f
(\))1012 1616 w
10 S f
(= m)1 162 1 1102 1616 t
10 I f
(*)1272 1616 w
10 R f
(and)1384 1616 w
10 I f
(u)1590 1616 w
10 S1 f
(_)1592 1548 w
10 R f
(\()1648 1616 w
10 I f
(t)1689 1616 w
10 R f
(,)1725 1616 w
10 I f
(t)1791 1616 w
10 S f
(\242)1827 1616 w
10 R f
(\))1860 1616 w
10 S f
(= \245)1 177 1 1950 1616 t
10 R f
(has value no more than \()5 1169 1 2189 1616 t
10 S f
(m)3366 1616 w
10 I f
(*)3432 1616 w
10 S f
(- m)1 162 1 3531 1616 t
10 R f
(\) +)1 151 1 3701 1616 t
10 I f
(value)3914 1616 w
10 R f
(\()4138 1616 w
10 I f
(F)4179 1616 w
7 R f
(3)4251 1636 w
10 R f
(\))4302 1616 w
10 S f
(<)4397 1616 w
10 R f
(\()4514 1616 w
10 S f
(m)4555 1616 w
10 I f
(*)4621 1616 w
10 S f
(- m)1 162 1 4720 1616 t
10 R f
(\) +)1 150 1 4890 1616 t
10 I f
({)720 1741 w
10 S f
(m +)1 162 1 768 1741 t
7 I f
(v)979 1841 w
7 S f
(\316)1038 1841 w
7 I f
(S)1116 1841 w
15 S f
(S)1020 1771 w
10 R f
([)1158 1741 w
10 S f
(- b)1 126 1 1207 1741 t
10 R f
(\()1341 1741 w
10 I f
(v)1382 1741 w
10 R f
(,)1434 1741 w
10 I f
(f)1508 1741 w
7 R f
(0)1547 1761 w
10 R f
(\) ])1 74 1 1598 1741 t
10 I f
(})1688 1741 w
10 R f
(=)1762 1741 w
10 S f
(m)1853 1741 w
10 I f
(*)1919 1741 w
10 S f
(+)2018 1741 w
7 I f
(v)2122 1841 w
7 S f
(\316)2181 1841 w
7 I f
(S)2259 1841 w
15 S f
(S)2163 1771 w
10 R f
([)2301 1741 w
10 S f
(- b)1 126 1 2350 1741 t
10 R f
(\()2484 1741 w
10 I f
(v)2525 1741 w
10 R f
(,)2577 1741 w
10 I f
(f)2651 1741 w
7 R f
(0)2690 1761 w
10 R f
( is, a maximum flow on)5 1000( That)1 243(\) ].)1 99 3 2741 1741 t
10 I f
(G)4118 1741 w
10 S1 f
(_ _)1 62 1 4128 1648 t
10 R f
(with)4225 1741 w
10 I f
(u)4438 1741 w
10 S1 f
(_)4440 1673 w
10 R f
(\()4496 1741 w
10 I f
(s)4537 1741 w
10 S f
(\242)4584 1741 w
10 R f
(,)4617 1741 w
10 I f
(s)4683 1741 w
10 R f
(\))4730 1741 w
10 S f
(= m)1 162 1 4820 1741 t
10 I f
(*)4990 1741 w
10 R f
(and)720 1946 w
10 I f
(u)902 1946 w
10 S1 f
(_)904 1878 w
10 R f
(\()960 1946 w
10 I f
(t)1001 1946 w
10 R f
(,)1037 1946 w
10 I f
(t)1103 1946 w
10 S f
(\242)1139 1946 w
10 R f
(\))1172 1946 w
10 S f
(= \245)1 177 1 1262 1946 t
10 R f
(has value less than)3 785 1 1477 1946 t
10 S f
(m)2300 1946 w
10 I f
(*)2366 1946 w
10 S f
(+)2465 1946 w
7 I f
(v)2569 2046 w
7 S f
(\316)2628 2046 w
7 I f
(S)2706 2046 w
15 S f
(S)2610 1976 w
10 R f
([)2748 1946 w
10 S f
(- b)1 126 1 2797 1946 t
10 R f
(\()2931 1946 w
10 I f
(v)2972 1946 w
10 R f
(,)3024 1946 w
10 I f
(f)3098 1946 w
7 R f
(0)3137 1966 w
10 R f
( of Lemma 3.2.)3 649( contradicts Part \(iii\))3 863( This)1 241(\) ].)1 99 4 3188 1946 t
(Therefore, if Algorithm 1 aborts at Step 8, there is no balancing preflow on)13 3016 1 720 2146 t
10 I f
(G)3762 2146 w
10 R f
( by Theorem 3.2, there is)5 1011(, and)1 195 2 3834 2146 t
(no feasible flow on)3 768 1 720 2266 t
10 I f
(G)1513 2266 w
10 R f
(.)1585 2266 w
( and 8,)2 273(From Steps 5)2 536 2 970 2422 t
10 I f
(F)1806 2422 w
7 R f
(3)1878 2442 w
10 R f
(saturates all the edges \()4 939 1 1948 2422 t
10 I f
(s)2895 2422 w
10 S f
(\242)2942 2422 w
10 R f
(,)2975 2422 w
10 I f
(v)3041 2422 w
10 R f
(\) and \()2 264 1 3093 2422 t
10 I f
(w)3365 2422 w
10 R f
(,)3440 2422 w
10 I f
(t)3506 2422 w
10 S f
(\242)3542 2422 w
10 R f
(\),)3575 2422 w
10 I f
(v)3660 2422 w
10 S f
(\316)3745 2422 w
10 I f
(S)3857 2422 w
10 R f
(and)3934 2422 w
10 I f
(w)4105 2422 w
10 S f
(\316)4213 2422 w
10 I f
(D)4325 2422 w
10 R f
( the)1 149(. Therefore,)1 494 2 4397 2422 t
(induced preflow)1 668 1 720 2542 t
10 I f
(f)1430 2542 w
7 I f
(b)1469 2562 w
10 R f
( the arguments in the previous paragraph,)6 1751( From)1 283( Step 9 is balancing.)4 872(constructed in)1 580 4 1554 2542 t
10 I f
(value)720 2662 w
10 R f
(\()944 2662 w
10 I f
(f)1001 2662 w
7 I f
(b)1040 2682 w
10 R f
(\))1091 2662 w
10 S f
( m)1 99( \243)1 81(= m)1 162 3 1181 2662 t
10 I f
(*)1531 2662 w
10 R f
(, where)1 295 1 1581 2662 t
10 S f
(m)1903 2662 w
10 I f
(*)1969 2662 w
10 R f
( Therefore,)1 469(is the value of a minimum balancing preflow.)7 1834 2 2046 2662 t
10 I f
(f)4376 2662 w
7 I f
(b)4415 2682 w
10 R f
(is a minimum)2 555 1 4485 2662 t
(balancing preflow.)1 748 1 720 2782 t
(It takes time)2 502 1 970 2938 t
10 I f
(O)1501 2938 w
10 R f
(\()1581 2938 w
10 I f
(mn)1622 2938 w
10 R f
(log)1785 2938 w
10 I f
(n)1954 2938 w
10 R f
(\) to construct a maximum flow using the Sleator-Tarjan algorithm in Steps)11 3028 1 2012 2938 t
( other steps take time)4 849( The)1 205(2, 4, and 7.)3 444 3 720 3058 t
10 I f
(O)2243 3058 w
10 R f
(\()2323 3058 w
10 I f
(m)2364 3058 w
10 R f
(\).)2444 3058 w
10 S1 f
()2552 3058 w
2552 3058 m
50 build_sq
2602 3058 m
10 B f
( minimum flow)2 650(3.4. Minimum-cost)1 827 2 720 3298 t
10 R f
( minimum-cost minimum flow for)4 1389(The algorithm for finding a)4 1105 2 970 3454 t
10 I f
(G)3493 3454 w
10 R f
(is based on the output of Algorithm)6 1446 1 3594 3454 t
(1, which computes the value of a minimum balancing preflow)9 2534 1 720 3574 t
10 S f
(m)3285 3574 w
10 R f
( pre-)1 190( for any balancing)3 741( Since)1 278(, if it exists.)3 488 4 3343 3574 t
(flow)720 3694 w
10 I f
(f)942 3694 w
7 I f
(b)981 3714 w
10 R f
(,)1024 3694 w
10 S f
(b)1088 3694 w
10 R f
(\()1151 3694 w
10 I f
(s)1192 3694 w
10 R f
(,)1239 3694 w
10 I f
(f)1313 3694 w
7 R f
(0)1352 3714 w
10 R f
(\))1403 3694 w
10 S f
(+ b)1 159 1 1493 3694 t
10 R f
(\()1660 3694 w
10 I f
(s)1701 3694 w
10 R f
(,)1748 3694 w
10 I f
(f)1822 3694 w
7 I f
(b)1861 3714 w
10 R f
(\) =)1 128 1 1912 3694 t
10 S f
(-)2079 3694 w
10 R f
([)2150 3694 w
10 S f
(b)2191 3694 w
10 R f
(\()2254 3694 w
10 I f
(t)2295 3694 w
10 R f
(,)2331 3694 w
10 I f
(f)2405 3694 w
7 R f
(0)2444 3714 w
10 R f
(\))2495 3694 w
10 S f
(+ b)1 159 1 2585 3694 t
10 R f
(\()2752 3694 w
10 I f
(t)2793 3694 w
10 R f
(,)2829 3694 w
10 I f
(f)2903 3694 w
7 I f
(b)2942 3714 w
10 R f
( let)1 139(\) ],)1 99 2 2993 3694 t
10 S f
(n)3270 3694 w
10 R f
(=)3361 3694 w
10 S f
(- b)1 126 1 3456 3694 t
10 R f
(\()3590 3694 w
10 I f
(t)3631 3694 w
10 R f
(,)3667 3694 w
10 I f
(f)3741 3694 w
7 I f
(b)3780 3714 w
10 R f
(\) =)1 129 1 3831 3694 t
10 S f
(m)4000 3694 w
10 R f
(+)4098 3694 w
10 S f
(b)4194 3694 w
10 R f
(\()4257 3694 w
10 I f
(s)4298 3694 w
10 R f
(,)4345 3694 w
10 I f
(f)4419 3694 w
7 R f
(0)4458 3714 w
10 R f
(\) +)1 129 1 4509 3694 t
10 S f
(b)4678 3694 w
10 R f
(\()4741 3694 w
10 I f
(t)4782 3694 w
10 R f
(,)4818 3694 w
10 I f
(f)4892 3694 w
7 R f
(0)4931 3714 w
10 R f
(\).)4982 3694 w
(From)720 3819 w
10 I f
(G)967 3819 w
10 R f
(we construct a balancing graph)4 1261 1 1069 3819 t
10 I f
(G)2360 3819 w
10 S1 f
(_ _)1 62 1 2370 3726 t
10 R f
( capacities of edges \()4 852(, in which the)3 559 2 2432 3819 t
10 I f
(s)3851 3819 w
10 S f
(\242)3898 3819 w
10 R f
(,)3931 3819 w
10 I f
(s)3997 3819 w
10 R f
(\) and \()2 268 1 4044 3819 t
10 I f
(t)4320 3819 w
10 R f
(,)4356 3819 w
10 I f
(t)4422 3819 w
10 S f
(\242)4458 3819 w
10 R f
(\) are)1 183 1 4491 3819 t
10 S f
(m)4703 3819 w
10 R f
(and)4790 3819 w
10 S f
(n)4963 3819 w
10 R f
(,)5015 3819 w
(respectively. Let)1 693 1 720 3944 t
10 I f
(F)1441 3944 w
10 R f
(be the derived flow on)4 910 1 1530 3944 t
10 I f
(G)2468 3944 w
10 S1 f
(_ _)1 62 1 2478 3851 t
10 R f
( Then)1 259( balancing preflow.)2 781(from a minimum)2 684 3 2568 3944 t
10 I f
(F)4321 3944 w
10 R f
(saturates all the)2 629 1 4411 3944 t
(edges incident to)2 681 1 720 4069 t
10 I f
(s)1428 4069 w
10 S f
(\242)1475 4069 w
10 R f
(or)1527 4069 w
10 I f
(t)1637 4069 w
10 S f
(\242)1673 4069 w
10 R f
( a flow)2 279(. Therefore,)1 493 2 1698 4069 t
10 I f
(F)2496 4069 w
10 R f
(on)2583 4069 w
10 I f
(G)2709 4069 w
10 S1 f
(_ _)1 62 1 2719 3976 t
10 R f
(is maximum if and only if all the edges incident to)10 2026 1 2807 4069 t
10 I f
(s)4859 4069 w
10 S f
(\242)4906 4069 w
10 R f
(or)4957 4069 w
10 I f
(t)720 4189 w
10 S f
(\242)756 4189 w
10 R f
( is the case if and only if the induced preflow)10 1872( This)1 234(are saturated.)1 537 3 812 4189 t
10 I f
(f)3487 4189 w
7 I f
(b)3526 4209 w
10 R f
(of)3601 4189 w
10 I f
(F)3716 4189 w
10 R f
(is balancing and has the mini-)5 1231 1 3809 4189 t
(mum value)1 447 1 720 4309 t
10 S f
(m)1192 4309 w
10 R f
(. Since)1 297 1 1250 4309 t
10 I f
(cost)1572 4309 w
10 R f
(\()1741 4309 w
10 I f
(F)1782 4309 w
10 R f
(\) =)1 114 1 1851 4309 t
10 I f
(cost)1990 4309 w
10 R f
(\()2159 4309 w
10 I f
(f)2216 4309 w
7 I f
(b)2255 4329 w
10 R f
(\), we have:)2 440 1 2306 4309 t
10 B f
(Lemma 3.3.)1 512 1 720 4470 t
10 R f
( preflow on)2 482(If there exists a balancing)4 1065 2 1267 4470 t
10 I f
(G)2850 4470 w
10 R f
(, then a flow)3 532 1 2922 4470 t
10 I f
(F)3490 4470 w
10 R f
(on)3587 4470 w
10 I f
(G)3723 4470 w
10 S1 f
(_ _)1 62 1 3733 4377 t
10 R f
(is a minimum-cost maximum)3 1209 1 3831 4470 t
(flow if and only if the induced preflow)7 1550 1 720 4590 t
10 I f
(f)2295 4590 w
7 I f
(b)2334 4610 w
10 R f
(on)2402 4590 w
10 I f
(G)2527 4590 w
10 R f
(is a minimum-cost minimum balancing preflow.)5 1933 1 2624 4590 t
10 S1 f
()720 4746 w
720 4746 m
50 build_sq
770 4746 m
10 R f
(To construct the minimum-cost maximum flows on)6 2127 1 970 4907 t
10 I f
(G)3133 4907 w
10 S1 f
(_ _)1 62 1 3143 4814 t
10 R f
( be used. Note that)4 794(, different techniques can)3 1041 2 3205 4907 t
(from the capacity constraints of the edges incident to)8 2113 1 720 5032 t
10 I f
(s)2858 5032 w
10 S f
(\242)2905 5032 w
10 R f
(in)2955 5032 w
10 I f
(G)3058 5032 w
10 S1 f
(_ _)1 62 1 3068 4939 t
10 R f
(, the flow values are bounded by)6 1300 1 3130 5032 t
10 I f
(L)4455 5032 w
10 S f
(=)4560 5032 w
7 I f
(e)4664 5132 w
7 S f
(\316)4723 5132 w
7 I f
(E)4801 5132 w
15 S f
(S)4709 5062 w
10 I f
(l)4843 5032 w
10 R f
(\()4879 5032 w
10 I f
(e)4920 5032 w
10 R f
(\).)4972 5032 w
10 B f
(Algorithm 2.)1 544 1 720 5268 t
10 I f
(Minimum-cost minimum flow)2 1172 1 1289 5268 t
10 R f
(.)2461 5268 w
( preflows on)2 519( Algorithm 1 to determine the existence of balancing)8 2180(1. Use)1 405 3 720 5424 t
10 I f
(G)3859 5424 w
10 R f
( there exist such flows,)4 958(. If)1 151 2 3931 5424 t
(compute the value)2 732 1 970 5544 t
10 S f
(m)1727 5544 w
10 R f
(of the minimum-balancing preflows, and)4 1634 1 1810 5544 t
10 S f
(n)3469 5544 w
10 R f
(as well.)1 305 1 3546 5544 t
(2. Construct)1 639 1 720 5705 t
10 I f
(G)1384 5705 w
10 S1 f
(_ _)1 62 1 1394 5612 t
10 R f
(, the balancing graph for)4 978 1 1456 5705 t
10 I f
(G)2459 5705 w
10 R f
(, setting)1 317 1 2531 5705 t
10 I f
(u)2873 5705 w
10 S1 f
(_)2875 5637 w
10 R f
(\()2931 5705 w
10 I f
(s)2972 5705 w
10 S f
(\242)3019 5705 w
10 R f
(,)3052 5705 w
10 I f
(s)3118 5705 w
10 R f
(\))3165 5705 w
10 S f
(= m)1 162 1 3255 5705 t
10 R f
(and)3442 5705 w
10 I f
(u)3611 5705 w
10 S1 f
(_)3613 5637 w
10 R f
(\()3669 5705 w
10 I f
(t)3710 5705 w
10 R f
(,)3746 5705 w
10 I f
(t)3812 5705 w
10 S f
(\242)3848 5705 w
10 R f
(\))3881 5705 w
10 S f
(= n)1 156 1 3971 5705 t
10 R f
(.)4127 5705 w
( a minimum-cost maximum flow)4 1317(3. Find)1 434 2 720 5866 t
10 I f
(F)2496 5866 w
10 R f
(on)2582 5866 w
10 I f
(G)2707 5866 w
10 S1 f
(_ _)1 62 1 2717 5773 t
10 R f
(.)2779 5866 w
(4. From)1 467 1 720 6022 t
10 I f
(F)1212 6022 w
10 R f
(construct the induced balancing preflow)4 1602 1 1298 6022 t
10 I f
(f)2925 6022 w
7 I f
(b)2964 6042 w
10 R f
(on)3032 6022 w
10 I f
(G)3157 6022 w
10 R f
(.)3229 6022 w
(5. Let)1 383 1 720 6178 t
10 I f
(f)1128 6178 w
10 S f
(=)1221 6178 w
10 I f
(f)1333 6178 w
7 I f
(b)1372 6198 w
10 S f
(+)1464 6178 w
10 I f
(f)1576 6178 w
7 R f
(0)1615 6198 w
10 R f
(, where)1 293 1 1658 6178 t
10 I f
(f)1976 6178 w
7 R f
(0)2015 6198 w
10 R f
(is the basic preflow;)3 807 1 2083 6178 t
10 I f
(f)2915 6178 w
10 R f
(is a minimum-cost minimum flow for)5 1509 1 2968 6178 t
10 I f
(G)4502 6178 w
10 R f
(.)4574 6178 w
10 S1 f
()4649 6178 w
4649 6178 m
50 build_sq
4699 6178 m
10 R f
(All steps except 3 take total time)6 1334 1 970 6334 t
10 I f
(O)2333 6334 w
10 R f
(\()2413 6334 w
10 I f
(mn)2454 6334 w
10 R f
(log)2617 6334 w
10 I f
(n)2786 6334 w
10 R f
( costs)1 230( algorithms for Step 3 yield different)6 1489(\). Different)1 477 3 2844 6334 t
( that single)2 475( Recall)1 323( shall use algorithms for finding the shortest paths frequently.)9 2621( We)1 206( 2.)1 118(for Algorithm)1 577 6 720 6454 t
(source shortest paths can be obtained in time)7 1950 1 720 6574 t
10 I f
(O)2718 6574 w
10 R f
(\()2798 6574 w
10 I f
(n)2839 6574 w
7 R f
(2)2900 6534 w
10 R f
(\),)2951 6574 w
10 I f
(O)3057 6574 w
10 R f
(\()3137 6574 w
10 I f
(m)3178 6574 w
10 R f
(log)3291 6574 w
7 I f
(m / n)2 115 1 3430 6594 t
7 S f
(+)3561 6594 w
7 R f
(2)3611 6594 w
10 I f
(n)3695 6574 w
10 R f
(\), and)1 250 1 3753 6574 t
10 I f
(O)4052 6574 w
10 R f
(\()4132 6574 w
10 I f
(m)4173 6574 w
10 S f
(+)4294 6574 w
10 I f
(n)4398 6574 w
10 R f
(log)4489 6574 w
10 I f
(n)4658 6574 w
10 R f
(\), using)1 324 1 4716 6574 t
( 1983], implicit heaps [Johnson, 1977; Tar-)6 1744(Dijkstra's algorithm [Aho, Hopcroft, and Ullman, 1974; Tarjan,)7 2576 2 720 6694 t
( first two algorithms are rel-)5 1128( The)1 206(jan, 1983], and Fibonacci heaps [Fredman and Tarjan, 1984], respectively.)9 2986 3 720 6814 t
( time bounds reported)3 898( The)1 213( the last one yields the best-known time bound.)8 1949(atively easy to implement, and)4 1260 4 720 6934 t
(here are from using Fibonacci heaps, and practitioners might want to choose the other two alternatives.)15 4112 1 720 7054 t
( minimum-cost maximum flow in Step 3, since)7 1947(If we use minimum-cost augmentation for finding a)7 2123 2 970 7210 t
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7 pagesetup
10 R f
(- 7 -)2 166 1 2797 480 t
(the flow value is bounded by)5 1262 1 720 840 t
10 I f
(L)2028 840 w
10 R f
( takes time)2 473(and each augmentation)2 962 2 2130 840 t
10 I f
(O)3610 840 w
10 R f
(\()3690 840 w
10 I f
(m)3731 840 w
10 S f
(+)3852 840 w
10 I f
(n)3956 840 w
10 R f
(log)4047 840 w
10 I f
(n)4216 840 w
10 R f
(\), the total cost is)4 766 1 4274 840 t
10 I f
(O)720 960 w
10 R f
(\( \()1 74 1 800 960 t
10 I f
(m)882 960 w
10 S f
(+)1003 960 w
10 I f
(n)1107 960 w
10 R f
(log)1198 960 w
10 I f
(n)1367 960 w
10 R f
(\))1425 960 w
10 I f
(L)1474 960 w
10 R f
(\).)1538 960 w
(Since the flow value is bounded by)6 1440 1 970 1116 t
10 I f
(L)2441 1116 w
10 R f
(, we can set the capacity upper bounds to)8 1686 1 2497 1116 t
10 I f
(L)4214 1116 w
10 R f
(for the edges with)3 739 1 4301 1116 t
(capacity upper bounds exceeding)3 1378 1 720 1236 t
10 I f
(L)2140 1236 w
10 R f
( A)1 138( 1972].)1 299(, and then use scaling on capacities [Edmonds and Karp,)9 2407 3 2196 1236 t
(minimum-cost maximum flow can be found in time)7 2069 1 720 1356 t
10 I f
(O)2814 1356 w
10 R f
(\()2894 1356 w
10 I f
(m)2935 1356 w
10 R f
(\()3015 1356 w
10 I f
(m)3056 1356 w
10 S f
(+)3177 1356 w
10 I f
(n)3281 1356 w
10 R f
(log)3372 1356 w
10 I f
(n)3541 1356 w
10 R f
(\) log)1 210 1 3599 1356 t
10 I f
(L)3850 1356 w
10 R f
(\).)3914 1356 w
10 B f
(Theorem 3.4.)1 667 1 720 1512 t
10 R f
( minimum-cost minimum flow)3 1547(Algorithm 2 constructs a)3 1303 2 1516 1512 t
10 I f
(f)4496 1512 w
10 R f
(in time)1 386 1 4654 1512 t
10 I f
(O)720 1632 w
10 R f
(\()800 1632 w
10 I f
(mn)841 1632 w
10 R f
(log)1004 1632 w
10 I f
(n)1173 1632 w
10 S f
(+)1272 1632 w
10 R f
(\()1376 1632 w
10 I f
(m)1417 1632 w
10 S f
(+)1538 1632 w
10 I f
(n)1642 1632 w
10 R f
(log)1733 1632 w
10 I f
(n)1902 1632 w
10 R f
(\))1960 1632 w
10 I f
(L)2009 1632 w
10 R f
( in Step 3, and in time)6 1145(\) if minimum-cost augmentation is used)5 1822 2 2073 1632 t
10 I f
(O)720 1752 w
10 R f
(\()800 1752 w
10 I f
(m)841 1752 w
10 R f
(\()921 1752 w
10 I f
(m)962 1752 w
10 S f
(+)1083 1752 w
10 I f
(n)1187 1752 w
10 R f
(log)1278 1752 w
10 I f
(n)1447 1752 w
10 R f
(\) log)1 210 1 1505 1752 t
10 I f
(L)1756 1752 w
10 R f
( if scaling is used, where)5 987(\) \))1 74 2 1820 1752 t
10 I f
(L)2906 1752 w
10 S f
(=)3011 1752 w
7 I f
(e)3115 1852 w
7 S f
(\316)3174 1852 w
7 I f
(E)3252 1852 w
15 S f
(S)3160 1782 w
10 I f
(l)3294 1752 w
10 R f
(\()3330 1752 w
10 I f
(e)3371 1752 w
10 R f
(\).)3423 1752 w
10 B f
(Proof.)720 1988 w
10 R f
(The correctness of the algorithm follows from Lemma 3.3 and Theorem 3.2.)11 3050 1 1008 1988 t
10 S1 f
()4108 1988 w
4108 1988 m
50 build_sq
4158 1988 m
10 B f
( flow)1 208(3.5. Minimum-cost)1 827 2 720 2228 t
10 R f
(We first reduce the minimum-cost balancing preflow problem on)8 2673 1 970 2384 t
10 I f
(G)3677 2384 w
10 R f
(to a problem on an augmented)5 1257 1 3783 2384 t
( balancing)1 426(graph, and then further reduce this problem to a minimum-cost maximum-flow problem on a)13 3894 2 720 2504 t
( we reduce the problem to a)6 1156(graph. Essentially,)1 774 2 720 2624 t
10 I f
(circulation)2683 2624 w
10 R f
( of practical)2 494( Because)1 391(problem \(see Section 6\).)3 1000 3 3155 2624 t
( identify)1 345( We)1 197( programming and circuit testing, we prefer to present solutions in this order.)12 3182(applications in)1 596 4 720 2744 t
( practical)1 368( For)1 189(the problems in the reduction only for the purpose of making the reduction conceptually clear.)14 3763 3 720 2864 t
( and then)2 378(purposes, one can construct the balancing graphs by a simple modification of the original graph,)14 3942 2 720 2984 t
(solve the final maximum-flow problems directly.)5 1965 1 720 3104 t
( graph)1 263(To construct the augmented)3 1142 2 970 3260 t
10 I f
(G *)1 130 1 2411 3260 t
10 R f
(from)2577 3260 w
10 I f
(G)2807 3260 w
10 R f
(, we add an edge \()5 780 1 2879 3260 t
10 I f
(t)3667 3260 w
10 R f
(,)3703 3260 w
10 I f
(s)3769 3260 w
10 R f
(\) with cost zero, and capacity)5 1224 1 3816 3260 t
(bounds)720 3380 w
10 I f
(l)1041 3380 w
10 R f
(\()1077 3380 w
10 I f
(t)1118 3380 w
10 R f
(,)1154 3380 w
10 I f
(s)1220 3380 w
10 R f
(\))1267 3380 w
10 S f
(=)1357 3380 w
10 R f
(0,)1461 3380 w
10 I f
(u)1568 3380 w
10 R f
(\()1626 3380 w
10 I f
(t)1667 3380 w
10 R f
(,)1703 3380 w
10 I f
(s)1769 3380 w
10 R f
(\))1816 3380 w
10 S f
(= \245)1 177 1 1906 3380 t
10 R f
( a balancing preflow)3 838(. Given)1 326 2 2083 3380 t
10 I f
(f)3279 3380 w
7 I f
(b)3318 3400 w
10 R f
(on)3392 3380 w
10 I f
(G)3523 3380 w
10 R f
(, we define a preflow)4 868 1 3595 3380 t
10 I f
(f)4494 3380 w
7 I f
(b)4533 3400 w
10 I f
(*)4584 3380 w
10 R f
(on)4665 3380 w
10 I f
(G *)1 130 1 4796 3380 t
10 R f
(as)4957 3380 w
(follows. Let)1 514 1 720 3500 t
10 I f
(f)1265 3500 w
7 I f
(b)1304 3520 w
10 I f
(*)1355 3500 w
10 R f
(\()1413 3500 w
10 I f
(e)1454 3500 w
10 R f
(\) =)1 120 1 1506 3500 t
10 I f
(f)1657 3500 w
7 I f
(b)1696 3520 w
10 R f
(\()1747 3500 w
10 I f
(e)1788 3500 w
10 R f
(\) for)1 180 1 1840 3500 t
10 I f
(e)2051 3500 w
10 S f
(\316)2136 3500 w
10 I f
(E)2248 3500 w
10 R f
( let)1 132(, and)1 200 2 2309 3500 t
10 I f
(f)2673 3500 w
7 I f
(b)2712 3520 w
10 I f
(*)2763 3500 w
10 R f
(\()2821 3500 w
10 I f
(t)2862 3500 w
10 R f
(,)2898 3500 w
10 I f
(s)2964 3500 w
10 R f
(\) =)1 121 1 3011 3500 t
10 S f
(b)3164 3500 w
10 R f
(\()3227 3500 w
10 I f
(s)3268 3500 w
10 R f
(,)3315 3500 w
10 I f
(f)3389 3500 w
7 R f
(0)3428 3520 w
10 R f
(\) +)1 121 1 3479 3500 t
10 S f
(b)3632 3500 w
10 R f
(\()3695 3500 w
10 I f
(s)3736 3500 w
10 R f
(,)3783 3500 w
10 I f
(f)3857 3500 w
7 I f
(b)3896 3520 w
10 R f
(\). Let)1 248 1 3947 3500 t
10 S f
(b)4227 3500 w
10 I f
(*)4290 3500 w
10 R f
(be the balancing)2 668 1 4372 3500 t
(index of vertices in)3 783 1 720 3620 t
10 I f
(G *)1 130 1 1533 3620 t
10 R f
( the balancing index of)4 935(. Obviously,)1 522 2 1663 3620 t
10 I f
(f)3150 3620 w
7 R f
(0)3189 3640 w
10 R f
( vertices in)2 446(for all)1 246 2 3262 3620 t
10 I f
(G *)1 130 1 3983 3620 t
10 R f
(remains the same as it)4 898 1 4142 3620 t
(was in)1 270 1 720 3740 t
10 I f
(G)1027 3740 w
10 R f
(, and)1 206 1 1099 3740 t
10 S f
(b)1343 3740 w
10 I f
(*)1406 3740 w
10 R f
(\()1464 3740 w
10 I f
(v)1505 3740 w
10 R f
(,)1557 3740 w
10 I f
(f)1631 3740 w
7 I f
(b)1670 3760 w
10 I f
(*)1721 3740 w
10 R f
(\) =)1 127 1 1779 3740 t
10 S f
(b)1944 3740 w
10 R f
(\()2007 3740 w
10 I f
(v)2048 3740 w
10 R f
(,)2100 3740 w
10 I f
(f)2174 3740 w
7 I f
(b)2213 3760 w
10 R f
(\) for)1 187 1 2264 3740 t
10 I f
(v)2489 3740 w
10 S f
(\316)2574 3740 w
10 I f
(S)2686 3740 w
10 S f
(\310)2777 3740 w
10 I f
(D)2895 3740 w
10 R f
( the other hand, since)4 903(. On)1 210 2 2967 3740 t
10 S f
(b)4118 3740 w
10 R f
(\()4181 3740 w
10 I f
(s)4222 3740 w
10 R f
(,)4269 3740 w
10 I f
(f)4343 3740 w
7 I f
(b)4382 3760 w
10 R f
(\) +)1 127 1 4433 3740 t
10 S f
(b)4598 3740 w
10 R f
(\()4661 3740 w
10 I f
(s)4702 3740 w
10 R f
(,)4749 3740 w
10 I f
(f)4823 3740 w
7 R f
(0)4862 3760 w
10 R f
(\) =)1 127 1 4913 3740 t
10 S f
(-)720 3860 w
10 R f
([)791 3860 w
10 S f
(b)832 3860 w
10 R f
(\()895 3860 w
10 I f
(t)936 3860 w
10 R f
(,)972 3860 w
10 I f
(f)1046 3860 w
7 I f
(b)1085 3880 w
10 R f
(\) +)1 120 1 1136 3860 t
10 S f
(b)1287 3860 w
10 R f
(\()1350 3860 w
10 I f
(t)1391 3860 w
10 R f
(,)1427 3860 w
10 I f
(f)1501 3860 w
7 R f
(0)1540 3880 w
10 R f
( we have)2 366(\) ],)1 99 2 1591 3860 t
10 S f
(b)2087 3860 w
10 I f
(*)2150 3860 w
10 R f
(\()2208 3860 w
10 I f
(t)2249 3860 w
10 R f
(,)2285 3860 w
10 I f
(f)2359 3860 w
7 I f
(b)2398 3880 w
10 I f
(*)2449 3860 w
10 R f
(\) =)1 120 1 2507 3860 t
10 S f
(b)2658 3860 w
10 R f
(\()2721 3860 w
10 I f
(t)2762 3860 w
10 R f
(,)2798 3860 w
10 I f
(f)2872 3860 w
7 I f
(b)2911 3880 w
10 R f
(\) +)1 120 1 2962 3860 t
10 I f
(f)3113 3860 w
7 I f
(b)3152 3880 w
10 I f
(*)3203 3860 w
10 R f
(\()3261 3860 w
10 I f
(t)3302 3860 w
10 R f
(,)3338 3860 w
10 I f
(s)3404 3860 w
10 R f
(\) =)1 120 1 3451 3860 t
10 S f
(b)3602 3860 w
10 R f
(\()3665 3860 w
10 I f
(t)3706 3860 w
10 R f
(,)3742 3860 w
10 I f
(f)3816 3860 w
7 I f
(b)3855 3880 w
10 R f
(\) + [)2 184 1 3906 3860 t
10 S f
(b)4098 3860 w
10 R f
(\()4161 3860 w
10 I f
(s)4202 3860 w
10 R f
(,)4249 3860 w
10 I f
(f)4323 3860 w
7 R f
(0)4362 3880 w
10 R f
(\) +)1 120 1 4413 3860 t
10 S f
(b)4564 3860 w
10 R f
(\()4627 3860 w
10 I f
(s)4668 3860 w
10 R f
(,)4715 3860 w
10 I f
(f)4789 3860 w
7 I f
(b)4828 3880 w
10 R f
( =)1 87(\) ])1 74 2 4879 3860 t
10 S f
(- b)1 126 1 720 3980 t
10 R f
(\()854 3980 w
10 I f
(t)895 3980 w
10 R f
(,)931 3980 w
10 I f
(f)1005 3980 w
7 R f
(0)1044 4000 w
10 R f
(\), and)1 248 1 1095 3980 t
10 S f
(b)1389 3980 w
10 I f
(*)1452 3980 w
10 R f
(\()1510 3980 w
10 I f
(s)1551 3980 w
10 R f
(,)1598 3980 w
10 I f
(f)1672 3980 w
7 I f
(b)1711 4000 w
10 I f
(*)1762 3980 w
10 R f
(\) =)1 136 1 1820 3980 t
10 S f
(b)2003 3980 w
10 R f
(\()2066 3980 w
10 I f
(s)2107 3980 w
10 R f
(,)2154 3980 w
10 I f
(f)2228 3980 w
7 I f
(b)2267 4000 w
10 R f
(\))2318 3980 w
10 S f
(-)2398 3980 w
10 I f
(f)2500 3980 w
7 I f
(b)2539 4000 w
10 I f
(*)2590 3980 w
10 R f
(\()2648 3980 w
10 I f
(t)2689 3980 w
10 R f
(,)2725 3980 w
10 I f
(s)2791 3980 w
10 R f
(\) =)1 136 1 2838 3980 t
10 S f
(b)3021 3980 w
10 R f
(\()3084 3980 w
10 I f
(s)3125 3980 w
10 R f
(,)3172 3980 w
10 I f
(f)3246 3980 w
7 I f
(b)3285 4000 w
10 R f
(\))3336 3980 w
10 S f
(-)3416 3980 w
10 R f
([)3518 3980 w
10 S f
(b)3559 3980 w
10 R f
(\()3622 3980 w
10 I f
(s)3663 3980 w
10 R f
(,)3710 3980 w
10 I f
(f)3784 3980 w
7 R f
(0)3823 4000 w
10 R f
(\) +)1 136 1 3874 3980 t
10 S f
(b)4057 3980 w
10 R f
(\()4120 3980 w
10 I f
(s)4161 3980 w
10 R f
(,)4208 3980 w
10 I f
(f)4282 3980 w
7 I f
(b)4321 4000 w
10 R f
( =)1 103(\) ])1 74 2 4372 3980 t
10 S f
(- b)1 126 1 4596 3980 t
10 R f
(\()4730 3980 w
10 I f
(s)4771 3980 w
10 R f
(,)4818 3980 w
10 I f
(f)4892 3980 w
7 R f
(0)4931 4000 w
10 R f
(\).)4982 3980 w
(Therefore, if)1 508 1 720 4100 t
10 I f
(f)1258 4100 w
7 I f
(b)1297 4120 w
10 R f
(is a balancing preflow on)4 1029 1 1370 4100 t
10 I f
(G)2429 4100 w
10 R f
(then)2531 4100 w
10 I f
(f)2733 4100 w
7 I f
(b)2772 4120 w
10 I f
(*)2823 4100 w
10 R f
( preflow on)2 468(is a balancing)2 559 2 2903 4100 t
10 I f
(G *)1 130 1 3959 4100 t
10 R f
(, where)1 297 1 4089 4100 t
10 I f
(s)4415 4100 w
10 R f
(and)4483 4100 w
10 I f
(t)4656 4100 w
10 R f
(are con-)1 327 1 4713 4100 t
(sidered as intermediate vertices, and the restriction of)7 2148 1 720 4220 t
10 I f
(f)2895 4220 w
7 I f
(b)2934 4240 w
10 I f
(*)2985 4220 w
10 R f
(to)3062 4220 w
10 I f
(G)3167 4220 w
10 R f
(is)3267 4220 w
10 I f
(f)3362 4220 w
7 I f
(b)3401 4240 w
10 R f
( the other hand, the restriction of a)7 1396(. On)1 200 2 3444 4220 t
(balancing preflow on)2 858 1 720 4340 t
10 I f
(G *)1 130 1 1608 4340 t
10 R f
(to)1768 4340 w
10 I f
(G)1876 4340 w
10 R f
( Since)1 276( preflow.)1 364(is a balancing)2 559 3 1978 4340 t
10 I f
(cost)3206 4340 w
10 R f
(\()3375 4340 w
10 I f
(f)3432 4340 w
7 I f
(b)3471 4360 w
10 I f
(*)3522 4340 w
10 R f
(\) =)1 118 1 3580 4340 t
10 I f
(cost)3727 4340 w
10 R f
(\()3896 4340 w
10 I f
(f)3953 4340 w
7 I f
(b)3992 4360 w
10 R f
(\), we only have to find a)6 997 1 4043 4340 t
(minimum-cost balancing preflow on)3 1457 1 720 4460 t
10 I f
(G *)1 130 1 2202 4460 t
10 R f
(.)2332 4460 w
(Note that)1 369 1 970 4616 t
10 I f
(t)1364 4616 w
10 R f
(is a surplus vertex and)4 893 1 1417 4616 t
10 I f
(s)2336 4616 w
10 R f
(is a deficient vertex in)4 891 1 2401 4616 t
10 I f
(G *)1 130 1 3318 4616 t
10 R f
( construct the balancing graph)4 1207(now. To)1 359 2 3474 4616 t
10 I f
(G)720 4741 w
10 S1 f
(_ _)1 62 1 730 4648 t
10 I f
(*)800 4741 w
10 R f
(, we add a new source vertex)6 1184 1 850 4741 t
10 I f
(s)2064 4741 w
10 S f
(\242)2111 4741 w
10 R f
(and a new sink vertex)4 890 1 2166 4741 t
10 I f
(t)3086 4741 w
10 S f
(\242)3122 4741 w
10 R f
(to)3177 4741 w
10 I f
(G *)1 130 1 3285 4741 t
10 R f
( are also added to)4 714( following edges)2 676(. The)1 235 3 3415 4741 t
10 I f
(G *)1 130 1 720 4861 t
10 R f
(:)850 4861 w
( each)1 207(i\) For)1 389 2 720 5022 t
10 I f
(v)1341 5022 w
10 R f
(in)1410 5022 w
10 I f
(S)1513 5022 w
10 S f
(\310)1604 5022 w
10 I f
({ t })2 124 1 1722 5022 t
10 R f
(, add the edge \()4 612 1 1846 5022 t
10 I f
(s)2466 5022 w
10 S f
(\242)2513 5022 w
10 R f
(,)2546 5022 w
10 I f
(v)2612 5022 w
10 R f
(\) with)1 236 1 2664 5022 t
10 I f
(u)2925 5022 w
10 S1 f
(_)2927 4954 w
10 R f
(\()2983 5022 w
10 I f
(s)3024 5022 w
10 S f
(\242)3071 5022 w
10 R f
(,)3104 5022 w
10 I f
(v)3170 5022 w
10 R f
(\) =)1 114 1 3222 5022 t
10 S f
(- b)1 126 1 3361 5022 t
10 R f
(\()3495 5022 w
10 I f
(v)3536 5022 w
10 R f
(,)3588 5022 w
10 I f
(f)3662 5022 w
7 R f
(0)3701 5042 w
10 R f
(\).)3752 5022 w
( each)1 207(ii\) For)1 389 2 720 5183 t
10 I f
(w)1341 5183 w
10 R f
(in)1433 5183 w
10 I f
(D)1536 5183 w
10 S f
(\310)1649 5183 w
10 I f
({ s })2 135 1 1767 5183 t
10 R f
(, add the edge \()4 612 1 1902 5183 t
10 I f
(w)2522 5183 w
10 R f
(,)2597 5183 w
10 I f
(t)2663 5183 w
10 S f
(\242)2699 5183 w
10 R f
(\) with)1 236 1 2732 5183 t
10 I f
(u)2993 5183 w
10 S1 f
(_)2995 5115 w
10 R f
(\()3051 5183 w
10 I f
(w)3092 5183 w
10 R f
(,)3167 5183 w
10 I f
(t)3233 5183 w
10 S f
(\242)3269 5183 w
10 R f
(\) =)1 114 1 3302 5183 t
10 S f
(b)3441 5183 w
10 R f
(\()3504 5183 w
10 I f
(w)3545 5183 w
10 R f
(,)3620 5183 w
10 I f
(f)3694 5183 w
7 R f
(0)3733 5203 w
10 R f
(\).)3784 5183 w
( each edge in)3 550(The cost of)2 465 2 720 5339 t
10 I f
(G)1769 5339 w
10 R f
( capacity)1 366( The)1 214(remains what is was, and the cost of each added edges is zero.)12 2585 3 1875 5339 t
(lower bound of each edge in)5 1208 1 720 5464 t
10 I f
(G)1968 5464 w
10 S1 f
(_ _)1 62 1 1978 5371 t
10 I f
(*)2048 5464 w
10 R f
( each edge)2 448(is zero, and for)3 643 2 2138 5464 t
10 I f
(e)3268 5464 w
10 R f
(in)3351 5464 w
10 I f
(G)3468 5464 w
10 R f
(, the capacity upper bound becomes)5 1500 1 3540 5464 t
10 I f
(u)720 5589 w
10 S1 f
(_)722 5521 w
10 R f
(\()778 5589 w
10 I f
(e)819 5589 w
10 R f
(\))871 5589 w
10 S f
(=)961 5589 w
10 I f
(u)1065 5589 w
10 R f
(\()1123 5589 w
10 I f
(e)1164 5589 w
10 R f
(\))1216 5589 w
10 S f
(-)1306 5589 w
10 I f
(l)1410 5589 w
10 R f
(\()1446 5589 w
10 I f
(e)1487 5589 w
10 R f
(\). Obviously,)1 550 1 1539 5589 t
10 I f
(G)2114 5589 w
10 S1 f
(_ _)1 62 1 2124 5496 t
10 I f
(*)2194 5589 w
10 R f
(has)2269 5589 w
10 I f
(O)2427 5589 w
10 R f
(\()2507 5589 w
10 I f
(m)2548 5589 w
10 R f
(\) edges and)2 454 1 2628 5589 t
10 I f
(O)3107 5589 w
10 R f
(\()3187 5589 w
10 I f
(n)3228 5589 w
10 R f
(\) vertices.)1 393 1 3286 5589 t
(Given a flow)2 525 1 970 5750 t
10 I f
(F)1522 5750 w
10 R f
(on)1610 5750 w
10 I f
(G)1737 5750 w
10 S1 f
(_ _)1 62 1 1747 5657 t
10 I f
(*)1817 5750 w
10 R f
(, the induced preflow)3 854 1 1867 5750 t
10 I f
(f)2748 5750 w
7 I f
(b)2787 5770 w
10 I f
(*)2838 5750 w
10 R f
( only if all the edges incident to)7 1284(is balancing if and)3 741 2 2915 5750 t
10 I f
(s)4968 5750 w
10 S f
(\242)5015 5750 w
10 R f
(or)720 5870 w
10 I f
(t)828 5870 w
10 S f
(\242)864 5870 w
10 R f
(are saturated, and in this case,)5 1194 1 914 5870 t
10 I f
(F)2133 5870 w
10 R f
( Thus)1 250(is a maximum flow.)3 800 2 2219 5870 t
10 B f
(Lemma 3.4.)1 508 1 720 6026 t
10 R f
(There exists a balancing preflow on)5 1457 1 1259 6026 t
10 I f
(G *)1 130 1 2748 6026 t
10 R f
(if and only if there exists a balancing preflow on)9 2001 1 2910 6026 t
10 I f
(G)4943 6026 w
10 R f
(.)5015 6026 w
(In this case,)2 484 1 720 6151 t
10 I f
(F)1234 6151 w
10 R f
(on)1325 6151 w
10 I f
(G)1455 6151 w
10 S1 f
(_ _)1 62 1 1465 6058 t
10 I f
(*)1535 6151 w
10 R f
( if and only if the induced preflow)7 1395(is a minimum-cost maximum flow)4 1404 2 1615 6151 t
10 I f
(f)4443 6151 w
7 I f
(b)4482 6171 w
10 I f
(*)4533 6151 w
10 R f
(on)4612 6151 w
10 I f
(G *)1 130 1 4741 6151 t
10 R f
(is a)1 140 1 4900 6151 t
( this is the case if and only if)8 1261(minimum-cost balancing preflow, and)3 1565 2 720 6271 t
10 I f
(f)3585 6271 w
7 I f
(b)3624 6291 w
10 R f
(, the restriction of)3 752 1 3667 6271 t
10 I f
(f)4458 6271 w
7 I f
(b)4497 6291 w
10 I f
(*)4548 6271 w
10 R f
(to)4637 6271 w
10 I f
(G)4754 6271 w
10 R f
(, is a)2 214 1 4826 6271 t
(minimum-cost balancing preflow on)3 1457 1 720 6391 t
10 I f
(G)2202 6391 w
10 R f
(.)2274 6391 w
10 S1 f
()720 6547 w
720 6547 m
50 build_sq
770 6547 m
10 B f
(Algorithm 3.)1 544 1 720 6703 t
10 I f
(Minimum-cost flow)1 775 1 1289 6703 t
10 R f
(.)2064 6703 w
(1. From)1 467 1 720 6859 t
10 I f
(G)1212 6859 w
10 R f
(construct the augmented graph)3 1228 1 1309 6859 t
10 I f
(G *)1 130 1 2562 6859 t
10 R f
(.)2692 6859 w
(2. From)1 467 1 720 7020 t
10 I f
(G *)1 130 1 1212 7020 t
10 R f
(construct the balancing graph)3 1178 1 1367 7020 t
10 I f
(G)2570 7020 w
10 S1 f
(_ _)1 62 1 2580 6927 t
10 I f
(*)2650 7020 w
10 R f
(.)2700 7020 w
( a minimum-cost maximum flow)4 1321(3. Find)1 434 2 720 7181 t
10 I f
(F)2501 7181 w
10 R f
(on)2588 7181 w
10 I f
(G)2714 7181 w
10 S1 f
(_ _)1 62 1 2724 7088 t
10 I f
(*)2794 7181 w
10 R f
( not all the edges incident to the source)8 1567(. If)1 142 2 2844 7181 t
10 I f
(s)4579 7181 w
10 S f
(\242)4626 7181 w
10 R f
(or sink)1 276 1 4677 7181 t
10 I f
(t)4979 7181 w
10 S f
(\242)5015 7181 w
10 R f
(are saturated, then abort. There is no feasible flow on)9 2125 1 970 7301 t
10 I f
(G)3120 7301 w
10 R f
(.)3192 7301 w
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8 pagesetup
10 R f
(- 8 -)2 166 1 2797 480 t
( the induced balancing preflow)4 1236(4. Compute)1 617 2 720 840 t
10 I f
(f)2598 840 w
7 I f
(b)2637 860 w
10 I f
(*)2688 840 w
10 R f
(from)2763 840 w
10 I f
(F)2982 840 w
10 R f
(on)3068 840 w
10 I f
(G *)1 130 1 3193 840 t
10 R f
(.)3323 840 w
(5. Let)1 383 1 720 996 t
10 I f
(f)1128 996 w
7 I f
(b)1167 1016 w
10 R f
(be the restriction of)3 779 1 1235 996 t
10 I f
(f)2039 996 w
7 I f
(b)2078 1016 w
10 I f
(*)2129 996 w
10 R f
(to)2204 996 w
10 I f
(G)2307 996 w
10 R f
(.)2379 996 w
(6. Let)1 383 1 720 1152 t
10 I f
(f)1128 1152 w
10 S f
(=)1221 1152 w
10 I f
(f)1333 1152 w
7 I f
(b)1372 1172 w
10 S f
(+)1464 1152 w
10 I f
(f)1576 1152 w
7 R f
(0)1615 1172 w
10 R f
(, where)1 293 1 1658 1152 t
10 I f
(f)1976 1152 w
7 R f
(0)2015 1172 w
10 R f
(is the basic preflow;)3 807 1 2083 1152 t
10 I f
(f)2915 1152 w
10 R f
(is a minimum-cost flow for)4 1094 1 2968 1152 t
10 I f
(G)4087 1152 w
10 R f
(.)4159 1152 w
10 S1 f
()4234 1152 w
4234 1152 m
50 build_sq
4284 1152 m
10 R f
(We can now prove the following analog of Theorem 3.4:)9 2273 1 720 1308 t
10 B f
(Theorem 3.5.)1 568 1 720 1464 t
10 R f
( a minimum-cost flow)3 904(Algorithm 3 constructs)2 932 2 1318 1464 t
10 I f
(f)3185 1464 w
10 R f
(in time)1 287 1 3244 1464 t
10 I f
(O)3562 1464 w
10 R f
(\()3642 1464 w
10 I f
(mn)3683 1464 w
10 R f
(log)3846 1464 w
10 I f
(n)4015 1464 w
10 S f
(+)4114 1464 w
10 R f
(\()4218 1464 w
10 I f
(m)4259 1464 w
10 S f
(+)4380 1464 w
10 I f
(n)4484 1464 w
10 R f
(log)4575 1464 w
10 I f
(n)4744 1464 w
10 R f
(\))4802 1464 w
10 I f
(L)4851 1464 w
10 R f
(\) if)1 125 1 4915 1464 t
(minimum-cost augmentation is used in Step 3, and in time)9 2379 1 720 1584 t
10 I f
(O)3129 1584 w
10 R f
(\()3209 1584 w
10 I f
(m)3250 1584 w
10 R f
(\()3330 1584 w
10 I f
(m)3371 1584 w
10 S f
(+)3492 1584 w
10 I f
(n)3596 1584 w
10 R f
(log)3687 1584 w
10 I f
(n)3856 1584 w
10 R f
(\) log)1 210 1 3914 1584 t
10 I f
(L)4165 1584 w
10 R f
( is used,)2 333( if scaling)2 404(\) \))1 74 3 4229 1584 t
(where)720 1704 w
10 I f
(L)988 1704 w
10 S f
(=)1093 1704 w
7 I f
(e)1197 1804 w
7 S f
(\316)1256 1804 w
7 I f
(E)1334 1804 w
15 S f
(S)1242 1734 w
10 I f
(l)1376 1704 w
10 R f
(\()1412 1704 w
10 I f
(e)1453 1704 w
10 R f
(\).)1505 1704 w
10 S1 f
()1613 1704 w
1613 1704 m
50 build_sq
1663 1704 m
10 B f
( Flow and Path-Covering Problems on Graphs with Capacity Upper Bounds)10 3252(4. Positive)1 458 2 720 2024 t
10 R f
(A flow)1 307 1 970 2180 t
10 I f
(f)1329 2180 w
10 R f
(is)1409 2180 w
10 I f
(positive)1528 2180 w
10 R f
(if)1891 2180 w
10 I f
(l)2004 2180 w
10 R f
(\()2040 2180 w
10 I f
(e)2081 2180 w
10 R f
(\))2133 2180 w
10 S f
(=)2223 2180 w
10 R f
(1 for all edges)3 649 1 2327 2180 t
10 I f
(e)3029 2180 w
10 R f
( now specialize results obtained for the)6 1726(. We)1 241 2 3073 2180 t
(minimum-flow problems to the following)4 1667 1 720 2300 t
10 I f
( low)1 153( f)1 44(positive -)1 352 3 2412 2300 t
10 R f
(problems:)2986 2300 w
10 B f
(Problem)720 2516 w
10 I f
(F)1111 2516 w
7 R f
(1)1177 2535 w
(\( 1 \))2 91 1 1177 2476 t
10 R f
(.)1276 2516 w
10 I f
(Minimum positive flow)2 917 1 1351 2516 t
10 R f
(.)2268 2516 w
(Find a positive flow for)4 944 1 970 2672 t
10 I f
(G)1939 2672 w
10 R f
(with a minimum flow value.)4 1136 1 2036 2672 t
10 B f
(Problem)720 2888 w
10 I f
(F)1111 2888 w
7 R f
(2)1177 2907 w
(\( 1 \))2 91 1 1177 2848 t
10 R f
(.)1276 2888 w
10 I f
(Minimum-cost minimum positive flow)3 1508 1 1351 2888 t
10 R f
(.)2859 2888 w
(Among the minimum positive flows for)5 1592 1 970 3044 t
10 I f
(G)2587 3044 w
10 R f
(, find one with minimum cost.)5 1209 1 2659 3044 t
10 B f
(Problem)720 3260 w
10 I f
(F)1111 3260 w
7 R f
(3)1177 3279 w
(\( 1 \))2 91 1 1177 3220 t
10 R f
(.)1276 3260 w
10 I f
(Minimum-cost positive flow)2 1111 1 1351 3260 t
10 R f
(.)2462 3260 w
(Find a positive flow for)4 944 1 970 3416 t
10 I f
(G)1939 3416 w
10 R f
(with minimum cost.)2 804 1 2036 3416 t
(Given a set of covering)4 931 1 970 3692 t
10 I f
(s)1926 3692 w
10 S f
(-)1989 3692 w
10 I f
(t)2060 3692 w
10 R f
(paths)2113 3692 w
10 I f
(P)2349 3692 w
10 R f
(, let)1 150 1 2410 3692 t
10 I f
(f)2585 3692 w
10 R f
(\()2629 3692 w
10 I f
(e)2670 3692 w
10 R f
(\) be the number of times)5 979 1 2722 3692 t
10 I f
(e)3726 3692 w
10 R f
(appears in)1 407 1 3795 3692 t
10 I f
(P)4228 3692 w
10 R f
(\(if)4315 3692 w
10 I f
(e)4435 3692 w
10 R f
(appears more)1 535 1 4505 3692 t
(than once on the same path, each occurrence of)8 1965 1 720 3812 t
10 I f
(e)2720 3812 w
10 R f
( Then)1 264(is counted\).)1 475 2 2799 3812 t
10 I f
(f)3572 3812 w
10 R f
(is a positive flow on)4 847 1 3634 3812 t
10 I f
(G)4515 3812 w
10 R f
(, called the)2 453 1 4587 3812 t
10 I f
(derived)720 3932 w
10 R f
(flow from)1 402 1 1044 3932 t
10 I f
(P)1471 3932 w
10 R f
( the cardinality of)3 707(. Obviously,)1 517 2 1532 3932 t
10 I f
(P)2781 3932 w
10 R f
(is)2867 3932 w
10 I f
(value)2959 3932 w
10 R f
(\()3183 3932 w
10 I f
(f)3240 3932 w
10 R f
(\), and)1 227 1 3292 3932 t
10 I f
(cost)3544 3932 w
10 R f
(\()3713 3932 w
10 I f
(f)3770 3932 w
10 R f
(\))3822 3932 w
10 S f
(=)3912 3932 w
10 I f
(cost)4016 3932 w
10 R f
(\()4185 3932 w
10 I f
(P)4226 3932 w
10 R f
(\).)4295 3932 w
( a positive flow)3 634(Conversely, given)1 731 2 970 4088 t
10 I f
(f)2365 4088 w
10 R f
(on)2423 4088 w
10 I f
(G)2553 4088 w
10 R f
(, one can successively reduce the flow along)7 1808 1 2625 4088 t
10 I f
(s)4463 4088 w
10 S f
(-)4526 4088 w
10 I f
(t)4597 4088 w
10 R f
(paths and)1 385 1 4655 4088 t
(cycles, constructing)1 796 1 720 4208 t
10 I f
(s)1544 4208 w
10 S f
(-)1607 4208 w
10 I f
(t)1678 4208 w
10 R f
( When)1 291(paths and cycles.)2 685 2 1734 4208 t
10 I f
(f)2738 4208 w
10 S f
(\272)2823 4208 w
10 R f
(0, we have constructed a set of)6 1245 1 2919 4208 t
10 I f
(s)4192 4208 w
10 S f
(-)4255 4208 w
10 I f
(t)4326 4208 w
10 R f
(paths and cycles)2 658 1 4382 4208 t
10 I f
(P)720 4328 w
10 R f
(such that the derived flow from)5 1296 1 813 4328 t
10 I f
(P)2142 4328 w
10 R f
(is)2236 4328 w
10 I f
(f)2336 4328 w
10 R f
(. Since)1 305 1 2364 4328 t
10 I f
(f)2702 4328 w
10 R f
(\()2746 4328 w
10 I f
(e)2787 4328 w
10 R f
(\))2839 4328 w
10 S f
(\263)2921 4328 w
10 R f
(1 for all)2 332 1 3017 4328 t
10 I f
(e)3382 4328 w
10 R f
(in)3459 4328 w
10 I f
(E)3570 4328 w
10 R f
(, we can append the cycles to)6 1214 1 3631 4328 t
10 I f
(s)4878 4328 w
10 S f
(-)4941 4328 w
10 I f
(t)5012 4328 w
10 R f
(paths, such that there are only)5 1192 1 720 4448 t
10 I f
(s)1937 4448 w
10 S f
(-)2000 4448 w
10 I f
(t)2071 4448 w
10 R f
(paths in)1 314 1 2124 4448 t
10 I f
(P)2463 4448 w
10 R f
(.)2524 4448 w
( by)1 132(On the other hand, the flow value is bounded)8 1848 2 970 4604 t
10 I f
(L)2982 4604 w
10 S f
(=)3087 4604 w
7 I f
(e)3191 4704 w
7 S f
(\316)3250 4704 w
7 I f
(E)3328 4704 w
15 S f
(S)3236 4634 w
10 I f
(l)3370 4604 w
10 R f
(\()3406 4604 w
10 I f
(e)3447 4604 w
10 R f
(\) =)1 121 1 3499 4604 t
10 I f
(m)3652 4604 w
10 R f
(, and each flow augmentation in)5 1316 1 3724 4604 t
( more than)2 447(Algorithms 1-3 is taken along a path of no)8 1781 2 720 4804 t
10 I f
(n)2983 4804 w
10 R f
( the value of)3 526(edges. Therefore,)1 729 2 3068 4804 t
7 I f
(e)4358 4904 w
7 S f
(\316)4417 4904 w
7 I f
(E)4495 4904 w
15 S f
(S)4403 4834 w
10 I f
(F)4537 4804 w
10 R f
(\()4606 4804 w
10 I f
(e)4647 4804 w
10 R f
(\) for the)2 341 1 4699 4804 t
(constructed flow)1 674 1 720 5004 t
10 I f
(F)1425 5004 w
10 R f
(is)1517 5004 w
10 I f
(O)1615 5004 w
10 R f
(\()1695 5004 w
10 I f
(mn)1736 5004 w
10 R f
( corresponding balancing preflow)3 1365(\), and that for the)4 714 2 1866 5004 t
7 I f
(e)3977 5104 w
7 S f
(\316)4036 5104 w
7 I f
(E)4114 5104 w
15 S f
(S)4022 5034 w
10 I f
(f)4172 5004 w
7 I f
(b)4211 5024 w
10 R f
(\()4262 5004 w
10 I f
(e)4303 5004 w
10 R f
(\) is)1 132 1 4355 5004 t
10 I f
(O)4519 5004 w
10 R f
(\()4599 5004 w
10 I f
(mn)4640 5004 w
10 R f
(\). The)1 270 1 4770 5004 t
(flows)720 5204 w
10 I f
(f)970 5204 w
10 R f
(computed by Algorithm 1-3 have the property:)6 1888 1 1026 5204 t
7 I f
(e)2941 5304 w
7 S f
(\316)3000 5304 w
7 I f
(E)3078 5304 w
15 S f
(S)2986 5234 w
10 I f
(f)3136 5204 w
10 R f
(\()3180 5204 w
10 I f
(e)3221 5204 w
10 R f
(\) =)1 116 1 3273 5204 t
10 I f
(O)3416 5204 w
10 R f
(\()3496 5204 w
10 I f
(mn)3537 5204 w
10 R f
( from such a positive)4 846(\). Therefore,)1 527 2 3667 5204 t
(flow, the corresponding covering paths can be constructed in time)9 2634 1 720 5404 t
10 I f
(O)3379 5404 w
10 R f
(\()3459 5404 w
10 I f
(mn)3500 5404 w
10 R f
( summarize:)1 491(\). We)1 246 2 3630 5404 t
10 B f
(Theorem 4.1.)1 569 1 720 5560 t
10 R f
(The covering-paths problems)2 1182 1 1320 5560 t
10 I f
(CP)2533 5560 w
7 I f
(i)2666 5579 w
7 R f
(\( 1 \))2 91 1 2666 5520 t
10 R f
( prob-)1 248(are equivalent to the corresponding positive-flow)5 1996 2 2796 5560 t
(lems)720 5680 w
10 I f
(F)936 5680 w
7 I f
(i)1002 5699 w
7 R f
(\( 1 \))2 91 1 1002 5640 t
10 R f
(,)1101 5680 w
10 I f
(i)1153 5680 w
10 S f
(=)1230 5680 w
10 R f
( solution of a positive-flow problem can be transformed to)9 2342( optimal)1 333( An)1 174( respectively.)1 534( 3,)1 116( ,)1 33( 2)1 91(1 ,)1 83 8 1334 5680 t
(an optimal solution of the corresponding covering-paths problem, and vice versa, in time)12 3551 1 720 5800 t
10 I f
(O)4296 5800 w
10 R f
(\()4376 5800 w
10 I f
(mn)4417 5800 w
10 R f
(\).)4547 5800 w
10 S1 f
()4655 5800 w
4655 5800 m
50 build_sq
4705 5800 m
10 R f
(We can use Algorithms 1)4 1031 1 970 5956 t
10 S f
(-)2017 5956 w
10 R f
( the positive-flow problems)3 1117(3 for)1 195 2 2088 5956 t
10 I f
(F)3430 5956 w
7 I f
(i)3496 5975 w
7 R f
(\( 1 \))2 91 1 3496 5916 t
10 R f
(,)3595 5956 w
10 I f
(i)3650 5956 w
10 S f
(=)3727 5956 w
10 R f
( and then obtain solu-)4 886( 3,)1 116( ,)1 33( 2)1 91(1 ,)1 83 5 3831 5956 t
(tions of the covering-paths problems.)4 1490 1 720 6076 t
(Since the flow value on the balancing graph)7 1797 1 970 6237 t
10 I f
(G)2798 6237 w
10 S1 f
(_ _)1 62 1 2808 6144 t
10 R f
(is bounded by)2 573 1 2901 6237 t
10 I f
(m)3505 6237 w
10 R f
( the Sleator-Tarjan)2 762(, instead of using)3 701 2 3577 6237 t
( that it takes)3 516( can easily check)3 711( One)1 227(algorithm, we can use Dinic's algorithm for finding maximum flows.)9 2866 4 720 6357 t
(time)720 6477 w
10 I f
(O)923 6477 w
10 R f
(\()1003 6477 w
10 I f
(mn)1044 6477 w
10 R f
( summarize:)1 491( We)1 188(\) [Gabow, 1985].)2 687 3 1174 6477 t
10 B f
(Theorem 4.2.)1 566 1 720 6633 t
10 R f
(The positive-flow problem)2 1077 1 1314 6633 t
10 I f
(F)2419 6633 w
7 R f
(1)2485 6652 w
(\( 1 \))2 91 1 2485 6593 t
10 R f
(and the covering-paths problem)3 1276 1 2612 6633 t
10 I f
(CP)3916 6633 w
7 R f
(1)4049 6652 w
(\( 1 \))2 91 1 4049 6593 t
10 R f
( solved in time)3 604(can be)1 260 2 4176 6633 t
10 I f
(O)720 6753 w
10 R f
(\()800 6753 w
10 I f
(mn)841 6753 w
10 R f
( positive-flow problems)2 967(\). The)1 269 2 971 6753 t
10 I f
(F)2238 6753 w
7 I f
(i)2304 6772 w
7 R f
(\( 1 \))2 91 1 2304 6713 t
10 R f
(and the covering-paths problems)3 1324 1 2434 6753 t
10 I f
(CP)3789 6753 w
7 I f
(i)3922 6772 w
7 R f
(\( 1 \))2 91 1 3922 6713 t
10 R f
(,)4021 6753 w
10 I f
(i)4077 6753 w
10 S f
(=)4154 6753 w
10 R f
( can be solved)3 583( 3,)1 116(2 ,)1 83 3 4258 6753 t
(in time)1 281 1 720 6873 t
10 I f
(O)1026 6873 w
10 R f
(\()1106 6873 w
10 I f
(m)1147 6873 w
10 R f
(\()1227 6873 w
10 I f
(m)1268 6873 w
10 S f
(+)1389 6873 w
10 I f
(n)1493 6873 w
10 R f
(log)1584 6873 w
10 I f
(n)1753 6873 w
10 R f
(\) \).)1 99 1 1811 6873 t
10 S1 f
()1960 6873 w
1960 6873 m
50 build_sq
2010 6873 m
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9 pagesetup
10 R f
(- 9 -)2 166 1 2797 480 t
10 B f
( and Covering-Paths Problems on Graphs without Capacity Upper Bounds)9 3194(5. Positive-Flow)1 702 2 720 840 t
10 R f
( of constructing testing sets, there is no a priori capacity upper bound on the edges.)15 3429(For the purpose)2 641 2 970 996 t
(Therefore, we can set)3 884 1 720 1116 t
10 I f
(u)1638 1116 w
10 R f
(\()1696 1116 w
10 I f
(e)1737 1116 w
10 R f
(\))1789 1116 w
10 S f
(= \245)1 177 1 1879 1116 t
10 R f
(for all)1 250 1 2090 1116 t
10 I f
(e)2373 1116 w
10 R f
(in)2450 1116 w
10 I f
(E)2561 1116 w
10 R f
(, and denote the corresponding positive-flow and covering-)7 2418 1 2622 1116 t
(paths problems by)2 751 1 720 1236 t
10 I f
(F)1505 1236 w
7 I f
(i)1571 1255 w
7 R f
(\( 2 \))2 91 1 1571 1196 t
10 R f
(and)1704 1236 w
10 I f
(CP)1882 1236 w
7 I f
(i)2015 1255 w
7 R f
(\( 2 \))2 91 1 2015 1196 t
10 R f
(,)2114 1236 w
10 I f
(i)2173 1236 w
10 S f
(=)2250 1236 w
10 R f
( we can apply Algorithms)4 1072( special cases,)2 580( As)1 170( respectively.)1 541( 3,)1 116( ,)1 33( 2)1 91(1 ,)1 83 8 2354 1236 t
( the exis-)2 376( since there is no capacity upper bound, it is easy to determine)12 2554( However,)1 446(1-3 for these problems.)3 944 4 720 1356 t
(tence of flows on)3 690 1 720 1476 t
10 I f
(G)1435 1476 w
10 R f
(:)1507 1476 w
10 B f
(Lemma 5.1.)1 510 1 720 1632 t
10 R f
(For a directed graph)3 830 1 1263 1632 t
10 I f
(G)2126 1632 w
10 R f
(without capacity upper bounds, the positive-flow and covering-paths)7 2809 1 2231 1632 t
(problems have a solution if and only if for every vertex)10 2207 1 720 1752 t
10 I f
(v)2952 1752 w
10 R f
(in)3021 1752 w
10 I f
(V)3124 1752 w
10 R f
(, there is an)3 460 1 3185 1752 t
10 I f
(s)3670 1752 w
10 S f
(-)3733 1752 w
10 I f
(v)3804 1752 w
10 R f
(and a)1 213 1 3873 1752 t
10 I f
(v)4111 1752 w
10 S f
(-)4179 1752 w
10 I f
(t)4250 1752 w
10 R f
(path in)1 275 1 4303 1752 t
10 I f
(G)4603 1752 w
10 R f
(.)4675 1752 w
10 S1 f
()4750 1752 w
4750 1752 m
50 build_sq
4800 1752 m
10 R f
(After checking the existence of feasible solutions, Algorithms 1)8 2549 1 970 1908 t
10 S f
(-)3535 1908 w
10 R f
(3 can be applied to these problems:)6 1409 1 3606 1908 t
10 B f
(Theorem 4.2)1 539 1 720 2064 t
10 S f
(\242)1259 2064 w
10 B f
(.)1284 2064 w
10 R f
(The positive-flow problem)2 1073 1 1335 2064 t
10 I f
(F)2434 2064 w
7 R f
(1)2500 2083 w
(\( 2 \))2 91 1 2500 2024 t
10 R f
( covering-paths problem)2 980(and the)1 292 2 2625 2064 t
10 I f
(CP)3924 2064 w
7 R f
(1)4057 2083 w
(\( 2 \))2 91 1 4057 2024 t
10 R f
(can be solved in time)4 857 1 4183 2064 t
10 I f
(O)720 2184 w
10 R f
(\()800 2184 w
10 I f
(mn)841 2184 w
10 R f
( positive-flow problems)2 967(\). The)1 269 2 971 2184 t
10 I f
(F)2238 2184 w
7 I f
(i)2304 2203 w
7 R f
(\( 2 \))2 91 1 2304 2144 t
10 R f
(and the covering-paths problems)3 1324 1 2434 2184 t
10 I f
(CP)3789 2184 w
7 I f
(i)3922 2203 w
7 R f
(\( 2 \))2 91 1 3922 2144 t
10 R f
(,)4021 2184 w
10 I f
(i)4077 2184 w
10 S f
(=)4154 2184 w
10 R f
( can be solved)3 583( 3,)1 116(2 ,)1 83 3 4258 2184 t
(in time)1 281 1 720 2304 t
10 I f
(O)1026 2304 w
10 R f
(\()1106 2304 w
10 I f
(m)1147 2304 w
10 R f
(\()1227 2304 w
10 I f
(m)1268 2304 w
10 S f
(+)1389 2304 w
10 I f
(n)1493 2304 w
10 R f
(log)1584 2304 w
10 I f
(n)1753 2304 w
10 R f
(\) \).)1 99 1 1811 2304 t
10 S1 f
()1960 2304 w
1960 2304 m
50 build_sq
2010 2304 m
10 R f
( bal-)1 182(By taking advantage of the fact that the capacity upper bounds are infinite for all the edges in the)18 3888 2 970 2460 t
( the source)2 448(ancing graphs except those incident to)5 1573 2 720 2580 t
10 I f
(s)2774 2580 w
10 S f
(\242)2821 2580 w
10 R f
(or sink)1 283 1 2879 2580 t
10 I f
(t)3195 2580 w
10 S f
(\242)3231 2580 w
10 R f
(, we can further reduce the time bounds for)8 1784 1 3256 2580 t
(Problems)720 2700 w
10 I f
(F)1123 2700 w
7 I f
(i)1189 2719 w
7 R f
(\( 2 \))2 91 1 1189 2660 t
10 R f
(and)1313 2700 w
10 I f
(CP)1482 2700 w
7 I f
(i)1615 2719 w
7 R f
(\( 2 \))2 91 1 1615 2660 t
10 R f
(,)1714 2700 w
10 I f
(i)1764 2700 w
10 S f
(=)1841 2700 w
10 R f
( 3.)1 116(2 ,)1 83 2 1945 2700 t
10 B f
(5.1.)720 2940 w
10 I f
(O)920 2940 w
10 R f
(\()1000 2940 w
10 I f
(mn)1041 2940 w
10 R f
(log)1204 2940 w
10 I f
(n)1373 2940 w
10 R f
(\))1431 2940 w
10 B f
(Algorithms)1489 2940 w
10 R f
( or sink in a balancing graph is)7 1334(Since the sum of capacity bounds of edges incident to the source)11 2736 2 970 3096 t
(bounded by)1 479 1 720 3216 t
10 I f
(m)1234 3216 w
10 R f
( bound, we can use capacity scaling)6 1481(, and the rest of the edges have infinite capacity upper)10 2253 2 1306 3216 t
( provides an)2 492( It)1 113( and Tarjan, 1987].)3 768([Edmonds and Karp, 1972; Gabow)4 1399 4 720 3336 t
10 I f
(O)3519 3336 w
10 R f
(\()3599 3336 w
10 I f
(n)3640 3336 w
10 R f
(\()3698 3336 w
10 I f
(m)3739 3336 w
10 S f
(+)3860 3336 w
10 I f
(n)3964 3336 w
10 R f
(log)4055 3336 w
10 I f
(n)4224 3336 w
10 R f
(\) log)1 210 1 4282 3336 t
10 I f
(n)4533 3336 w
10 R f
(\) algorithm)1 449 1 4591 3336 t
(for finding a minimum-cost maximum flow on a balancing graph.)9 2631 1 720 3456 t
(If)970 3612 w
10 I f
(m)1069 3612 w
10 S f
(\263)1182 3612 w
10 I f
(n)1278 3612 w
10 R f
(log)1369 3612 w
10 I f
(n)1538 3612 w
10 R f
( bound is)2 385(, then this)2 408 2 1588 3612 t
10 I f
(O)2415 3612 w
10 R f
(\()2495 3612 w
10 I f
(mn)2536 3612 w
10 R f
(log)2699 3612 w
10 I f
(n)2868 3612 w
10 R f
( we can use minimum-cost augmenta-)5 1562(\). Otherwise,)1 552 2 2926 3612 t
(tion for finding a minimum-cost maximum flow with cost)8 2429 1 720 3732 t
10 I f
(O)3187 3732 w
10 R f
(\()3267 3732 w
10 I f
(m)3308 3732 w
10 R f
(\()3388 3732 w
10 I f
(m)3429 3732 w
10 S f
(+)3550 3732 w
10 I f
(n)3654 3732 w
10 R f
(log)3745 3732 w
10 I f
(n)3914 3732 w
10 R f
( which is)2 387(\) \),)1 99 2 3972 3732 t
10 I f
(O)4496 3732 w
10 R f
(\()4576 3732 w
10 I f
(mn)4617 3732 w
10 R f
(log)4780 3732 w
10 I f
(n)4949 3732 w
10 R f
(\))5007 3732 w
(when)720 3852 w
10 I f
(m)961 3852 w
10 S f
(<)1082 3852 w
10 I f
(n)1186 3852 w
10 R f
(log)1244 3852 w
10 I f
(n)1380 3852 w
10 R f
( summarize:)1 491(. We)1 213 2 1430 3852 t
10 B f
(Theorem 5.1.)1 573 1 720 4008 t
10 R f
(The positive-flow problems)2 1130 1 1328 4008 t
10 I f
(F)2493 4008 w
7 I f
(i)2559 4027 w
7 R f
(\( 2 \))2 91 1 2559 3968 t
10 R f
(and the covering-paths problems)3 1336 1 2693 4008 t
10 I f
(CP)4064 4008 w
7 I f
(i)4197 4027 w
7 R f
(\( 2 \))2 91 1 4197 3968 t
10 R f
(,)4296 4008 w
10 I f
(i)4356 4008 w
10 S f
(=)4433 4008 w
10 R f
( can be)2 304( 3,)1 116(2 ,)1 83 3 4537 4008 t
(solved in time)2 567 1 720 4128 t
10 I f
(O)1312 4128 w
10 R f
(\()1392 4128 w
10 I f
(mn)1433 4128 w
10 R f
(log)1596 4128 w
10 I f
(n)1765 4128 w
10 R f
(\).)1823 4128 w
10 S1 f
()1931 4128 w
1931 4128 m
50 build_sq
1981 4128 m
10 R f
(Next, we use)2 518 1 970 4284 t
10 I f
(partial scaling)1 587 1 1513 4284 t
10 R f
(to further improve the efficiency of the algorithms.)7 2035 1 2125 4284 t
10 B f
( Scaling:)1 370(5.2. Partial)1 494 2 720 4524 t
10 I f
(O)1609 4524 w
10 R f
(\()1689 4524 w
10 I f
(n)1730 4524 w
10 R f
(\()1788 4524 w
10 I f
(m)1829 4524 w
10 S f
(+)1950 4524 w
10 I f
(n)2054 4524 w
10 R f
(log)2145 4524 w
10 I f
(n)2314 4524 w
10 R f
( \()1 41(\) log)1 210 2 2372 4524 t
10 I f
(m / n)2 166 1 2631 4524 t
10 R f
(\) \))1 74 1 2805 4524 t
10 B f
(Algorithms)2904 4524 w
10 R f
( positive-flow problems)2 993(In Algorithms 2 and 3, the)5 1145 2 970 4680 t
10 I f
(F)3152 4680 w
7 I f
(i)3218 4699 w
7 R f
(\( 2 \))2 91 1 3218 4640 t
10 R f
(,)3317 4680 w
10 I f
(i)3386 4680 w
10 S f
(=)3463 4680 w
10 R f
( are reduced to minimum-cost)4 1274( 3,)1 116(2 ,)1 83 3 3567 4680 t
( on balancing graphs, where the edges incident to the source or sink have bounded)14 3299(maximum-flow problems)1 1021 2 720 4800 t
( of scaling all the edges)5 970( Instead)1 344( have an infinite capacity upper bound.)6 1591(capacities and the rest of the edges)6 1415 4 720 4920 t
(after setting the capacity bounds of some of the edges to)10 2256 1 720 5040 t
10 I f
(m)3002 5040 w
10 R f
( previous subsection, we now)4 1179(, as was done in the)5 787 2 3074 5040 t
(use)720 5160 w
10 I f
(partial scaling)1 590 1 881 5160 t
10 R f
( only scale the edges incident to)6 1294( We)1 191(to further improve the efficiency of the algorithms.)7 2056 3 1499 5160 t
( used by Edmonds and Karp [1972] for the)8 1756( approach is a generalization of scaling)6 1599( Our)1 212(the source or sink.)3 753 4 720 5280 t
( interested readers are referred to the original)7 1838( The)1 210(transportation problem, and we adopt their terminology.)6 2272 3 720 5400 t
(paper.)720 5520 w
( maximum flow on a balancing graph)6 1624(We construct a minimum-cost)3 1267 2 970 5681 t
10 I f
(G)3907 5681 w
10 S1 f
(_ _)1 62 1 3917 5588 t
10 S f
(=)4028 5681 w
10 R f
(\()4132 5681 w
10 I f
(V)4173 5681 w
10 S1 f
(_ _)1 51 1 4183 5588 t
10 S f
(\310)4275 5681 w
10 I f
({ s)1 87 1 4393 5681 t
10 S f
(\242)4488 5681 w
10 R f
(,)4521 5681 w
10 I f
(t)4587 5681 w
10 S f
(\242)4623 5681 w
10 I f
(})4656 5681 w
10 R f
(,)4704 5681 w
10 I f
(E)4770 5681 w
10 S1 f
(_ _)1 51 1 4780 5588 t
10 R f
(\). A)1 201 1 4839 5681 t
10 I f
( unction)1 308(labeling f)1 405 2 720 5806 t
10 R f
( the set of vertices of)5 859(is a mapping from)3 748 2 1464 5806 t
10 I f
(G)3101 5806 w
10 S1 f
(_ _)1 62 1 3111 5713 t
10 R f
( that for)2 326( Note)1 249(to the real numbers.)3 808 3 3203 5806 t
10 I f
(v)4616 5806 w
10 R f
(,)4668 5806 w
10 I f
(w)4734 5806 w
10 S f
(\316)4842 5806 w
10 I f
(V)4954 5806 w
10 S1 f
(_ _)1 51 1 4964 5713 t
10 R f
(,)5015 5806 w
(the capacity of edge \()4 862 1 720 5931 t
10 I f
(v)1590 5931 w
10 R f
(,)1642 5931 w
10 I f
(w)1708 5931 w
10 R f
( flow)1 210( A)1 124(\) is infinity.)2 472 3 1783 5931 t
10 I f
(f)2616 5931 w
10 R f
(on)2671 5931 w
10 I f
(G)2798 5931 w
10 S1 f
(_ _)1 62 1 2808 5838 t
10 R f
(is)2897 5931 w
10 I f
(extreme)2991 5931 w
10 R f
(if and only if there is a labeling function)8 1625 1 3333 5931 t
10 S f
(p)4985 5931 w
10 R f
(such that for every edge \()5 1046 1 720 6051 t
10 I f
(v)1774 6051 w
10 R f
(,)1826 6051 w
10 I f
(w)1892 6051 w
10 R f
(\) in the residual graph)4 900 1 1967 6051 t
10 I f
(R)2898 6051 w
10 R f
(\()2967 6051 w
10 I f
(f)3024 6051 w
10 R f
(\),)3076 6051 w
10 S f
(p)3165 6051 w
10 R f
(\()3228 6051 w
10 I f
(v)3269 6051 w
10 R f
(\))3321 6051 w
10 S f
(- p)1 159 1 3411 6051 t
10 R f
(\()3578 6051 w
10 I f
(w)3619 6051 w
10 R f
(\))3694 6051 w
10 S f
(+ D)1 165 1 3784 6051 t
10 R f
(\()3957 6051 w
10 I f
(v)3998 6051 w
10 R f
(,)4050 6051 w
10 I f
(w)4116 6051 w
10 R f
(\))4191 6051 w
10 S f
(\263)4255 6051 w
10 R f
(0, where)1 349 1 4351 6051 t
10 S f
(D)4730 6051 w
10 R f
(is the)1 219 1 4821 6051 t
( flow)1 213( A)1 127( defined in Section 3.3.)4 947(cost function on the edges in the residual graph as)9 2030 4 720 6176 t
10 I f
(f)4067 6176 w
10 R f
(is extreme on)2 548 1 4125 6176 t
10 I f
(G)4703 6176 w
10 S1 f
(_ _)1 62 1 4713 6083 t
10 R f
(if and)1 235 1 4805 6176 t
(only if there exists)3 741 1 720 6296 t
10 S f
(p)1486 6296 w
10 R f
(such that the following six conditions hold:)6 1734 1 1566 6296 t
(\(i\) For)1 389 1 720 6457 t
10 I f
(v)1134 6457 w
10 R f
(,)1186 6457 w
10 I f
(w)1252 6457 w
10 S f
(\316)1360 6457 w
10 I f
(V)1472 6457 w
10 S1 f
(_ _)1 51 1 1482 6364 t
10 R f
(and \()1 202 1 1558 6457 t
10 I f
(v)1768 6457 w
10 R f
(,)1820 6457 w
10 I f
(w)1886 6457 w
10 R f
(\))1961 6457 w
10 S f
(\316)2043 6457 w
10 I f
(E)2155 6457 w
10 S1 f
(_ _)1 51 1 2165 6364 t
10 R f
(,)2216 6457 w
10 S f
(p)2266 6457 w
10 R f
(\()2329 6457 w
10 I f
(v)2370 6457 w
10 R f
(\))2422 6457 w
10 S f
(- p)1 159 1 2512 6457 t
10 R f
(\()2679 6457 w
10 I f
(w)2720 6457 w
10 R f
(\))2795 6457 w
10 S f
(+)2885 6457 w
10 I f
(cost)2989 6457 w
10 R f
(\()3158 6457 w
10 I f
(v)3199 6457 w
10 R f
(,)3251 6457 w
10 I f
(w)3317 6457 w
10 R f
(\))3392 6457 w
10 S f
(\263)3474 6457 w
10 R f
(0;)3570 6457 w
(\(ii\) For)1 389 1 720 6618 t
10 I f
(v)1134 6618 w
10 R f
(,)1186 6618 w
10 I f
(w)1252 6618 w
10 S f
(\316)1360 6618 w
10 I f
(V)1472 6618 w
10 S1 f
(_ _)1 51 1 1482 6525 t
10 R f
(and \()1 202 1 1558 6618 t
10 I f
(v)1768 6618 w
10 R f
(,)1820 6618 w
10 I f
(w)1886 6618 w
10 R f
(\))1961 6618 w
10 S f
(\316)2043 6618 w
10 I f
(E)2155 6618 w
10 S1 f
(_ _)1 51 1 2165 6525 t
10 R f
(,)2216 6618 w
10 S f
(p)2266 6618 w
10 R f
(\()2329 6618 w
10 I f
(v)2370 6618 w
10 R f
(\))2422 6618 w
10 S f
(- p)1 159 1 2512 6618 t
10 R f
(\()2679 6618 w
10 I f
(w)2720 6618 w
10 R f
(\))2795 6618 w
10 S f
(+)2885 6618 w
10 I f
(cost)2989 6618 w
10 R f
(\()3158 6618 w
10 I f
(v)3199 6618 w
10 R f
(,)3251 6618 w
10 I f
(w)3317 6618 w
10 R f
(\))3392 6618 w
10 S f
(>)3482 6618 w
10 R f
(0 implies)1 370 1 3586 6618 t
10 I f
(f)3981 6618 w
10 R f
(\()4025 6618 w
10 I f
(v)4066 6618 w
10 R f
(,)4118 6618 w
10 I f
(w)4184 6618 w
10 R f
(\))4259 6618 w
10 S f
(=)4349 6618 w
10 R f
(0;)4453 6618 w
(\(iii\) For)1 389 1 720 6779 t
10 I f
(v)1134 6779 w
10 S f
(\316)1219 6779 w
10 I f
(S)1331 6779 w
10 S1 f
(_)1336 6686 w
10 R f
(,)1381 6779 w
10 S f
(p)1431 6779 w
10 R f
(\()1494 6779 w
10 I f
(s)1535 6779 w
10 S f
(\242)1582 6779 w
10 R f
(\))1615 6779 w
10 S f
(> p)1 159 1 1705 6779 t
10 R f
(\()1872 6779 w
10 I f
(v)1913 6779 w
10 R f
(\) implies)1 353 1 1965 6779 t
10 I f
(f)2343 6779 w
10 R f
(\()2387 6779 w
10 I f
(s)2428 6779 w
10 S f
(\242)2475 6779 w
10 R f
(,)2508 6779 w
10 I f
(v)2574 6779 w
10 R f
(\))2626 6779 w
10 S f
(=)2716 6779 w
10 R f
(0;)2820 6779 w
(\(iv\) For)1 389 1 720 6940 t
10 I f
(v)1134 6940 w
10 S f
(\316)1219 6940 w
10 I f
(S)1331 6940 w
10 S1 f
(_)1336 6847 w
10 R f
(,)1381 6940 w
10 S f
(p)1431 6940 w
10 R f
(\()1494 6940 w
10 I f
(s)1535 6940 w
10 S f
(\242)1582 6940 w
10 R f
(\))1615 6940 w
10 S f
(< p)1 159 1 1705 6940 t
10 R f
(\()1872 6940 w
10 I f
(v)1913 6940 w
10 R f
(\) implies)1 353 1 1965 6940 t
10 I f
(f)2343 6940 w
10 R f
(\()2387 6940 w
10 I f
(s)2428 6940 w
10 S f
(\242)2475 6940 w
10 R f
(,)2508 6940 w
10 I f
(v)2574 6940 w
10 R f
(\))2626 6940 w
10 S f
(=)2716 6940 w
10 I f
(u)2820 6940 w
10 S1 f
(_)2822 6872 w
10 R f
(\()2878 6940 w
10 I f
(s)2919 6940 w
10 S f
(\242)2966 6940 w
10 R f
(,)2999 6940 w
10 I f
(v)3065 6940 w
10 R f
(\);)3117 6940 w
(\(v\) For)1 389 1 720 7101 t
10 I f
(w)1134 7101 w
10 S f
(\316)1242 7101 w
10 I f
(D)1354 7101 w
10 S1 f
(_ _)1 62 1 1364 7008 t
10 R f
(,)1426 7101 w
10 S f
(p)1476 7101 w
10 R f
(\()1539 7101 w
10 I f
(w)1580 7101 w
10 R f
(\))1655 7101 w
10 S f
(> p)1 159 1 1745 7101 t
10 R f
(\()1912 7101 w
10 I f
(t)1953 7101 w
10 S f
(\242)1989 7101 w
10 R f
(\) implies)1 353 1 2022 7101 t
10 I f
(f)2400 7101 w
10 R f
(\()2444 7101 w
10 I f
(w)2485 7101 w
10 R f
(,)2560 7101 w
10 I f
(t)2626 7101 w
10 S f
(\242)2662 7101 w
10 R f
(\))2695 7101 w
10 S f
(=)2785 7101 w
10 R f
(0;)2889 7101 w
(\(vi\) For)1 389 1 720 7262 t
10 I f
(w)1134 7262 w
10 S f
(\316)1242 7262 w
10 I f
(D)1354 7262 w
10 S1 f
(_ _)1 62 1 1364 7169 t
10 R f
(,)1426 7262 w
10 S f
(p)1476 7262 w
10 R f
(\()1539 7262 w
10 I f
(w)1580 7262 w
10 R f
(\))1655 7262 w
10 S f
(< p)1 159 1 1745 7262 t
10 R f
(\()1912 7262 w
10 I f
(t)1953 7262 w
10 S f
(\242)1989 7262 w
10 R f
(\) implies)1 353 1 2022 7262 t
10 I f
(f)2400 7262 w
10 R f
(\()2444 7262 w
10 I f
(w)2485 7262 w
10 R f
(,)2560 7262 w
10 I f
(t)2626 7262 w
10 S f
(\242)2662 7262 w
10 R f
(\))2695 7262 w
10 S f
(=)2785 7262 w
10 I f
(u)2889 7262 w
10 S1 f
(_)2891 7194 w
10 R f
(\()2947 7262 w
10 I f
(w)2988 7262 w
10 R f
(,)3063 7262 w
10 I f
(t)3129 7262 w
10 S f
(\242)3165 7262 w
10 R f
(\).)3198 7262 w
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10 pagesetup
10 R f
(- 10 -)2 216 1 2772 480 t
(A flow is)2 386 1 970 840 t
10 I f
(pseudo-extreme)1388 840 w
10 R f
( in)1 111( As)1 169(if there exists a labeling function satisfying conditions \(i\) and \(ii\).)10 2708 3 2052 840 t
( be shown that a pseudo-extreme maximum flow on)8 2123(Edmonds and Karp [1972], it can)5 1366 2 720 965 t
10 I f
(G)4240 965 w
10 S1 f
(_ _)1 62 1 4250 872 t
10 R f
(is extreme, i.e., a)3 697 1 4343 965 t
( maximum flow)2 655( problem is now reduced to finding a pseudo-extreme)8 2191( The)1 212(minimum-cost maximum flow.)2 1262 4 720 1085 t
(on)720 1210 w
10 I f
(G)845 1210 w
10 S1 f
(_ _)1 62 1 855 1117 t
10 R f
(.)917 1210 w
(Choose a positive integer)3 1022 1 970 1366 t
10 I f
(l)2020 1366 w
10 R f
( capacity upper bounds of the edges incident to the source)10 2352(such that the)2 511 2 2076 1366 t
10 I f
(s)4968 1366 w
10 S f
(\242)5015 1366 w
10 R f
(or sink)1 279 1 720 1486 t
10 I f
(t)1028 1486 w
10 S f
(\242)1064 1486 w
10 R f
( most)1 223(have at)1 289 2 1118 1486 t
10 I f
(l)1658 1486 w
10 R f
(digits in their binary expansions. Obviously,)5 1790 1 1714 1486 t
10 I f
(l)3532 1486 w
10 S f
(\243 \351)1 146 1 3601 1486 t
10 R f
(log)3755 1486 w
10 I f
(m)3924 1486 w
10 S f
(\371)4004 1486 w
10 R f
(, where)1 296 1 4042 1486 t
10 S f
(\351)4366 1486 w
10 R f
(.)4424 1456 w
10 S f
(\371)4457 1486 w
10 R f
(is the ceiling)2 517 1 4523 1486 t
( Problem)1 372(function. For)1 555 2 720 1611 t
10 I f
(p)1680 1611 w
10 R f
(, where 0)2 384 1 1730 1611 t
10 S f
(\243)2155 1611 w
10 I f
(p)2251 1611 w
10 S f
(\243)2342 1611 w
10 I f
(l)2438 1611 w
10 R f
(, edge \()2 312 1 2466 1611 t
10 I f
(s)2786 1611 w
10 S f
(\242)2833 1611 w
10 R f
(,)2866 1611 w
10 I f
(v)2932 1611 w
10 R f
(\) has capacity)2 564 1 2984 1611 t
10 S f
(\351)3581 1611 w
10 I f
(u)3639 1611 w
10 S1 f
(_)3641 1543 w
10 R f
(\()3697 1611 w
10 I f
(s)3738 1611 w
10 S f
(\242)3785 1611 w
10 R f
(,)3818 1611 w
10 I f
(v)3884 1611 w
10 R f
(\))3936 1611 w
10 I f
(/)3977 1611 w
10 R f
(2)4013 1611 w
7 I f
(p)4068 1571 w
10 S f
(\371)4119 1611 w
10 R f
( \()1 67(and edge)1 365 2 4190 1611 t
10 I f
(w)4630 1611 w
10 R f
(,)4705 1611 w
10 I f
(t)4771 1611 w
10 S f
(\242)4807 1611 w
10 R f
(\) has)1 200 1 4840 1611 t
(capacity)720 1736 w
10 S f
(\351)1077 1736 w
10 I f
(u)1135 1736 w
10 S1 f
(_)1137 1668 w
10 R f
(\()1193 1736 w
10 I f
(w)1234 1736 w
10 R f
(,)1309 1736 w
10 I f
(t)1375 1736 w
10 S f
(\242)1411 1736 w
10 R f
(\))1444 1736 w
10 I f
(/)1485 1736 w
10 R f
(2)1521 1736 w
7 I f
(p)1576 1696 w
10 S f
(\371)1627 1736 w
10 R f
(, where)1 293 1 1665 1736 t
10 I f
(v)1983 1736 w
10 S f
(\316)2068 1736 w
10 I f
(S)2180 1736 w
10 S1 f
(_)2185 1643 w
10 R f
(and)2255 1736 w
10 I f
(w)2424 1736 w
10 S f
(\316)2532 1736 w
10 I f
(D)2644 1736 w
10 S1 f
(_ _)1 62 1 2654 1643 t
10 R f
( capacity of the other edges remains infinite.)7 1774(. The)1 230 2 2716 1736 t
( scaling method computes maximum pseudo-extreme flows successively for Problems)9 3692(Then the)1 378 2 970 1892 t
10 I f
(l)720 2012 w
10 R f
(,)756 2012 w
10 I f
(l)822 2012 w
10 S f
(-)899 2012 w
10 R f
( can be easily shown that if)6 1099( It)1 113( 0.)1 132( . . . ,)4 200(1 ,)1 83 5 1003 2012 t
10 I f
(f)2656 2012 w
10 R f
(is a maximum pseudo-extreme flow computed in Problem)7 2330 1 2710 2012 t
10 I f
(p)720 2132 w
10 R f
(and)798 2132 w
10 S f
(p)970 2132 w
10 R f
( labeling function, then 2)4 1018(is the associated)2 655 2 1053 2132 t
10 I f
(f)2742 2132 w
10 R f
(can be taken as its initial pseudo-extreme flow in Prob-)9 2241 1 2799 2132 t
(lem)720 2252 w
10 I f
(p)895 2252 w
10 S f
(-)994 2252 w
10 R f
(1 with)1 253 1 1098 2252 t
10 S f
(p)1376 2252 w
10 R f
(as its associated labeling function.)4 1368 1 1456 2252 t
( Karp [1972], and we omit it. How-)7 1487(The detailed implementation is similar to that of Edmonds and)9 2583 2 970 2408 t
( that each minimum-cost augmentation)4 1564(ever, to guarantee the correctness of the algorithm, we have to prove)11 2756 2 720 2528 t
(over a pseudo-extreme flow still gives a pseudo-extreme flow.)8 2492 1 720 2648 t
(Since)970 2809 w
10 I f
(f)1240 2809 w
7 I f
(k)1290 2769 w
10 R f
(is a pseudo-extreme flow and)4 1267 1 1377 2809 t
10 S f
(p)2692 2809 w
7 I f
(k)2752 2769 w
10 R f
( labeling function, for \()4 1025(is the associated)2 695 2 2839 2809 t
10 I f
(v)4567 2809 w
10 R f
(,)4619 2809 w
10 I f
(w)4685 2809 w
10 R f
(\))4760 2809 w
10 S f
(\316)4842 2809 w
10 I f
(E)4954 2809 w
10 S1 f
(_ _)1 51 1 4964 2716 t
10 R f
(,)5015 2809 w
10 I f
(v)720 2934 w
10 R f
(,)772 2934 w
10 I f
(w)838 2934 w
10 S f
(\316)946 2934 w
10 I f
(V)1058 2934 w
10 S1 f
(_ _)1 51 1 1068 2841 t
10 R f
(,)1119 2934 w
10 S f
(D)1183 2934 w
7 I f
(k)1249 2894 w
10 R f
(\()1296 2934 w
10 I f
(v)1337 2934 w
10 R f
(,)1389 2934 w
10 I f
(w)1455 2934 w
10 R f
(\) =)1 128 1 1530 2934 t
10 S f
(p)1697 2934 w
7 I f
(k)1757 2894 w
10 R f
(\()1804 2934 w
10 I f
(v)1845 2934 w
10 R f
(\))1897 2934 w
10 S f
(- p)1 159 1 2010 2934 t
7 I f
(k)2174 2894 w
10 R f
(\()2221 2934 w
10 I f
(w)2262 2934 w
10 R f
(\))2337 2934 w
10 S f
(+ D)1 165 1 2427 2934 t
10 R f
(\()2600 2934 w
10 I f
(v)2641 2934 w
10 R f
(,)2693 2934 w
10 I f
(w)2759 2934 w
10 R f
(\))2834 2934 w
10 S f
(= p)1 159 1 2906 2934 t
7 I f
(k)3070 2894 w
10 R f
(\()3117 2934 w
10 I f
(v)3158 2934 w
10 R f
(\))3210 2934 w
10 S f
(- p)1 159 1 3282 2934 t
7 I f
(k)3446 2894 w
10 R f
(\()3493 2934 w
10 I f
(w)3534 2934 w
10 R f
(\))3609 2934 w
10 S f
(+)3699 2934 w
10 I f
(cost)3803 2934 w
10 R f
(\()3972 2934 w
10 I f
(v)4013 2934 w
10 R f
(,)4065 2934 w
10 I f
(w)4131 2934 w
10 R f
(\))4206 2934 w
10 S f
(\263)4278 2934 w
10 R f
( the other)2 405(0. On)1 261 2 4374 2934 t
(hand, for)1 399 1 720 3059 t
10 I f
(v)1183 3059 w
10 R f
(,)1235 3059 w
10 I f
(w)1301 3059 w
10 S f
(\316)1409 3059 w
10 I f
(V)1521 3059 w
10 S1 f
(_ _)1 51 1 1531 2966 t
10 R f
(, \()1 122 1 1582 3059 t
10 I f
(v)1712 3059 w
10 R f
(,)1764 3059 w
10 I f
(w)1830 3059 w
10 R f
(\))1905 3059 w
10 S f
(\316)1987 3059 w
10 I f
(E)2099 3059 w
10 S1 f
(_ _)1 51 1 2109 2966 t
10 R f
(, and \()2 330 1 2160 3059 t
10 I f
(w)2498 3059 w
10 R f
(,)2573 3059 w
10 I f
(v)2639 3059 w
10 R f
(\))2691 3059 w
10 S f
(\316)2773 3059 w
10 I f
(R)2885 3059 w
10 R f
(\()2954 3059 w
10 I f
(f)3011 3059 w
7 I f
(k)3061 3019 w
10 R f
(\),)3108 3059 w
10 S f
(D)3231 3059 w
7 I f
(k)3297 3019 w
10 R f
(\()3344 3059 w
10 I f
(w)3385 3059 w
10 R f
(,)3460 3059 w
10 I f
(v)3526 3059 w
10 R f
(\))3578 3059 w
10 S f
(= p)1 159 1 3668 3059 t
7 I f
(k)3832 3019 w
10 R f
(\()3879 3059 w
10 I f
(w)3920 3059 w
10 R f
(\))3995 3059 w
10 S f
(- p)1 159 1 4085 3059 t
7 I f
(k)4249 3019 w
10 R f
(\()4296 3059 w
10 I f
(v)4337 3059 w
10 R f
(\))4389 3059 w
10 S f
(+ D)1 165 1 4479 3059 t
10 R f
(\()4652 3059 w
10 I f
(w)4693 3059 w
10 R f
(,)4768 3059 w
10 I f
(v)4834 3059 w
10 R f
(\) =)1 154 1 4886 3059 t
10 S f
(-)720 3179 w
10 R f
([)791 3179 w
10 S f
(p)832 3179 w
7 I f
(k)892 3139 w
10 R f
(\()939 3179 w
10 I f
(v)980 3179 w
10 R f
(\))1032 3179 w
10 S f
(- p)1 159 1 1122 3179 t
7 I f
(k)1286 3139 w
10 R f
(\()1333 3179 w
10 I f
(w)1374 3179 w
10 R f
(\))1449 3179 w
10 S f
(+)1539 3179 w
10 I f
(cost)1643 3179 w
10 R f
(\()1812 3179 w
10 I f
(v)1853 3179 w
10 R f
(,)1905 3179 w
10 I f
(w)1971 3179 w
10 R f
( If)1 139(\) ].)1 99 2 2046 3179 t
10 S f
(D)2332 3179 w
7 I f
(k)2398 3139 w
10 R f
(\()2445 3179 w
10 I f
(w)2486 3179 w
10 R f
(,)2561 3179 w
10 I f
(v)2627 3179 w
10 R f
(\))2679 3179 w
10 S f
(<)2769 3179 w
10 R f
(0, then)1 295 1 2873 3179 t
10 S f
(p)3216 3179 w
7 I f
(k)3276 3139 w
10 R f
(\()3323 3179 w
10 I f
(v)3364 3179 w
10 R f
(\))3416 3179 w
10 S f
(- p)1 159 1 3506 3179 t
7 I f
(k)3670 3139 w
10 R f
(\()3717 3179 w
10 I f
(w)3758 3179 w
10 R f
(\))3833 3179 w
10 S f
(+)3923 3179 w
10 I f
(cost)4027 3179 w
10 R f
(\()4196 3179 w
10 I f
(v)4237 3179 w
10 R f
(,)4289 3179 w
10 I f
(w)4355 3179 w
10 R f
(\))4430 3179 w
10 S f
(>)4520 3179 w
10 R f
(0; that is,)2 416 1 4624 3179 t
10 I f
(f)720 3299 w
7 I f
(k)770 3259 w
10 R f
(\()817 3299 w
10 I f
(v)858 3299 w
10 R f
(,)910 3299 w
10 I f
(w)976 3299 w
10 R f
(\))1051 3299 w
10 S f
(=)1141 3299 w
10 R f
(0, since)1 325 1 1245 3299 t
10 I f
(f)1615 3299 w
7 I f
(k)1665 3259 w
10 R f
( \()1 79( Thus,)1 296(is a pseudo-extreme flow.)3 1091 3 1749 3299 t
10 I f
(w)3223 3299 w
10 R f
(,)3298 3299 w
10 I f
(v)3364 3299 w
10 R f
(\) cannot be in)3 609 1 3416 3299 t
10 I f
(R)4071 3299 w
10 R f
(\()4140 3299 w
10 I f
(f)4197 3299 w
7 I f
(k)4247 3259 w
10 R f
(\), a contradiction.)2 746 1 4294 3299 t
(Therefore,)720 3424 w
10 S f
(D)1162 3424 w
7 I f
(k)1228 3384 w
10 R f
(\()1275 3424 w
10 I f
(v)1316 3424 w
10 R f
(,)1368 3424 w
10 I f
(w)1434 3424 w
10 R f
(\))1509 3424 w
10 S f
(\263)1591 3424 w
10 R f
(0 for all \()3 374 1 1687 3424 t
10 I f
(v)2069 3424 w
10 R f
(,)2121 3424 w
10 I f
(w)2187 3424 w
10 R f
(\))2262 3424 w
10 S f
(\316)2344 3424 w
10 I f
(R)2456 3424 w
10 R f
(\()2525 3424 w
10 I f
(f)2582 3424 w
7 I f
(k)2632 3384 w
10 R f
(\), where)1 326 1 2679 3424 t
10 I f
(v)3030 3424 w
10 R f
(,)3082 3424 w
10 I f
(w)3148 3424 w
10 S f
(\316)3256 3424 w
10 I f
(V)3368 3424 w
10 S1 f
(_ _)1 51 1 3378 3331 t
10 R f
(.)3429 3424 w
(Starting with)1 545 1 970 3580 t
10 S f
(p)1565 3580 w
7 R f
(0)1625 3540 w
10 S f
(\272)1709 3580 w
10 R f
(0 and)1 244 1 1805 3580 t
10 I f
(f)2099 3580 w
7 R f
(0)2149 3540 w
10 S f
(\272)2233 3580 w
10 R f
( a)1 95( Given)1 320( minimum-cost augmentations.)2 1294(0, we do the following)4 1002 4 2329 3580 t
(pseudo-extreme flow)1 848 1 720 3700 t
10 I f
(f)1596 3700 w
7 I f
(k)1646 3660 w
10 R f
(with its associated labeling function)4 1450 1 1713 3700 t
10 S f
(p)3191 3700 w
7 I f
(k)3251 3660 w
10 R f
( minimum-cost path from)3 1031(, augment along a)3 719 2 3290 3700 t
10 I f
(s)720 3820 w
10 S f
(\242)767 3820 w
10 R f
(to)822 3820 w
10 I f
(t)930 3820 w
10 S f
(\242)966 3820 w
10 R f
(in the residual graph)3 833 1 1021 3820 t
10 I f
(R)1884 3820 w
10 R f
(\()1953 3820 w
10 I f
(f)2010 3820 w
7 I f
(k)2060 3780 w
10 R f
(\), with respect to the costs)5 1068 1 2107 3820 t
10 S f
(D)3205 3820 w
7 I f
(k)3271 3780 w
10 R f
(\()3318 3820 w
10 I f
(v)3359 3820 w
10 R f
(,)3411 3820 w
10 I f
(w)3477 3820 w
10 R f
(\))3552 3820 w
10 S f
(= p)1 159 1 3642 3820 t
7 I f
(k)3806 3780 w
10 R f
(\()3853 3820 w
10 I f
(v)3894 3820 w
10 R f
(\))3946 3820 w
10 S f
(- p)1 159 1 4036 3820 t
7 I f
(k)4200 3780 w
10 R f
(\()4247 3820 w
10 I f
(w)4288 3820 w
10 R f
(\))4363 3820 w
10 S f
(+ D)1 165 1 4453 3820 t
10 R f
(\()4626 3820 w
10 I f
(v)4667 3820 w
10 R f
(,)4719 3820 w
10 I f
(w)4785 3820 w
10 R f
(\) for)1 180 1 4860 3820 t
(all edges \()2 414 1 720 3940 t
10 I f
(v)1142 3940 w
10 R f
(,)1194 3940 w
10 I f
(w)1260 3940 w
10 R f
(\) in)1 138 1 1335 3940 t
10 I f
(R)1500 3940 w
10 R f
(\()1569 3940 w
10 I f
(f)1626 3940 w
7 I f
(k)1676 3900 w
10 R f
(\). If)1 176 1 1723 3940 t
10 S f
(s)1926 3940 w
7 I f
(k)1991 3900 w
10 R f
(\()2038 3940 w
10 I f
(v)2079 3940 w
10 R f
(\) denotes the cost of a shortest path from)8 1641 1 2131 3940 t
10 I f
(s)3799 3940 w
10 S f
(\242)3846 3940 w
10 R f
(to)3898 3940 w
10 I f
(v)4003 3940 w
10 R f
( the costs)2 374(with respect to)2 592 2 4074 3940 t
10 S f
(D)720 4060 w
7 I f
(k)786 4020 w
10 R f
(, set)1 167 1 825 4060 t
10 S f
(p)1023 4060 w
7 I f
(k)1083 4020 w
7 S f
(+)1130 4020 w
7 R f
(1)1180 4020 w
10 R f
(\()1231 4060 w
10 I f
(v)1272 4060 w
10 R f
(\) =)1 120 1 1324 4060 t
10 S f
(p)1475 4060 w
7 I f
(k)1535 4020 w
10 R f
(\()1582 4060 w
10 I f
(v)1623 4060 w
10 R f
(\))1675 4060 w
10 S f
(+ s)1 164 1 1765 4060 t
7 I f
(k)1934 4020 w
10 R f
(\()1981 4060 w
10 I f
(v)2022 4060 w
10 R f
(\). Obviously,)1 556 1 2074 4060 t
10 S f
(D)2661 4060 w
7 I f
(k)2727 4020 w
10 R f
(\()2774 4060 w
10 I f
(v)2815 4060 w
10 R f
(,)2867 4060 w
10 I f
(w)2933 4060 w
10 R f
(\))3008 4060 w
10 S f
(\263)3090 4060 w
10 R f
(0 for all \()3 392 1 3186 4060 t
10 I f
(v)3586 4060 w
10 R f
(,)3638 4060 w
10 I f
(w)3704 4060 w
10 R f
(\))3779 4060 w
10 S f
(\316)3861 4060 w
10 I f
(R)3973 4060 w
10 R f
(\()4042 4060 w
10 I f
(f)4099 4060 w
7 I f
(k)4149 4020 w
10 R f
(\). Since)1 336 1 4196 4060 t
10 S f
(p)4564 4060 w
7 R f
(0)4624 4020 w
10 R f
(\()4675 4060 w
10 I f
(s)4716 4060 w
10 S f
(\242)4763 4060 w
10 R f
(\))4796 4060 w
10 S f
(=)4886 4060 w
10 R f
(0)4990 4060 w
(and)720 4180 w
10 S f
(D)889 4180 w
7 I f
(k)955 4140 w
10 R f
(is nonnegative,)1 605 1 1019 4180 t
10 S f
(p)1649 4180 w
7 I f
(k)1709 4140 w
10 R f
(\()1756 4180 w
10 I f
(s)1797 4180 w
10 S f
(\242)1844 4180 w
10 R f
(\))1877 4180 w
10 S f
(=)1967 4180 w
10 R f
(0 for all)2 316 1 2071 4180 t
10 I f
(k)2412 4180 w
10 R f
( now have:)2 438(. We)1 213 2 2456 4180 t
10 B f
(Lemma 5.2.)1 510 1 720 4341 t
10 R f
(For each)1 354 1 1263 4341 t
10 I f
(v)1651 4341 w
10 R f
(in)1729 4341 w
10 I f
(V)1841 4341 w
10 S1 f
(_ _)1 51 1 1851 4248 t
10 R f
(,)1902 4341 w
10 S f
(p)1961 4341 w
7 I f
(k)2021 4301 w
7 S f
(+)2068 4301 w
7 R f
(1)2118 4301 w
10 R f
(\()2169 4341 w
10 I f
(v)2210 4341 w
10 R f
(\) gives the cost of a shortest path from)8 1603 1 2262 4341 t
10 I f
(s)3899 4341 w
10 S f
(\242)3946 4341 w
10 R f
(to)4005 4341 w
10 I f
(v)4117 4341 w
10 R f
(in the residual graph)3 845 1 4195 4341 t
10 I f
(R)720 4461 w
10 R f
(\()789 4461 w
10 I f
(f)846 4461 w
7 I f
(k)896 4421 w
10 R f
(\) with respect to the cost function)6 1373 1 943 4461 t
10 S f
(D)2347 4461 w
10 R f
(. If)1 122 1 2408 4461 t
10 I f
(f)2561 4461 w
7 I f
(k)2611 4421 w
10 R f
( after a minimum-cost aug-)4 1107(is a pseudo-extreme flow, then)4 1252 2 2681 4461 t
(mentation in)1 504 1 720 4581 t
10 I f
(R)1250 4581 w
10 R f
(\()1319 4581 w
10 I f
(f)1376 4581 w
7 I f
(k)1426 4541 w
10 R f
(\) with respect to the cost)5 984 1 1473 4581 t
10 S f
(D)2483 4581 w
7 I f
(k)2549 4541 w
10 R f
(,)2588 4581 w
10 I f
(f)2639 4581 w
7 I f
(k)2689 4541 w
7 S f
(+)2736 4541 w
7 R f
(1)2786 4541 w
10 R f
(is still a pseudo-extreme flow with)5 1395 1 2855 4581 t
10 S f
(p)4277 4581 w
7 I f
(k)4337 4541 w
7 S f
(+)4384 4541 w
7 R f
(1)4434 4541 w
10 R f
(as the associ-)2 536 1 4504 4581 t
(ated labeling function.)2 896 1 720 4701 t
10 B f
(Proof.)720 4857 w
10 R f
(Let)1009 4857 w
10 I f
(p)1168 4857 w
10 R f
(\()1226 4857 w
10 I f
(v)1267 4857 w
10 R f
(\) be a path from)4 641 1 1319 4857 t
10 I f
(s)1986 4857 w
10 S f
(\242)2033 4857 w
10 R f
(to)2084 4857 w
10 I f
(v)2188 4857 w
10 R f
(in)2258 4857 w
10 I f
(R)2362 4857 w
10 R f
(\()2431 4857 w
10 I f
(f)2488 4857 w
7 I f
(k)2538 4817 w
10 R f
(\). Then)1 289 1 2585 4857 t
7 I f
(p)2900 4957 w
7 R f
(\()2940 4957 w
7 I f
(v)2968 4957 w
7 R f
(\))3004 4957 w
15 S f
(S)2919 4887 w
10 S f
(D)3035 4857 w
7 I f
(k)3101 4817 w
10 R f
(\()3148 4857 w
(.)3189 4827 w
(,)3222 4857 w
(.)3288 4827 w
(\) =)1 115 1 3321 4857 t
10 S f
(p)3462 4857 w
7 I f
(k)3522 4817 w
10 R f
(\()3569 4857 w
10 I f
(s)3610 4857 w
10 S f
(\242)3657 4857 w
10 R f
(\))3690 4857 w
10 S f
(- p)1 159 1 3780 4857 t
7 I f
(k)3944 4817 w
10 R f
(\()3991 4857 w
10 I f
(v)4032 4857 w
10 R f
(\))4084 4857 w
10 S f
(+)4143 4857 w
7 I f
(p)4247 4957 w
7 R f
(\()4287 4957 w
7 I f
(v)4315 4957 w
7 R f
(\))4351 4957 w
15 S f
(S)4266 4887 w
10 S f
(D)4382 4857 w
10 R f
(\()4451 4857 w
(.)4492 4827 w
(,)4525 4857 w
(.)4591 4827 w
(\), where)1 328 1 4624 4857 t
10 S f
(D)4979 4857 w
10 R f
( in)1 103(is the cost function)3 761 2 720 5057 t
10 I f
(R)1609 5057 w
10 R f
(\()1678 5057 w
10 I f
(f)1735 5057 w
7 I f
(k)1785 5017 w
10 R f
( an)1 119(\). Therefore,)1 525 2 1832 5057 t
10 I f
(s)2501 5057 w
10 S f
(\242 -)1 96 1 2548 5057 t
10 I f
(v)2660 5057 w
10 R f
(path in)1 275 1 2729 5057 t
10 I f
(R)3029 5057 w
10 R f
(\()3098 5057 w
10 I f
(f)3155 5057 w
7 I f
(k)3205 5017 w
10 R f
(\) is of minimum cost with respect to)7 1447 1 3252 5057 t
10 S f
(D)4724 5057 w
10 R f
(if and)1 230 1 4810 5057 t
(only if it is of minimum cost with respect to)9 1759 1 720 5177 t
10 S f
(D)2504 5177 w
7 I f
(k)2570 5137 w
10 R f
(.)2609 5177 w
(Let)970 5333 w
10 I f
(p *)1 108 1 1131 5333 t
10 R f
(\()1247 5333 w
10 I f
(v)1288 5333 w
10 R f
(\) be a shortest path from)5 988 1 1340 5333 t
10 I f
(s)2356 5333 w
10 S f
(\242)2403 5333 w
10 R f
(to)2456 5333 w
10 I f
(v)2562 5333 w
10 R f
( cost function)2 552(with respect to the)3 744 2 2634 5333 t
10 S f
(D)3959 5333 w
7 I f
(k)4025 5293 w
10 R f
(\(and)4093 5333 w
10 S f
(D)4299 5333 w
10 R f
(\). Then)1 317 1 4360 5333 t
10 S f
(p)4706 5333 w
7 I f
(k)4766 5293 w
7 S f
(+)4813 5293 w
7 R f
(1)4863 5293 w
10 R f
(\()4914 5333 w
10 I f
(v)4955 5333 w
10 R f
(\))5007 5333 w
(=)720 5453 w
10 S f
(p)809 5453 w
7 I f
(k)869 5413 w
10 R f
(\()916 5453 w
10 I f
(v)957 5453 w
10 R f
(\) +)1 122 1 1009 5453 t
10 S f
(s)1163 5453 w
7 I f
(k)1228 5413 w
10 R f
(\()1275 5453 w
10 I f
(v)1316 5453 w
10 R f
(\) =)1 121 1 1368 5453 t
10 S f
(p)1521 5453 w
7 I f
(k)1581 5413 w
10 R f
(\()1628 5453 w
10 I f
(v)1669 5453 w
10 R f
(\) + [)2 186 1 1721 5453 t
10 S f
(p)1915 5453 w
7 I f
(k)1975 5413 w
10 R f
(\()2022 5453 w
10 I f
(s)2063 5453 w
10 S f
(\242)2110 5453 w
10 R f
(\))2143 5453 w
10 S f
(- p)1 159 1 2208 5453 t
7 I f
(k)2372 5413 w
10 R f
(\()2419 5453 w
10 I f
(v)2460 5453 w
10 R f
(\) +)1 121 1 2512 5453 t
7 I f
(p *)1 75 1 2665 5553 t
7 R f
(\()2745 5553 w
7 I f
(v)2773 5553 w
7 R f
(\))2809 5553 w
15 S f
(S)2704 5483 w
10 S f
(D)2840 5453 w
10 R f
(\()2909 5453 w
(.)2950 5423 w
(,)2983 5453 w
(.)3049 5423 w
( =)1 88(\) ])1 74 2 3082 5453 t
10 S f
(p)3276 5453 w
7 I f
(k)3336 5413 w
10 R f
(\()3383 5453 w
10 I f
(s)3424 5453 w
10 S f
(\242)3471 5453 w
10 R f
(\) +)1 121 1 3504 5453 t
7 I f
(p *)1 75 1 3657 5553 t
7 R f
(\()3737 5553 w
7 I f
(v)3765 5553 w
7 R f
(\))3801 5553 w
15 S f
(S)3696 5483 w
10 S f
(D)3840 5453 w
10 R f
(\()3909 5453 w
(.)3950 5423 w
(,)3983 5453 w
(.)4049 5423 w
(\) =)1 121 1 4082 5453 t
7 I f
(p *)1 75 1 4235 5553 t
7 R f
(\()4315 5553 w
7 I f
(v)4343 5553 w
7 R f
(\))4379 5553 w
15 S f
(S)4274 5483 w
10 S f
(D)4418 5453 w
10 R f
(\()4487 5453 w
(.)4528 5423 w
(,)4561 5453 w
(.)4627 5423 w
(\). There-)1 380 1 4660 5453 t
(fore,)720 5653 w
10 S f
(p)935 5653 w
7 I f
(k)995 5613 w
7 S f
(+)1042 5613 w
7 R f
(1)1092 5613 w
10 R f
(\()1143 5653 w
10 I f
(v)1184 5653 w
10 R f
(\) gives the cost of a shortest path from)8 1579 1 1236 5653 t
10 I f
(s)2846 5653 w
10 S f
(\242)2893 5653 w
10 R f
(to)2949 5653 w
10 I f
(v)3058 5653 w
10 R f
(in the residual graph)3 836 1 3133 5653 t
10 I f
(R)4000 5653 w
10 R f
(\()4069 5653 w
10 I f
(f)4126 5653 w
7 I f
(k)4176 5613 w
10 R f
(\) with respect to the)4 817 1 4223 5653 t
(cost function)1 519 1 720 5773 t
10 S f
(D)1264 5773 w
10 R f
( first part of the lemma is proved.)7 1337(. The)1 230 2 1325 5773 t
(To show that)2 580 1 970 5929 t
10 I f
(f)1604 5929 w
7 I f
(k)1654 5889 w
7 S f
(+)1701 5889 w
7 R f
(1)1751 5889 w
10 R f
( for)1 171(is pseudo-extreme, we have to show that conditions \(i\) and \(ii\) hold)11 3021 2 1848 5929 t
(\()720 6054 w
10 I f
(v)761 6054 w
10 R f
(,)813 6054 w
10 I f
(w)879 6054 w
10 R f
(\))954 6054 w
10 S f
(\316)1036 6054 w
10 I f
(E)1148 6054 w
10 S1 f
(_ _)1 51 1 1158 5961 t
10 R f
(, where)1 293 1 1209 6054 t
10 I f
(v)1527 6054 w
10 R f
(,)1579 6054 w
10 I f
(w)1645 6054 w
10 S f
(\316)1753 6054 w
10 I f
(V)1865 6054 w
10 S1 f
(_ _)1 51 1 1875 5961 t
10 R f
(.)1926 6054 w
(We have)1 376 1 970 6210 t
10 S f
(p)1396 6210 w
7 I f
(k)1456 6170 w
7 S f
(+)1503 6170 w
7 R f
(1)1553 6170 w
10 R f
(\()1604 6210 w
10 I f
(v)1645 6210 w
10 R f
(\))1697 6210 w
10 S f
(- p)1 159 1 1787 6210 t
7 I f
(k)1951 6170 w
7 S f
(+)1998 6170 w
7 R f
(1)2048 6170 w
10 R f
(\()2099 6210 w
10 I f
(w)2140 6210 w
10 R f
(\))2215 6210 w
10 S f
(+)2305 6210 w
10 I f
(cost)2409 6210 w
10 R f
(\()2578 6210 w
10 I f
(v)2619 6210 w
10 R f
(,)2671 6210 w
10 I f
(w)2737 6210 w
10 R f
(\) =)1 139 1 2812 6210 t
7 I f
(p *)1 75 1 3001 6310 t
7 R f
(\()3081 6310 w
7 I f
(v)3109 6310 w
7 R f
(\))3145 6310 w
15 S f
(S)3040 6240 w
10 S f
(D)3176 6210 w
10 R f
(\()3245 6210 w
(.)3286 6180 w
(,)3319 6210 w
(.)3385 6180 w
(\))3418 6210 w
10 S f
(-)3501 6210 w
7 I f
(p *)1 75 1 3605 6310 t
7 R f
(\()3685 6310 w
7 I f
(w)3713 6310 w
7 R f
(\))3765 6310 w
15 S f
(S)3652 6240 w
10 S f
(D)3796 6210 w
10 R f
(\()3865 6210 w
(.)3906 6180 w
(,)3939 6210 w
(.)4005 6180 w
(\) +)1 139 1 4038 6210 t
10 S f
(D)4227 6210 w
10 R f
(\()4296 6210 w
10 I f
(v)4337 6210 w
10 R f
(,)4389 6210 w
10 I f
(w)4455 6210 w
10 R f
(\))4530 6210 w
10 S f
(\263)4613 6210 w
10 R f
(0, since)1 331 1 4709 6210 t
7 I f
(p *)1 75 1 720 6510 t
7 R f
(\()800 6510 w
7 I f
(v)828 6510 w
7 R f
(\))864 6510 w
15 S f
(S)759 6440 w
10 S f
(D)903 6410 w
10 R f
(\()972 6410 w
(.)1013 6380 w
(,)1046 6410 w
(.)1112 6380 w
(\) +)1 118 1 1145 6410 t
10 S f
(D)1292 6410 w
10 R f
(\()1361 6410 w
10 I f
(v)1402 6410 w
10 R f
(,)1454 6410 w
10 I f
(w)1520 6410 w
10 R f
(\) is the cost of an)5 705 1 1595 6410 t
10 I f
(s)2329 6410 w
10 S f
(\242 -)1 96 1 2376 6410 t
10 I f
(w)2488 6410 w
10 R f
(path in)1 279 1 2584 6410 t
10 I f
(R)2892 6410 w
10 R f
(\()2961 6410 w
10 I f
(f)3018 6410 w
7 I f
(k)3068 6370 w
10 R f
(\) and)1 206 1 3115 6410 t
7 I f
(p *)1 75 1 3350 6510 t
7 R f
(\()3430 6510 w
7 I f
(w)3458 6510 w
7 R f
(\))3510 6510 w
15 S f
(S)3397 6440 w
10 S f
(D)3549 6410 w
10 R f
(\()3618 6410 w
(.)3659 6380 w
(,)3692 6410 w
(.)3758 6380 w
(\) is the cost of a shortest)6 995 1 3791 6410 t
10 I f
(s)4814 6410 w
10 S f
(\242 -)1 96 1 4861 6410 t
10 I f
(w)4973 6410 w
10 R f
(path in)1 275 1 720 6610 t
10 I f
(R)1020 6610 w
10 R f
(\()1089 6610 w
10 I f
(f)1146 6610 w
7 I f
(k)1196 6570 w
10 R f
(\). Thus \(i\) is proved.)4 821 1 1243 6610 t
(If)970 6766 w
10 S f
(p)1061 6766 w
7 I f
(k)1121 6726 w
7 S f
(+)1168 6726 w
7 R f
(1)1218 6726 w
10 R f
(\()1269 6766 w
10 I f
(v)1310 6766 w
10 R f
(\))1362 6766 w
10 S f
(- p)1 159 1 1452 6766 t
7 I f
(k)1616 6726 w
7 S f
(+)1663 6726 w
7 R f
(1)1713 6726 w
10 R f
(\()1764 6766 w
10 I f
(w)1805 6766 w
10 R f
(\))1880 6766 w
10 S f
(+)1970 6766 w
10 I f
(cost)2074 6766 w
10 R f
(\()2243 6766 w
10 I f
(v)2284 6766 w
10 R f
(,)2336 6766 w
10 I f
(w)2402 6766 w
10 R f
(\))2477 6766 w
10 S f
(>)2535 6766 w
10 R f
(0, then in)2 375 1 2639 6766 t
10 I f
(R)3039 6766 w
10 R f
(\()3108 6766 w
10 I f
(f)3165 6766 w
7 I f
(k)3215 6726 w
10 R f
(\),)3262 6766 w
7 I f
(p *)1 75 1 3345 6866 t
7 R f
(\()3425 6866 w
7 I f
(v)3453 6866 w
7 R f
(\))3489 6866 w
15 S f
(S)3384 6796 w
10 S f
(D)3520 6766 w
10 R f
(\()3589 6766 w
(.)3630 6736 w
(,)3663 6766 w
(.)3729 6736 w
(\))3762 6766 w
10 S f
(-)3821 6766 w
7 I f
(p *)1 75 1 3925 6866 t
7 R f
(\()4005 6866 w
7 I f
(w)4033 6866 w
7 R f
(\))4085 6866 w
15 S f
(S)3972 6796 w
10 S f
(D)4116 6766 w
10 R f
(\()4185 6766 w
(.)4226 6736 w
(,)4259 6766 w
(.)4325 6736 w
(\) +)1 115 1 4358 6766 t
10 S f
(D)4499 6766 w
10 R f
(\()4568 6766 w
10 I f
(v)4609 6766 w
10 R f
(,)4661 6766 w
10 I f
(w)4727 6766 w
10 R f
(\))4802 6766 w
10 S f
(>)4861 6766 w
10 R f
(0,)4965 6766 w
(i.e.,)720 6966 w
7 I f
(p *)1 75 1 897 7066 t
7 R f
(\()977 7066 w
7 I f
(v)1005 7066 w
7 R f
(\))1041 7066 w
15 S f
(S)936 6996 w
10 S f
(D)1080 6966 w
10 R f
(\()1149 6966 w
(.)1190 6936 w
(,)1223 6966 w
(.)1289 6936 w
(\) +)1 119 1 1322 6966 t
10 S f
(D)1471 6966 w
10 R f
(\()1540 6966 w
10 I f
(v)1581 6966 w
10 R f
(,)1633 6966 w
10 I f
(w)1699 6966 w
10 R f
(\))1774 6966 w
10 S f
(>)1837 6966 w
7 I f
(p *)1 75 1 1941 7066 t
7 R f
(\()2021 7066 w
7 I f
(w)2049 7066 w
7 R f
(\))2101 7066 w
15 S f
(S)1988 6996 w
10 S f
(D)2132 6966 w
10 R f
(\()2201 6966 w
(.)2242 6936 w
(,)2275 6966 w
(.)2341 6936 w
( implies that \()3 568(\). This)1 291 2 2374 6966 t
10 I f
(v)3241 6966 w
10 R f
(,)3293 6966 w
10 I f
(w)3359 6966 w
10 R f
( minimum-cost augmenting)2 1114(\) is not on a)4 492 2 3434 6966 t
(path in)1 275 1 720 7166 t
10 I f
(R)1020 7166 w
10 R f
(\()1089 7166 w
10 I f
(f)1146 7166 w
7 I f
(k)1196 7126 w
10 R f
(\), and, therefore,)2 661 1 1243 7166 t
10 I f
(f)1929 7166 w
7 I f
(k)1979 7126 w
7 S f
(+)2026 7126 w
7 R f
(1)2076 7126 w
10 R f
(\()2127 7166 w
10 I f
(v)2168 7166 w
10 R f
(,)2220 7166 w
10 I f
(w)2286 7166 w
10 R f
(\))2361 7166 w
10 S f
(=)2451 7166 w
10 I f
(f)2563 7166 w
7 I f
(k)2613 7126 w
10 R f
(\()2660 7166 w
10 I f
(v)2701 7166 w
10 R f
(,)2753 7166 w
10 I f
(w)2819 7166 w
10 R f
(\).)2894 7166 w
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11 pagesetup
10 R f
(- 11 -)2 216 1 2772 480 t
( the contrary that)3 1078(Assume on)1 579 2 970 840 t
10 I f
(f)2785 840 w
7 I f
(k)2835 800 w
10 R f
(\()2882 840 w
10 I f
(v)2923 840 w
10 R f
(,)2975 840 w
10 I f
(w)3041 840 w
10 R f
(\))3116 840 w
10 S f
(>)3206 840 w
10 R f
( \()1 191(0. Then)1 463 2 3310 840 t
10 I f
(w)3972 840 w
10 R f
(,)4047 840 w
10 I f
(v)4113 840 w
10 R f
(\))4165 840 w
10 S f
(\316)4247 840 w
10 I f
(R)4359 840 w
10 R f
(\()4428 840 w
10 I f
(f)4485 840 w
7 I f
(k)4535 800 w
10 R f
(\). From)1 458 1 4582 840 t
10 S f
(p)720 960 w
7 I f
(k)780 920 w
7 S f
(+)827 920 w
7 R f
(1)877 920 w
10 R f
(\()928 960 w
10 I f
(v)969 960 w
10 R f
(\))1021 960 w
10 S f
(- p)1 159 1 1111 960 t
7 I f
(k)1275 920 w
7 S f
(+)1322 920 w
7 R f
(1)1372 920 w
10 R f
(\()1423 960 w
10 I f
(w)1464 960 w
10 R f
(\))1539 960 w
10 S f
(+)1629 960 w
10 I f
(cost)1733 960 w
10 R f
(\()1902 960 w
10 I f
(v)1943 960 w
10 R f
(,)1995 960 w
10 I f
(w)2061 960 w
10 R f
(\))2136 960 w
10 S f
(>)2226 960 w
10 R f
(0, we have)2 509 1 2330 960 t
10 S f
(p)2904 960 w
7 I f
(k)2964 920 w
7 S f
(+)3011 920 w
7 R f
(1)3061 920 w
10 R f
(\()3112 960 w
10 I f
(w)3153 960 w
10 R f
(\))3228 960 w
10 S f
(-)3318 960 w
10 I f
(cost)3422 960 w
10 R f
(\()3591 960 w
10 I f
(v)3632 960 w
10 R f
(,)3684 960 w
10 I f
(w)3750 960 w
10 R f
(\) =)1 154 1 3825 960 t
10 S f
(p)4044 960 w
7 I f
(k)4104 920 w
7 S f
(+)4151 920 w
7 R f
(1)4201 920 w
10 R f
(\()4252 960 w
10 I f
(w)4293 960 w
10 R f
(\) +)1 153 1 4368 960 t
10 S f
(D)4585 960 w
10 R f
(\()4654 960 w
10 I f
(w)4695 960 w
10 R f
(,)4770 960 w
10 I f
(v)4836 960 w
10 R f
(\))4888 960 w
10 S f
(<)4985 960 w
(p)720 1080 w
7 I f
(k)780 1040 w
7 S f
(+)827 1040 w
7 R f
(1)877 1040 w
10 R f
(\()928 1080 w
10 I f
(v)969 1080 w
10 R f
(\), where)1 343 1 1021 1080 t
10 S f
(D)1406 1080 w
10 R f
(\()1475 1080 w
10 I f
(w)1516 1080 w
10 R f
(,)1591 1080 w
10 I f
(v)1657 1080 w
10 R f
(\) is the value of edge \()6 994 1 1709 1080 t
10 I f
(w)2711 1080 w
10 R f
(,)2786 1080 w
10 I f
(v)2852 1080 w
10 R f
( graph)1 270(\) of the cost function in the residual)7 1542 2 2904 1080 t
10 I f
(R)4759 1080 w
10 R f
(\()4828 1080 w
10 I f
(f)4885 1080 w
7 I f
(k)4935 1040 w
10 R f
(\).)4982 1080 w
(Therefore,)720 1200 w
10 S f
(p)1164 1200 w
7 I f
(k)1224 1160 w
7 S f
(+)1271 1160 w
7 R f
(1)1321 1160 w
10 R f
(\()1372 1200 w
10 I f
(v)1413 1200 w
10 R f
(\) is not the cost of a shortest path from)9 1558 1 1465 1200 t
10 I f
(s)3049 1200 w
10 S f
(\242)3096 1200 w
10 R f
(to)3147 1200 w
10 I f
(v)3251 1200 w
10 R f
(in the residual graph)3 821 1 3321 1200 t
10 I f
(R)4168 1200 w
10 R f
(\()4237 1200 w
10 I f
(f)4294 1200 w
7 I f
(k)4344 1160 w
10 R f
(\) with respect to)3 649 1 4391 1200 t
(the cost function)2 730 1 720 1320 t
10 S f
(D)1507 1320 w
10 R f
(of)1625 1320 w
10 I f
(R)1765 1320 w
10 R f
(\()1834 1320 w
10 I f
(f)1891 1320 w
7 I f
(k)1941 1280 w
10 R f
( Therefore,)1 500( contradiction to the first part of the lemma.)8 2009( is a)2 225(\). This)1 318 4 1988 1320 t
10 I f
(f)720 1440 w
7 I f
(k)770 1400 w
7 S f
(+)817 1400 w
7 R f
(1)867 1400 w
10 R f
(\()918 1440 w
10 I f
(v)959 1440 w
10 R f
(,)1011 1440 w
10 I f
(w)1077 1440 w
10 R f
(\))1152 1440 w
10 S f
(=)1242 1440 w
10 I f
(f)1354 1440 w
7 I f
(k)1404 1400 w
10 R f
(\()1451 1440 w
10 I f
(v)1492 1440 w
10 R f
(,)1544 1440 w
10 I f
(w)1610 1440 w
10 R f
(\))1685 1440 w
10 S f
(=)1775 1440 w
10 R f
(0, and this proves \(ii\).)4 877 1 1879 1440 t
10 S1 f
()2806 1440 w
2806 1440 m
50 build_sq
2856 1440 m
10 R f
(The total number of flow augmentations is)6 1704 1 970 1596 t
10 I f
(max)1878 1961 w
10 R f
(\()2085 1961 w
10 S f
(\357)2118 1978 w
10 I f
(S)2166 1961 w
10 S1 f
(_)2171 1868 w
10 S f
(\357)2216 1978 w
10 R f
(,)2264 1961 w
10 S f
(\357)2322 1978 w
10 I f
(D)2370 1961 w
10 S1 f
(_ _)1 62 1 2380 1868 t
10 S f
(\357)2442 1978 w
10 R f
(\))2490 1961 w
10 S f
(\354)2580 1774 w
(\357)2580 1874 w
(\355)2580 1974 w
(\357)2580 2074 w
(\356)2580 2174 w
10 R f
(2)2629 1961 w
10 S f
(+)2728 1961 w
(\351)2840 1824 w
(\357)2840 1924 w
(\357)2840 2024 w
(\353)2840 2124 w
10 R f
(log)2890 2024 w
10 I f
(max)3084 2119 w
10 R f
(\()3291 2119 w
10 S f
(\357)3324 2136 w
10 I f
(S)3372 2119 w
10 S1 f
(_)3377 2026 w
10 S f
(\357)3422 2136 w
10 R f
(,)3470 2119 w
10 S f
(\357)3528 2136 w
10 I f
(D)3576 2119 w
10 S1 f
(_ _)1 62 1 3586 2026 t
10 S f
(\357)3648 2136 w
10 R f
(\))3696 2119 w
7 I f
(v)3156 1964 w
7 S f
(\316)3215 1964 w
7 I f
(S)3293 1964 w
7 S1 f
(_)3299 1892 w
15 S f
(S)3197 1877 w
10 I f
(u)3327 1847 w
10 S1 f
(_)3329 1779 w
10 R f
(\()3385 1847 w
10 I f
(s)3426 1847 w
10 S f
(\242)3473 1847 w
10 R f
(,)3506 1847 w
10 I f
(v)3572 1847 w
10 R f
(\))3624 1847 w
10 S1 f
(_ _____________)1 675 1 3069 1994 t
10 S f
(\371)3754 1824 w
(\357)3754 1924 w
(\357)3754 2024 w
(\373)3754 2124 w
(\374)3792 1774 w
(\357)3792 1874 w
(\375)3792 1974 w
(\357)3792 2074 w
(\376)3792 2174 w
10 I f
(.)3857 1961 w
10 R f
(Since)720 2351 w
7 I f
(v)967 2468 w
7 S f
(\316)1026 2468 w
7 I f
(S)1104 2468 w
7 S1 f
(_)1110 2396 w
15 S f
(S)1008 2381 w
10 I f
(u)1138 2351 w
10 S1 f
(_)1140 2283 w
10 R f
(\()1196 2351 w
10 I f
(s)1237 2351 w
10 S f
(\242)1284 2351 w
10 R f
(,)1317 2351 w
10 I f
(v)1383 2351 w
10 R f
(\))1435 2351 w
10 S f
(=)1525 2351 w
10 I f
(O)1629 2351 w
10 R f
(\()1709 2351 w
10 I f
(m)1750 2351 w
10 R f
(\) and)1 202 1 1830 2351 t
10 I f
(max)2057 2351 w
10 R f
(\()2264 2351 w
10 S f
(\357)2297 2368 w
10 I f
(S)2345 2351 w
10 S1 f
(_)2350 2258 w
10 S f
(\357)2395 2368 w
10 R f
(,)2443 2351 w
10 S f
(\357)2501 2368 w
10 I f
(D)2549 2351 w
10 S1 f
(_ _)1 62 1 2559 2258 t
10 S f
(\357)2621 2368 w
10 R f
(\))2669 2351 w
10 S f
(=)2759 2351 w
10 I f
(O)2863 2351 w
10 R f
(\()2943 2351 w
10 I f
(n)2984 2351 w
10 R f
(\),)3042 2351 w
10 I f
(max)1533 2813 w
10 R f
(\()1740 2813 w
10 S f
(\357)1773 2830 w
10 I f
(S)1821 2813 w
10 S1 f
(_)1826 2720 w
10 S f
(\357)1871 2830 w
10 R f
(,)1919 2813 w
10 S f
(\357)1977 2830 w
10 I f
(D)2025 2813 w
10 S1 f
(_ _)1 62 1 2035 2720 t
10 S f
(\357)2097 2830 w
10 R f
(\))2145 2813 w
10 S f
(\354)2235 2626 w
(\357)2235 2726 w
(\355)2235 2826 w
(\357)2235 2926 w
(\356)2235 3026 w
10 R f
(2)2284 2813 w
10 S f
(+)2383 2813 w
(\351)2495 2676 w
(\357)2495 2776 w
(\357)2495 2876 w
(\353)2495 2976 w
10 R f
(log)2545 2876 w
10 I f
(max)2739 2971 w
10 R f
(\()2946 2971 w
10 S f
(\357)2979 2988 w
10 I f
(S)3027 2971 w
10 S1 f
(_)3032 2878 w
10 S f
(\357)3077 2988 w
10 R f
(,)3125 2971 w
10 S f
(\357)3183 2988 w
10 I f
(D)3231 2971 w
10 S1 f
(_ _)1 62 1 3241 2878 t
10 S f
(\357)3303 2988 w
10 R f
(\))3351 2971 w
7 I f
(v)2811 2816 w
7 S f
(\316)2870 2816 w
7 I f
(S)2948 2816 w
7 S1 f
(_)2954 2744 w
15 S f
(S)2852 2729 w
10 I f
(u)2982 2699 w
10 S1 f
(_)2984 2631 w
10 R f
(\()3040 2699 w
10 I f
(s)3081 2699 w
10 S f
(\242)3128 2699 w
10 R f
(,)3161 2699 w
10 I f
(v)3227 2699 w
10 R f
(\))3279 2699 w
10 S1 f
(_ _____________)1 675 1 2724 2846 t
10 S f
(\371)3409 2676 w
(\357)3409 2776 w
(\357)3409 2876 w
(\373)3409 2976 w
(\374)3447 2626 w
(\357)3447 2726 w
(\375)3447 2826 w
(\357)3447 2926 w
(\376)3447 3026 w
(=)3553 2813 w
10 I f
(O)3657 2813 w
10 R f
(\()3737 2813 w
10 I f
(n)3778 2813 w
10 R f
(log)3869 2813 w
10 I f
(n)4074 2883 w
(m)4063 2753 w
10 S1 f
(_ __)1 102 1 4048 2783 t
10 R f
(\).)4168 2813 w
(Each augmentation takes time)3 1231 1 720 3198 t
10 I f
(O)1986 3198 w
10 R f
(\()2066 3198 w
10 I f
(m)2107 3198 w
10 S f
(+)2228 3198 w
10 I f
(n)2332 3198 w
10 R f
(log)2423 3198 w
10 I f
(n)2592 3198 w
10 R f
( the total cost for solving the)6 1210(\) using Fibonacci heaps, and)4 1180 2 2650 3198 t
(min-cost max-flow problem is)3 1213 1 720 3318 t
10 I f
(O)1958 3318 w
10 R f
(\()2038 3318 w
10 I f
(n)2079 3318 w
10 R f
(\()2137 3318 w
10 I f
(m)2178 3318 w
10 S f
(+)2299 3318 w
10 I f
(n)2403 3318 w
10 R f
(log)2494 3318 w
10 I f
(n)2663 3318 w
10 R f
( \()1 41(\) log)1 210 2 2721 3318 t
10 I f
(m / n)2 166 1 2980 3318 t
10 R f
( Therefore,)1 442(\) \).)1 99 2 3154 3318 t
10 B f
(Theorem 5.2.)1 565 1 720 3474 t
10 R f
(The positive-flow and the covering-paths problems)5 2054 1 1312 3474 t
10 I f
(F)3394 3474 w
7 I f
(i)3460 3493 w
7 R f
(\( 2 \))2 91 1 3460 3434 t
10 R f
(and)3587 3474 w
10 I f
(P)3759 3474 w
7 I f
(i)3825 3493 w
7 R f
(\( 2 \))2 91 1 3825 3434 t
10 R f
(,)3924 3474 w
10 I f
(i)3977 3474 w
10 S f
(=)4054 3474 w
10 R f
( can be solved in)4 683( 3,)1 116(2 ,)1 83 3 4158 3474 t
(time)720 3594 w
10 I f
(O)923 3594 w
10 R f
(\()1003 3594 w
10 I f
(n)1044 3594 w
10 R f
(\()1102 3594 w
10 I f
(m)1143 3594 w
10 S f
(+)1264 3594 w
10 I f
(n)1368 3594 w
10 R f
(log)1459 3594 w
10 I f
(n)1628 3594 w
10 R f
( \()1 41(\) log)1 210 2 1686 3594 t
10 I f
(m / n)2 166 1 1945 3594 t
10 R f
(\) \).)1 99 1 2119 3594 t
10 S1 f
()2268 3594 w
2268 3594 m
50 build_sq
2318 3594 m
10 R f
(The following lemma shows that this bound is even tighter than)10 2589 1 970 3750 t
10 I f
(O)3589 3750 w
10 R f
(\()3669 3750 w
10 I f
(mn)3710 3750 w
10 R f
(log)3873 3750 w
10 I f
(n)4042 3750 w
10 R f
(\), and we omit the rou-)5 940 1 4100 3750 t
(tine proof.)1 416 1 720 3870 t
10 B f
(Lemma 5.3.)1 563 1 720 4030 t
10 R f
(Let)1369 4030 w
10 I f
(m)1588 4030 w
10 R f
(be of order)2 559 1 1746 4030 t
10 I f
(n)2391 4030 w
7 S f
(l)2452 3976 w
4 R f
(0)2497 3990 w
10 R f
(\()2538 4030 w
10 I f
(log n)1 219 1 2579 4030 t
10 R f
(\))2806 4030 w
7 S f
(l)2850 3976 w
4 R f
(1)2895 3990 w
10 R f
(\()2936 4030 w
10 I f
(log)2977 4030 w
10 R f
(log)3146 4030 w
10 I f
(n)3315 4030 w
10 R f
(\))3373 4030 w
7 S f
(l)3417 3976 w
4 R f
(1)3462 3990 w
10 R f
(. . .)2 91 1 3503 4000 t
(, where 1)2 490 1 3594 4030 t
10 S f
(\243 l)1 151 1 4125 4030 t
7 R f
(0)4287 4050 w
10 S f
(\243)4371 4030 w
10 R f
( \(i\))1 181(2. Then)1 392 2 4467 4030 t
10 I f
(n)720 4150 w
10 R f
(\()778 4150 w
10 I f
(m)819 4150 w
10 S f
(+)940 4150 w
10 I f
(n)1044 4150 w
10 R f
(log)1135 4150 w
10 I f
(n)1304 4150 w
10 R f
( \()1 41(\) log)1 210 2 1362 4150 t
10 I f
(m / n)2 166 1 1621 4150 t
10 R f
(\) is of order)3 528 1 1795 4150 t
10 I f
(O)2368 4150 w
10 R f
(\()2448 4150 w
10 I f
(mn)2489 4150 w
10 R f
(log)2652 4150 w
10 I f
(n)2821 4150 w
10 R f
(\); \(ii\) for)2 389 1 2879 4150 t
10 S f
(l)3313 4150 w
7 R f
(0)3379 4170 w
10 S f
(>)3471 4150 w
10 R f
(1,)3575 4150 w
10 I f
(n)3694 4150 w
10 R f
(\()3752 4150 w
10 I f
(m)3793 4150 w
10 S f
(+)3914 4150 w
10 I f
(n)4018 4150 w
10 R f
(log)4109 4150 w
10 I f
(n)4278 4150 w
10 R f
( \()1 41(\) log)1 210 2 4336 4150 t
10 I f
(m / n)2 166 1 4595 4150 t
10 R f
(\) is of)2 271 1 4769 4150 t
(order)720 4270 w
10 S f
(W)955 4270 w
10 R f
(\()1040 4270 w
10 I f
(mn)1081 4270 w
10 R f
(log)1244 4270 w
10 I f
(n)1413 4270 w
10 R f
(\); and \(iii\) for)3 546 1 1471 4270 t
10 S f
(l)2042 4270 w
7 R f
(0)2108 4290 w
10 S f
(=)2200 4270 w
10 R f
(1,)2304 4270 w
7 I f
(n)2120 4690 w
7 S f
(\256 \245)1 148 1 2183 4690 t
10 R f
(lim)2158 4620 w
10 I f
(mn)2679 4690 w
10 R f
(log)2842 4690 w
10 I f
(n)3011 4690 w
(n)2363 4490 w
10 R f
(\()2421 4490 w
10 I f
(m)2462 4490 w
10 S f
(+)2583 4490 w
10 I f
(n)2687 4490 w
10 R f
(log)2778 4490 w
10 I f
(n)2947 4490 w
10 R f
(\) log)1 210 1 3005 4490 t
10 I f
(n)3292 4560 w
(m)3281 4430 w
10 S1 f
(_ __)1 102 1 3266 4460 t
(_ ____________________)1 1045 1 2348 4590 t
10 S f
(=)3452 4620 w
10 R f
(0)3556 4620 w
10 I f
(.)3614 4620 w
10 S1 f
()720 4850 w
720 4850 m
50 build_sq
770 4850 m
10 B f
( Problems)1 430(6. Minimum-Circulation)1 1075 2 720 5090 t
10 R f
( flow and covering-paths problems on a directed graph)8 2192(We now study)2 577 2 970 5246 t
10 I f
(G)3765 5246 w
7 I f
(O)3848 5266 w
10 R f
(without a source)2 662 1 3932 5246 t
10 I f
(s)4620 5246 w
10 R f
(or sink)1 276 1 4685 5246 t
10 I f
(t)4987 5246 w
10 R f
(.)5015 5246 w
( circulation on)2 593( A)1 130( in Section 1.)3 552(Otherwise, the capacity bounds and costs are the same as)9 2359 4 720 5366 t
10 I f
(G)4387 5366 w
10 R f
(is a preflow)2 487 1 4492 5366 t
10 I f
(f)5012 5366 w
10 R f
(such that)1 358 1 720 5486 t
10 I f
(l)1103 5486 w
10 R f
(\()1139 5486 w
10 I f
(e)1180 5486 w
10 R f
(\))1232 5486 w
10 S f
(\243)1314 5486 w
10 I f
(f)1418 5486 w
10 R f
(\()1462 5486 w
10 I f
(e)1503 5486 w
10 R f
(\))1555 5486 w
10 S f
(\243)1637 5486 w
10 I f
(u)1733 5486 w
10 R f
(\()1791 5486 w
10 I f
(e)1832 5486 w
10 R f
(\) for all)2 301 1 1884 5486 t
10 I f
(e)2211 5486 w
10 R f
(in)2281 5486 w
10 I f
(E)2385 5486 w
10 R f
(and such that the balancing index)5 1339 1 2472 5486 t
10 S f
(b)3837 5486 w
10 R f
(\()3900 5486 w
10 I f
(v)3941 5486 w
10 R f
(,)3993 5486 w
10 I f
(f)4067 5486 w
10 R f
(\))4119 5486 w
10 S f
(=)4209 5486 w
10 R f
(0 for all)2 318 1 4313 5486 t
10 I f
(v)4657 5486 w
10 R f
(in)4727 5486 w
10 I f
(V)4831 5486 w
10 R f
(. A)1 148 1 4892 5486 t
(distinguished vertex)1 833 1 720 5606 t
10 I f
(O)1603 5606 w
10 R f
(is called the)2 527 1 1725 5606 t
10 I f
(origin)2301 5606 w
10 R f
( flow value out of)4 806(. The)1 254 2 2546 5606 t
10 I f
(O)3655 5606 w
10 R f
(\(or equivalently, into)2 889 1 3776 5606 t
10 I f
(O)4714 5606 w
10 R f
(\), i.e.,)1 254 1 4786 5606 t
7 R f
(\()720 5826 w
7 I f
(O)748 5826 w
7 R f
(,)803 5826 w
7 I f
(v)849 5826 w
7 R f
(\))885 5826 w
7 S f
(\316)942 5826 w
7 I f
(E)1020 5826 w
15 S f
(S)847 5756 w
10 I f
(f)1079 5726 w
10 R f
(\()1123 5726 w
10 I f
(O)1164 5726 w
10 R f
(,)1244 5726 w
10 I f
(v)1310 5726 w
10 R f
(\), is called the)3 560 1 1362 5726 t
10 I f
(repetition)1947 5726 w
10 R f
(of the circulation. We study the following circulation problems:)8 2550 1 2361 5726 t
10 B f
(Problem)720 6022 w
10 I f
(C)1111 6022 w
7 R f
(1)1183 6041 w
(\( 0 \))2 91 1 1183 5982 t
10 R f
(.)1282 6022 w
10 I f
(Minimum circulation)1 847 1 1357 6022 t
10 R f
(.)2229 6022 w
(Find a circulation for)3 846 1 970 6178 t
10 I f
(G)1841 6178 w
10 R f
(with minimum repetition.)2 1026 1 1938 6178 t
10 B f
(Problem)720 6394 w
10 I f
(C)1111 6394 w
7 R f
(2)1183 6413 w
(\( 0 \))2 91 1 1183 6354 t
10 R f
(.)1282 6394 w
10 I f
(Minimum-cost minimum circulation)2 1438 1 1357 6394 t
10 R f
(.)2795 6394 w
(Among the minimum circulations of)4 1461 1 970 6550 t
10 I f
(G)2456 6550 w
10 R f
(, find one with minimum cost.)5 1209 1 2528 6550 t
10 B f
(Problem)720 6766 w
10 I f
(C)1111 6766 w
7 R f
(3)1183 6785 w
(\( 0 \))2 91 1 1183 6726 t
10 R f
(.)1282 6766 w
10 I f
(Minimum-cost circulation)1 1041 1 1357 6766 t
10 R f
(.)2398 6766 w
(Find a circulation for)3 846 1 970 6922 t
10 I f
(G)1841 6922 w
10 R f
(with minimum cost.)2 804 1 1938 6922 t
( algorithms exist for Problem)4 1184(Strongly polynomial)1 828 2 970 7198 t
10 I f
(C)3010 7198 w
7 R f
(3)3082 7217 w
(\( 0 \))2 91 1 3082 7158 t
10 R f
( gives an algo-)3 594( Balancing)1 458(; see Tardos [1985].)3 807 3 3181 7198 t
( algorithms for the)3 744( it yields better)3 603( However,)1 442(rithm with a run time depending on the capacity lower bounds.)10 2531 4 720 7318 t
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12 pagesetup
10 R f
(- 12 -)2 216 1 2772 480 t
(postman-tour problems.)1 955 1 720 840 t
(We can replace the origin)4 1056 1 970 996 t
10 I f
(O)2059 996 w
10 R f
(by two vertices)2 626 1 2164 996 t
10 I f
(s)2824 996 w
10 R f
(and)2897 996 w
10 I f
(t)3075 996 w
10 R f
(, and replace edges \()4 852 1 3103 996 t
10 I f
(O)3963 996 w
10 R f
(,)4043 996 w
10 I f
(v)4109 996 w
10 R f
(\) by \()2 234 1 4161 996 t
10 I f
(s)4403 996 w
10 R f
(,)4450 996 w
10 I f
(v)4516 996 w
10 R f
(\) and edges)2 472 1 4568 996 t
(\()720 1116 w
10 I f
(w)761 1116 w
10 R f
(,)836 1116 w
10 I f
(O)902 1116 w
10 R f
( \()1 59(\) by)1 160 2 982 1116 t
10 I f
(w)1209 1116 w
10 R f
(,)1284 1116 w
10 I f
(t)1350 1116 w
10 R f
( this way, the minimum-circulation problems)5 1810( In)1 134(\) with the same capacity bounds and costs.)7 1710 3 1386 1116 t
10 I f
(C)720 1236 w
7 I f
(i)792 1255 w
7 R f
(\( 0 \))2 91 1 792 1196 t
10 R f
(can be reduced to the minimum-flow problems)6 1881 1 917 1236 t
10 I f
(F)2824 1236 w
7 I f
(i)2890 1255 w
7 R f
(\( 0 \))2 91 1 2890 1196 t
10 R f
(,)2989 1236 w
10 I f
(i)3040 1236 w
10 S f
(=)3117 1236 w
10 R f
( and Algorithms 1)3 728( 3,)1 116( ,)1 33( 2)1 91(1 ,)1 83 5 3221 1236 t
10 S f
(-)4288 1236 w
10 R f
( be applied.)2 467(3 can)1 214 2 4359 1236 t
(Thus,)720 1356 w
10 B f
(Theorem 6.1.)1 602 1 720 1512 t
10 R f
(The minimum-circulation problem)2 1466 1 1386 1512 t
10 I f
(C)2916 1512 w
7 R f
(1)2988 1531 w
(\( 0 \))2 91 1 2988 1472 t
10 R f
(can be solved in time)4 1009 1 3152 1512 t
10 I f
(O)4226 1512 w
10 R f
(\()4306 1512 w
10 I f
(mn)4347 1512 w
10 R f
(log)4510 1512 w
10 I f
(n)4679 1512 w
10 R f
(\). The)1 303 1 4737 1512 t
(minimum-cost minimum-circulation problem)2 1835 1 720 1632 t
10 I f
(C)2589 1632 w
7 R f
(2)2661 1651 w
(\( 0 \))2 91 1 2661 1592 t
10 R f
( circulation problem)2 826(and the minimum-cost)2 918 2 2794 1632 t
10 I f
(C)4571 1632 w
7 R f
(3)4643 1651 w
(\( 0 \))2 91 1 4643 1592 t
10 R f
(can be)1 265 1 4775 1632 t
(solved in time)2 617 1 720 1752 t
10 I f
(O)1387 1752 w
10 R f
(\()1467 1752 w
10 I f
(mn)1508 1752 w
10 R f
(log)1671 1752 w
10 I f
(n)1840 1752 w
10 S f
(+)1939 1752 w
10 R f
(\()2043 1752 w
10 I f
(m)2084 1752 w
10 S f
(+)2205 1752 w
10 I f
(n)2309 1752 w
10 R f
(log)2400 1752 w
10 I f
(n)2569 1752 w
10 R f
(\))2627 1752 w
10 I f
(L)2676 1752 w
10 R f
( and in time)3 553(\) if minimum-cost augmentation is used,)5 1747 2 2740 1752 t
10 I f
(O)720 1872 w
10 R f
(\()800 1872 w
10 I f
(m)841 1872 w
10 R f
(\()921 1872 w
10 I f
(m)962 1872 w
10 S f
(+)1083 1872 w
10 I f
(n)1187 1872 w
10 R f
(log)1278 1872 w
10 I f
(n)1447 1872 w
10 R f
(\) log)1 210 1 1505 1872 t
10 I f
(L)1756 1872 w
10 R f
( if scaling is used, where)5 987(\) \))1 74 2 1820 1872 t
10 I f
(L)2906 1872 w
10 S f
(=)3011 1872 w
7 I f
(e)3115 1972 w
7 S f
(\316)3174 1972 w
7 I f
(E)3252 1972 w
15 S f
(S)3160 1902 w
10 I f
(l)3294 1872 w
10 R f
(\()3330 1872 w
10 I f
(e)3371 1872 w
10 R f
(\).)3423 1872 w
10 S1 f
()3531 1872 w
3531 1872 m
50 build_sq
3581 1872 m
10 B f
( Capacity Upper)2 709( Positive-Circulation and Postman-Tour Problems on Graphs with or without)9 3318(7. The)1 293 3 720 2192 t
(Bounds)720 2312 w
10 R f
( the minimum-circulation problems, we can define positive-circulation)7 2887(Similarly, as special cases of)4 1183 2 970 2468 t
(problems, where the capacity lower bound of every edge)8 2295 1 720 2588 t
10 I f
(e)3044 2588 w
10 R f
(in)3117 2588 w
10 I f
(E)3224 2588 w
10 R f
(is)3313 2588 w
10 I f
(l)3408 2588 w
10 R f
(\()3444 2588 w
10 I f
(e)3485 2588 w
10 R f
(\))3537 2588 w
10 S f
(=)3627 2588 w
10 R f
( denote the corresponding)3 1043(1. We)1 266 2 3731 2588 t
(positive-circulation problems by)2 1299 1 720 2708 t
10 I f
(C)2044 2708 w
7 I f
(i)2116 2727 w
7 R f
(\( 1 \))2 91 1 2116 2668 t
10 R f
(,)2215 2708 w
10 I f
(i)2265 2708 w
10 S f
(=)2342 2708 w
10 R f
( 3.)1 116( ,)1 33( 2)1 91(1 ,)1 83 4 2446 2708 t
(Let)970 2864 w
10 I f
(G)1128 2864 w
7 I f
(O)1211 2884 w
10 R f
(be a directed graph with origin)5 1228 1 1294 2864 t
10 I f
(O)2547 2864 w
10 R f
(. A)1 147 1 2619 2864 t
10 I f
(closed)2791 2864 w
10 R f
(path in)1 275 1 3071 2864 t
10 I f
(G)3371 2864 w
7 I f
(O)3454 2884 w
10 R f
(is one in which the beginning and the)7 1503 1 3537 2864 t
(ending vertex is the same origin)5 1284 1 720 2984 t
10 I f
(O)2030 2984 w
10 R f
(. A)1 123 1 2102 2984 t
10 I f
(postman tour)1 547 1 2251 2984 t
10 R f
( uses every edge at least once in the)8 1424(is a closed path that)4 792 2 2824 2984 t
( to the covering-paths problems)4 1277( Similar)1 348( the edges [Kwan, 1960; Edmonds and Johnson, 1973].)8 2230(direction of)1 465 4 720 3104 t
10 I f
(CP)720 3224 w
7 I f
(i)853 3243 w
7 R f
(\( 1 \))2 91 1 853 3184 t
10 R f
(,)952 3224 w
10 I f
(i)1002 3224 w
10 S f
(=)1079 3224 w
10 R f
( we can define the following postman-tour problems on)8 2219( 3,)1 116( ,)1 33( 2)1 91(1 ,)1 83 5 1183 3224 t
10 I f
(G)3750 3224 w
7 I f
(O)3833 3244 w
10 R f
(:)3891 3224 w
10 B f
(Problem)720 3380 w
10 I f
(PT)1111 3380 w
7 R f
(1)1233 3399 w
(\( 1 \))2 91 1 1233 3340 t
10 R f
(.)1332 3380 w
10 I f
(Least-repetitious postman tour)2 1234 1 1407 3380 t
10 R f
(.)2641 3380 w
(Find a postman tour with minimum repetition.)6 1854 1 970 3536 t
10 B f
(Problem)720 3692 w
10 I f
(PT)1111 3692 w
7 R f
(2)1233 3711 w
(\( 1 \))2 91 1 1233 3652 t
10 R f
(.)1332 3692 w
10 I f
(Minimum-cost least-repetitious postman tour)3 1808 1 1382 3692 t
10 R f
(.)3190 3692 w
(Among the least-repetitious postman tours, find one with minimum cost.)9 2908 1 970 3848 t
10 B f
(Problem)720 4004 w
10 I f
(PT)1111 4004 w
7 R f
(3)1233 4023 w
(\( 1 \))2 91 1 1233 3964 t
10 R f
(.)1332 4004 w
10 I f
(Minimum-cost postman tour)2 1133 1 1407 4004 t
10 R f
(.)2540 4004 w
(Find a postman tour with minimum cost.)6 1632 1 970 4160 t
( positive-circulation prob-)2 1059(As special cases of the positive-circulation problems, we can consider the)10 3011 2 970 4436 t
( \(or)1 153(lems on graphs without capacity upper bound)6 1898 2 720 4556 t
10 I f
(u)2808 4556 w
10 R f
(\()2866 4556 w
10 I f
(e)2907 4556 w
10 R f
(\))2959 4556 w
10 S f
(= \245)1 177 1 3049 4556 t
10 R f
(for all)1 253 1 3263 4556 t
10 I f
(e)3553 4556 w
10 R f
(in)3634 4556 w
10 I f
(E)3749 4556 w
10 R f
( denote the three corre-)4 972(\). We)1 258 2 3810 4556 t
( by)1 132(sponding problems)1 770 2 720 4676 t
10 I f
(C)1654 4676 w
7 I f
(i)1726 4695 w
7 R f
(\( 2 \))2 91 1 1726 4636 t
10 R f
(,)1825 4676 w
10 I f
(i)1882 4676 w
10 S f
(=)1959 4676 w
10 R f
( can also define postman-tour problems on such graphs, and)9 2459( We)1 195( 3.)1 116( ,)1 33( 2)1 91(1 ,)1 83 6 2063 4676 t
(denote the corresponding problems by)4 1531 1 720 4796 t
10 I f
(PT)2276 4796 w
7 I f
(i)2398 4815 w
7 R f
(\( 2 \))2 91 1 2398 4756 t
10 R f
(,)2497 4796 w
10 I f
(i)2547 4796 w
10 S f
(=)2624 4796 w
10 R f
( 3.)1 116(2 ,)1 83 2 2728 4796 t
( equivalent to the corresponding)4 1311(Similarly, it can be shown that the positive-circulation problems are)9 2759 2 970 4952 t
( corresponding positive-circulation and)3 1570(postman-tour problems, and the transformation of solutions between)7 2750 2 720 5072 t
(postman-tour problems can be done in time)6 1737 1 720 5192 t
10 I f
(O)2482 5192 w
10 R f
(\()2562 5192 w
10 I f
(mn)2603 5192 w
10 R f
(\).)2733 5192 w
( to that for the positive-flow and the covering-paths problems, we omit)11 2896(Since the approach is similar)4 1174 2 970 5348 t
( summarize:)1 491( We)1 188(the details.)1 433 3 720 5468 t
10 B f
(Theorem 7.1.)1 564 1 720 5624 t
10 R f
( and the postman-tour problems)4 1279(The positive-circulation)1 958 2 1310 5624 t
10 I f
(C)3574 5624 w
7 R f
(1)3646 5643 w
(\()3646 5584 w
7 I f
(j)3685 5584 w
7 R f
(\))3710 5584 w
10 R f
(and)3768 5624 w
10 I f
(PT)3939 5624 w
7 R f
(1)4061 5643 w
(\()4061 5584 w
7 I f
(j)4100 5584 w
7 R f
(\))4125 5584 w
10 R f
(can be solved in time)4 857 1 4183 5624 t
10 I f
(O)720 5744 w
10 R f
(\()800 5744 w
10 I f
(mn)841 5744 w
10 R f
(\),)971 5744 w
10 I f
(j)1055 5744 w
10 S f
(=)1132 5744 w
10 R f
( postman-tour problems)2 955( positive-circulation and the)3 1121( The)1 206( 2.)1 116(1 ,)1 83 5 1236 5744 t
10 I f
(C)3742 5744 w
7 I f
(i)3814 5763 w
7 R f
(\()3814 5704 w
7 I f
(j)3853 5704 w
7 R f
(\))3878 5704 w
10 R f
(and)3934 5744 w
10 I f
(PT)4103 5744 w
7 I f
(i)4225 5763 w
7 R f
(\()4225 5704 w
7 I f
(j)4264 5704 w
7 R f
(\))4289 5704 w
10 R f
(,)4320 5744 w
10 I f
(i)4370 5744 w
10 S f
(=)4447 5744 w
10 R f
( can be)2 282( ,)1 33( 3)1 91(2 ,)1 83 4 4551 5744 t
(solved in time)2 567 1 720 5864 t
10 I f
(O)1312 5864 w
10 R f
(\()1392 5864 w
10 I f
(m)1433 5864 w
10 R f
(\()1513 5864 w
10 I f
(m)1554 5864 w
10 S f
(+)1675 5864 w
10 I f
(n)1779 5864 w
10 R f
(log)1870 5864 w
10 I f
(n)2039 5864 w
10 R f
( if)1 86(\) \))1 74 2 2097 5864 t
10 I f
(j)2282 5864 w
10 S f
(=)2359 5864 w
10 R f
(1, and in time)3 550 1 2463 5864 t
10 I f
(O)3038 5864 w
10 R f
(\()3118 5864 w
10 I f
(n)3159 5864 w
10 R f
(\()3217 5864 w
10 I f
(m)3258 5864 w
10 S f
(+)3379 5864 w
10 I f
(n)3483 5864 w
10 R f
(log)3574 5864 w
10 I f
(n)3743 5864 w
10 R f
( \()1 41(\) log)1 210 2 3801 5864 t
10 I f
(m / n)2 166 1 4060 5864 t
10 R f
( if)1 86(\) \))1 74 2 4234 5864 t
10 I f
(j)4419 5864 w
10 S f
(=)4496 5864 w
10 R f
(2.)4600 5864 w
10 S1 f
()720 6020 w
720 6020 m
50 build_sq
770 6020 m
10 R f
(Problem)970 6176 w
10 I f
(PT)1355 6176 w
7 R f
(3)1477 6195 w
(\( 2 \))2 91 1 1477 6136 t
10 R f
( was used to)3 557( Balancing)1 477(is the classical Chinese-postman problem [Kwan, 1960].)6 2384 3 1622 6176 t
( a minimum-cost maximum-flow problem on a balancing graph [Gibbons, 1985], and)11 3438(reduce the problem to)3 882 2 720 6296 t
( ran in time)3 485(the previously best-known algorithm)3 1498 2 720 6416 t
10 I f
(O)2737 6416 w
10 R f
(\()2817 6416 w
10 I f
(mn)2858 6416 w
10 R f
(log)3021 6416 w
10 I f
(n)3190 6416 w
10 R f
( Lemma)1 339( From)1 276(\) [Gabow and Tarjan, 1987].)4 1177 3 3248 6416 t
(5.3 and Theorem 7.1, our algorithm runs in time)8 1969 1 720 6536 t
10 I f
(O)2719 6536 w
10 R f
(\()2799 6536 w
10 I f
(n)2840 6536 w
10 R f
(\()2898 6536 w
10 I f
(m)2939 6536 w
10 S f
(+)3060 6536 w
10 I f
(n)3164 6536 w
10 R f
(log)3222 6536 w
10 I f
(n)3358 6536 w
10 R f
(\))3416 6536 w
10 I f
(log)3465 6536 w
10 R f
(\()3601 6536 w
10 I f
(m / n)2 166 1 3642 6536 t
10 R f
( is asymptotically fas-)3 892(\), which)1 332 2 3816 6536 t
(ter.)720 6656 w
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13 pagesetup
10 R f
(- 13 -)2 216 1 2772 480 t
10 B f
( Path-Covering Problems)2 1082(8. General)1 469 2 720 840 t
10 R f
( the most general covering-paths and postman-tour problems on directed)9 2979(We have not yet discussed)4 1091 2 970 996 t
( general, we have)3 697( In)1 133(graphs with arbitrary capacity upper and lower bounds.)7 2206 3 720 1116 t
10 B f
(Theorem 8.1.)1 563 1 720 1272 t
10 R f
(Problems)1308 1272 w
10 I f
(CP)1711 1272 w
7 I f
(i)1844 1291 w
7 R f
(\( 0 \))2 91 1 1844 1232 t
10 R f
(and)1968 1272 w
10 I f
(PT)2137 1272 w
7 I f
(i)2259 1291 w
7 R f
(\( 0 \))2 91 1 2259 1232 t
10 R f
(, for)1 166 1 2358 1272 t
10 I f
(i)2549 1272 w
10 S f
(=)2626 1272 w
10 R f
( are NP-complete.)2 723( 3,)1 116(2 ,)1 83 3 2730 1272 t
10 B f
(Proof.)720 1428 w
10 R f
(We first show that Problem)4 1099 1 1008 1428 t
10 I f
(PT)2132 1428 w
7 R f
(3)2254 1447 w
(\( 0 \))2 91 1 2254 1388 t
10 R f
( one in which all)4 670( special case of this problem is)6 1226( A)1 122(is NP-complete.)1 644 4 2378 1428 t
( some edges have a zero capacity lower bound, and some edges)11 2532(edges have an infinite capacity upper bound,)6 1788 2 720 1548 t
( special case is the well-known NP-complete)6 1833( This)1 235( bound.)1 307(have a unit capacity lower)4 1071 4 720 1668 t
10 I f
(rural - postman)2 594 1 4198 1668 t
10 R f
(prob-)4824 1668 w
(lem [Lenstra and Kan, 1976].)4 1175 1 720 1788 t
(Given Problem)1 611 1 970 1944 t
10 I f
(PT)1609 1944 w
7 R f
(3)1731 1963 w
(\( 0 \))2 91 1 1731 1904 t
10 R f
(on a graph)2 427 1 1858 1944 t
10 I f
(G)2313 1944 w
10 R f
(with origin)1 445 1 2413 1944 t
10 I f
(O)2886 1944 w
10 R f
( augment the graph by adding the edges \()8 1679(, we)1 169 2 2958 1944 t
10 I f
(s)4814 1944 w
10 R f
(,)4861 1944 w
10 I f
(O)4927 1944 w
10 R f
(\))5007 1944 w
(and \()1 202 1 720 2064 t
10 I f
(O)930 2064 w
10 R f
(,)1010 2064 w
10 I f
(t)1076 2064 w
10 R f
(\), giving them a cost of zero, and a capacity upper bound of infinity and a capacity lower bound of)19 3928 1 1112 2064 t
( provides)1 373( a minimum-cost \(minimum-cardinality\) set of covering paths on the augmented graph)11 3493(zero. Then)1 454 3 720 2184 t
(a solution for Problem)3 933 1 720 2304 t
10 I f
(PT)1690 2304 w
7 R f
(3)1812 2323 w
(\( 0 \))2 91 1 1812 2264 t
10 R f
( Problem)1 376(. Thus)1 287 2 1911 2304 t
10 I f
(PT)2611 2304 w
7 R f
(3)2733 2323 w
(\( 0 \))2 91 1 2733 2264 t
10 R f
(can be reduced to Problem)4 1112 1 2869 2304 t
10 I f
(CP)4018 2304 w
7 R f
(2)4151 2323 w
(\( 0 \))2 91 1 4151 2264 t
10 R f
(or Problem)1 458 1 4286 2304 t
10 I f
(CP)4780 2304 w
7 R f
(3)4913 2323 w
(\( 0 \))2 91 1 4913 2264 t
10 R f
(;)5012 2304 w
(therefore, they too are NP-complete.)4 1457 1 720 2424 t
(Given Problem)1 619 1 970 2580 t
10 I f
(CP)1625 2580 w
7 R f
(2)1758 2599 w
(\( 0 \))2 91 1 1758 2540 t
10 R f
(on)1893 2580 w
10 I f
(G)2029 2580 w
10 R f
( \()1 70(, we add an edge)4 711 2 2101 2580 t
10 I f
(t)2890 2580 w
10 R f
(,)2926 2580 w
10 I f
(s)2992 2580 w
10 R f
(\) with a cost of zero, a capacity lower bound of)10 2001 1 3039 2580 t
( covering-paths problem now becomes a minimum-cost)6 2273( The)1 213(zero, and a capacity upper bound of infinity.)7 1834 3 720 2700 t
( problem)1 361(least-repetitious postman-tour)1 1198 2 720 2820 t
10 I f
(PT)2307 2820 w
7 R f
(2)2429 2839 w
(\( 0 \))2 91 1 2429 2780 t
10 R f
(on the augmented graph with)4 1177 1 2556 2820 t
10 I f
(s)3761 2820 w
10 R f
( Problem)1 367( Therefore,)1 470(as origin.)1 375 3 3828 2820 t
10 I f
(PT)720 2940 w
7 R f
(2)842 2959 w
(\( 0 \))2 91 1 842 2900 t
10 R f
(is also NP-complete.)2 830 1 966 2940 t
10 S1 f
()1846 2940 w
1846 2940 m
50 build_sq
1896 2940 m
10 R f
( to be devised for)4 717(In practice, however, additional constraints may allow polynomial-time algorithms)8 3353 2 970 3096 t
( example, if the subgraph consisting of edges with positive capacity lower bound is)13 3480( For)1 201(these problems.)1 639 3 720 3216 t
( can be done in)4 644(connected, then the problem is equivalent to a minimum-flow problem, and the reduction)12 3676 2 720 3336 t
(time)720 3456 w
10 I f
(O)923 3456 w
10 R f
(\()1003 3456 w
10 I f
(mn)1044 3456 w
10 R f
(\).)1174 3456 w
10 B f
( on Mixed Graphs)3 776(9. Problems)1 530 2 720 3696 t
10 R f
( A)1 142( corresponding problems on mixed graphs.)5 1809(We now discuss briefly the)4 1163 3 970 3852 t
10 I f
(mixed)4129 3852 w
10 R f
(graph has both)2 628 1 4412 3852 t
( also be formulated)3 780( optimization problems considered in this paper can)7 2102( The)1 210(directed and undirected edges.)3 1228 4 720 3972 t
(for mixed graphs, and we have:)5 1258 1 720 4092 t
10 B f
(Theorem 9.1.)1 574 1 720 4248 t
10 R f
(For mixed graphs, Problems)3 1166 1 1330 4248 t
10 I f
(X)2532 4248 w
7 I f
(i)2598 4267 w
7 R f
(\()2598 4208 w
7 I f
(j)2637 4208 w
7 R f
(\))2662 4208 w
10 R f
( NP-complete, where)2 869(, are)1 182 2 2693 4248 t
10 I f
(X)3781 4248 w
10 S f
(=)3891 4248 w
10 I f
(C)3995 4248 w
10 R f
(,)4070 4248 w
10 I f
(F)4136 4248 w
10 R f
(,)4205 4248 w
10 I f
(CP)4271 4248 w
10 R f
(,)4407 4248 w
10 I f
(PT)4473 4248 w
10 R f
(,)4598 4248 w
10 I f
(i)4660 4248 w
10 S f
(=)4737 4248 w
10 R f
( 3,)1 116(2 ,)1 83 2 4841 4248 t
(and)720 4368 w
10 I f
(j)889 4368 w
10 S f
(=)966 4368 w
10 R f
( 2.)1 116( ,)1 33( 1)1 91(0 ,)1 83 4 1070 4368 t
10 B f
(Proof.)720 4524 w
10 R f
(We first note that Problem)4 1108 1 1020 4524 t
10 I f
(PT)2165 4524 w
7 R f
(3)2287 4543 w
(\( 2 \))2 91 1 2287 4484 t
10 R f
(can be reduced to Problems)4 1151 1 2423 4524 t
10 I f
(CP)3611 4524 w
7 R f
(3)3744 4543 w
(\( 2 \))2 91 1 3744 4484 t
10 R f
(and)3880 4524 w
10 I f
(CP)4061 4524 w
7 R f
(2)4194 4543 w
(\( 2 \))2 91 1 4194 4484 t
10 R f
(and that Problem)2 709 1 4331 4524 t
10 I f
(CP)720 4644 w
7 R f
(2)853 4663 w
(\( 2 \))2 91 1 853 4604 t
10 R f
(can be reduced to Problem)4 1076 1 980 4644 t
10 I f
(PT)2084 4644 w
7 R f
(2)2206 4663 w
(\( 2 \))2 91 1 2206 4604 t
10 R f
( Problem)1 366(. Since)1 299 2 2305 4644 t
10 I f
(PT)2997 4644 w
7 R f
(3)3119 4663 w
(\( 2 \))2 91 1 3119 4604 t
10 R f
(is)3245 4644 w
10 I f
(NP)3339 4644 w
10 R f
(-complete [Papadimitriou, 1976], all of)4 1573 1 3467 4644 t
( reduction is the same as that in the proof of Theorem 8.1, and we omit the details.)17 3290( The)1 205(them are.)1 371 3 720 4764 t
(Since Problems)1 627 1 970 4920 t
10 I f
(F)1624 4920 w
7 I f
(i)1690 4939 w
7 R f
(\( 2 \))2 91 1 1690 4880 t
10 R f
(\()1816 4920 w
10 I f
(C)1849 4920 w
7 I f
(i)1921 4939 w
7 R f
(\( 2 \))2 91 1 1921 4880 t
10 R f
(\) and)1 204 1 2020 4920 t
10 I f
(CP)2251 4920 w
7 I f
(i)2384 4939 w
7 R f
(\( 2 \))2 91 1 2384 4880 t
10 R f
(\()2510 4920 w
10 I f
(PT)2543 4920 w
7 I f
(i)2665 4939 w
7 R f
(\( 2 \))2 91 1 2665 4880 t
10 R f
( they are special cases of Problems)6 1409(\) are equivalent,)2 649 2 2764 4920 t
10 I f
(X)4850 4920 w
7 I f
(i)4916 4939 w
7 R f
(\( 1 \))2 91 1 4916 4880 t
10 R f
(,)5015 4920 w
( cases of Problems)3 749(which in turn are special)4 989 2 720 5040 t
10 I f
(X)2484 5040 w
7 I f
(i)2550 5059 w
7 R f
(\( 0 \))2 91 1 2550 5000 t
10 R f
(, respectively, for)2 700 1 2649 5040 t
10 I f
(X)3375 5040 w
10 S f
(=)3485 5040 w
10 I f
(F)3589 5040 w
10 R f
(,)3658 5040 w
10 I f
(C)3724 5040 w
10 R f
(,)3799 5040 w
10 I f
(P)3865 5040 w
10 R f
(,)3934 5040 w
10 I f
(p)4000 5040 w
10 R f
(,)4050 5040 w
10 I f
(i)4101 5040 w
10 S f
(=)4178 5040 w
10 R f
( theorem)1 353( The)1 206( 3.)1 116(2 ,)1 83 4 4282 5040 t
(follows.)720 5160 w
10 S1 f
()1095 5160 w
1095 5160 m
50 build_sq
1145 5160 m
10 B f
(10. Conclusions)1 693 1 720 5400 t
10 R f
( showed that)2 531( We)1 200(We started by considering basic questions arising in program and circuit testing.)11 3339 3 970 5556 t
( several natural classes of network-optimization problems, and we derived a)10 3055(these questions are instances of)4 1265 2 720 5676 t
( work provides a unifying framework for these optimiza-)8 2289( Our)1 207(general technique for solving these problems.)5 1824 3 720 5796 t
( testing and Chinese-)3 883( few of the problems have been studied in isolation, such as the)12 2707( A)1 137(tion problems.)1 593 4 720 5916 t
(postman problems, but our methods yield the fastest-known solutions even for these problems.)12 3786 1 720 6036 t
( symbols)1 367( The)1 213( summarized in the following three tables.)6 1732(The results are)2 601 4 970 6192 t
10 I f
(C)3916 6192 w
7 I f
(i)3988 6211 w
7 R f
(\()3988 6152 w
7 I f
(j)4027 6152 w
7 R f
(\))4052 6152 w
10 R f
(,)4083 6192 w
10 I f
(F)4141 6192 w
7 I f
(i)4207 6211 w
7 R f
(\()4207 6152 w
7 I f
(j)4246 6152 w
7 R f
(\))4271 6152 w
10 R f
(,)4302 6192 w
10 I f
(CP)4360 6192 w
7 I f
(i)4493 6211 w
7 R f
(\()4493 6152 w
7 I f
(j)4532 6152 w
7 R f
(\))4557 6152 w
10 R f
(, and)1 202 1 4588 6192 t
10 I f
(PT)4823 6192 w
7 I f
(i)4945 6211 w
7 R f
(\()4945 6152 w
7 I f
(j)4984 6152 w
7 R f
(\))5009 6152 w
10 R f
( sub-)1 203( The)1 211( problems: circulation, flow, covering paths, and postman tours.)8 2601(refer to the four main classes of)6 1305 4 720 6312 t
(script,)720 6432 w
10 I f
(i)997 6432 w
10 S f
(=)1074 6432 w
10 R f
( forms of each problem: minimizing the cardinality, minimizing)8 2605( distinguishes the three)3 934( 3,)1 116( ,)1 33( 2)1 91(1 ,)1 83 6 1178 6432 t
( superscript,)1 490( The)1 207( minimizing only the cost.)4 1056(the cost and cardinality, and)4 1135 4 720 6552 t
10 I f
(j)3635 6552 w
10 S f
(=)3712 6552 w
10 R f
( classifies the capacity)3 901( 2,)1 116( ,)1 33( 1)1 91(0 ,)1 83 5 3816 6552 t
( unit lower bound and an arbitrary)6 1442(bounds on the edges: 0 for arbitrary lower and upper bounds, 1 for a)13 2878 2 720 6672 t
( the tables,)2 442( In)1 140(upper bound, and 2 for a unit lower bound and an infinite upper bound.)13 2934 3 720 6792 t
10 I f
(m)4267 6792 w
10 R f
(is the number of)3 670 1 4370 6792 t
(edges,)720 6912 w
10 I f
(n)997 6912 w
10 R f
(the number of nodes, and)4 1012 1 1072 6912 t
10 I f
(L)2109 6912 w
10 S f
(=)2214 6912 w
7 I f
(e)2318 7012 w
7 S f
(\316)2377 7012 w
7 I f
(E)2455 7012 w
15 S f
(S)2363 6942 w
10 I f
(l)2497 6912 w
10 R f
(\()2533 6912 w
10 I f
(e)2574 6912 w
10 R f
(\) where)1 301 1 2626 6912 t
10 I f
(l)2952 6912 w
10 R f
(\()2988 6912 w
10 I f
(e)3029 6912 w
10 R f
(\) is the capacity lower bound on edge)7 1494 1 3081 6912 t
10 I f
(e)4600 6912 w
10 R f
(.)4644 6912 w
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14 pagesetup
10 R f
(- 14 -)2 216 1 2772 480 t
10 S f
(_ _____________________________________________)1 2274 1 1743 790 t
(_ _____________________________________________)1 2274 1 1743 810 t
10 R f
(P)1815 920 w
8 R f
(ROBLEM)1871 920 w
10 R f
(C)2586 920 w
8 R f
(OST)2653 920 w
10 R f
(P)3185 920 w
8 R f
(ROBLEM)3241 920 w
10 R f
(C)3798 920 w
8 R f
(OST)3865 920 w
10 S f
(_ _____________________________________________)1 2274 1 1743 940 t
10 I f
(C)1801 1060 w
7 R f
(1)1873 1079 w
(\( 0 \))2 91 1 1873 1020 t
10 R f
(,)1988 1060 w
10 I f
(F)2054 1060 w
7 R f
(1)2120 1079 w
(\( 0 \))2 91 1 2120 1020 t
10 I f
(mn)2537 1060 w
10 R f
(log)2667 1060 w
10 I f
(n CP)1 438 1 2803 1060 t
7 R f
(1)3246 1079 w
(\( 0 \))2 91 1 3246 1020 t
10 R f
(,)3361 1060 w
10 I f
(PT)3427 1060 w
7 R f
(1)3549 1079 w
(\( 0 \))2 91 1 3549 1020 t
10 R f
(?)3885 1060 w
10 I f
(C)1801 1180 w
7 R f
(1)1873 1199 w
(\( 1 \))2 91 1 1873 1140 t
10 R f
(,)1988 1180 w
10 I f
(F)2054 1180 w
7 R f
(1)2120 1199 w
(\( 1 \))2 91 1 2120 1140 t
10 I f
(mn CP)1 607 1 2634 1180 t
7 R f
(1)3246 1199 w
(\( 1 \))2 91 1 3246 1140 t
10 R f
(,)3361 1180 w
10 I f
(PT)3427 1180 w
7 R f
(1)3549 1199 w
(\( 1 \))2 91 1 3549 1140 t
10 I f
(mn)3846 1180 w
(C)1801 1300 w
7 R f
(1)1873 1319 w
(\( 2 \))2 91 1 1873 1260 t
10 R f
(,)1988 1300 w
10 I f
(F)2054 1300 w
7 R f
(1)2120 1319 w
(\( 2 \))2 91 1 2120 1260 t
10 I f
(mn CP)1 607 1 2634 1300 t
7 R f
(1)3246 1319 w
(\( 2 \))2 91 1 3246 1260 t
10 R f
(,)3361 1300 w
10 I f
(PT)3427 1300 w
7 R f
(1)3549 1319 w
(\( 2 \))2 91 1 3549 1260 t
10 I f
(mn)3846 1300 w
10 S f
( \347)1 -1664(_ _____________________________________________)1 2274 2 1743 1320 t
(\347)2353 1310 w
(\347)2353 1210 w
(\347)2353 1110 w
(\347)2353 1010 w
(\347)2353 910 w
(\347)3028 1320 w
(\347)3028 1310 w
(\347)3028 1210 w
(\347)3028 1110 w
(\347)3028 1010 w
(\347)3028 910 w
(\347)3048 1320 w
(\347)3048 1310 w
(\347)3048 1210 w
(\347)3048 1110 w
(\347)3048 1010 w
(\347)3048 910 w
(\347)3723 1320 w
(\347)3723 1310 w
(\347)3723 1210 w
(\347)3723 1110 w
(\347)3723 1010 w
(\347)3723 910 w
10 B f
(Table 1.)1 345 1 2041 1560 t
10 R f
(Minimum-cardinality problems.)1 1283 1 2436 1560 t
10 S f
(_ _____________________________________________________________________________________)1 4257 1 751 1870 t
(_ _____________________________________________________________________________________)1 4257 1 751 1890 t
10 R f
(P)764 2000 w
8 R f
(ROBLEM)820 2000 w
10 R f
(C)2066 2000 w
8 R f
(OST)2133 2000 w
10 R f
(P)3254 2000 w
8 R f
(ROBLEM)3310 2000 w
10 R f
(C)4328 2000 w
8 R f
(OST)4395 2000 w
10 S f
(_ _____________________________________________________________________________________)1 4257 1 751 2020 t
10 I f
(C)751 2140 w
7 R f
(2)823 2159 w
(\( 0 \))2 91 1 823 2100 t
10 R f
(,)938 2140 w
10 I f
(F)1004 2140 w
7 R f
(2)1070 2159 w
(\( 0 \))2 91 1 1070 2100 t
10 R f
(min)1319 2140 w
10 I f
({)1516 2140 w
10 R f
(\()1564 2140 w
10 I f
(m)1605 2140 w
10 S f
(+)1701 2140 w
10 I f
(n)1772 2140 w
10 R f
(log)1830 2140 w
10 I f
(n)1966 2140 w
10 R f
(\))2024 2140 w
10 I f
(L)2073 2140 w
10 R f
(,)2137 2140 w
10 I f
(m)2203 2140 w
10 R f
(\()2283 2140 w
10 I f
(m)2324 2140 w
10 S f
(+)2420 2140 w
10 I f
(n)2491 2140 w
10 R f
(log)2549 2140 w
10 I f
(n)2685 2140 w
10 R f
(\) log)1 177 1 2743 2140 t
10 I f
( CP)1 278(L })1 104 2 2928 2140 t
7 R f
(2)3315 2159 w
(\( 0 \))2 91 1 3315 2100 t
10 R f
(,)3430 2140 w
10 I f
(PT)3496 2140 w
7 R f
(2)3618 2159 w
(\( 0 \))2 91 1 3618 2100 t
10 R f
(NP-complete)4174 2140 w
10 I f
(C)751 2260 w
7 R f
(2)823 2279 w
(\( 1 \))2 91 1 823 2220 t
10 R f
(,)938 2260 w
10 I f
(F)1004 2260 w
7 R f
(2)1070 2279 w
(\( 1 \))2 91 1 1070 2220 t
10 I f
(m)1889 2260 w
10 R f
(\()1969 2260 w
10 I f
(m)2010 2260 w
10 S f
(+)2106 2260 w
10 I f
(n)2177 2260 w
10 R f
(log)2235 2260 w
10 I f
(n)2371 2260 w
10 R f
(\))2429 2260 w
10 I f
(CP)3182 2260 w
7 R f
(2)3315 2279 w
(\( 1 \))2 91 1 3315 2220 t
10 R f
(,)3430 2260 w
10 I f
(PT)3496 2260 w
7 R f
(2)3618 2279 w
(\( 1 \))2 91 1 3618 2220 t
10 I f
(m)4151 2260 w
10 R f
(\()4231 2260 w
10 I f
(m)4272 2260 w
10 S f
(+)4368 2260 w
10 I f
(n)4439 2260 w
10 R f
(log)4497 2260 w
10 I f
(n)4633 2260 w
10 R f
(\))4691 2260 w
10 I f
(C)751 2380 w
7 R f
(2)823 2399 w
(\( 2 \))2 91 1 823 2340 t
10 R f
(,)938 2380 w
10 I f
(F)1004 2380 w
7 R f
(2)1070 2399 w
(\( 2 \))2 91 1 1070 2340 t
10 I f
(n)1605 2380 w
10 R f
(\()1663 2380 w
10 I f
(m)1704 2380 w
10 S f
(+)1825 2380 w
10 I f
(n)1929 2380 w
10 R f
(log)2020 2380 w
10 I f
(n)2189 2380 w
10 R f
( \()1 74(\) log)1 210 2 2247 2380 t
10 I f
(m / n)2 166 1 2539 2380 t
10 R f
(\))2713 2380 w
10 I f
(CP)3182 2380 w
7 R f
(2)3315 2399 w
(\( 2 \))2 91 1 3315 2340 t
10 R f
(,)3430 2380 w
10 I f
(PT)3496 2380 w
7 R f
(2)3618 2399 w
(\( 2 \))2 91 1 3618 2340 t
10 I f
(n)3867 2380 w
10 R f
(\()3925 2380 w
10 I f
(m)3966 2380 w
10 S f
(+)4087 2380 w
10 I f
(n)4191 2380 w
10 R f
(log)4282 2380 w
10 I f
(n)4451 2380 w
10 R f
( \()1 74(\) log)1 210 2 4509 2380 t
10 I f
(m / n)2 166 1 4801 2380 t
10 R f
(\))4975 2380 w
10 S f
( \347)1 -3764(_ _____________________________________________________________________________________)1 4257 2 751 2400 t
(\347)1244 2390 w
(\347)1244 2290 w
(\347)1244 2190 w
(\347)1244 2090 w
(\347)1244 1990 w
(\347)3097 2400 w
(\347)3097 2390 w
(\347)3097 2290 w
(\347)3097 2190 w
(\347)3097 2090 w
(\347)3097 1990 w
(\347)3117 2400 w
(\347)3117 2390 w
(\347)3117 2290 w
(\347)3117 2190 w
(\347)3117 2090 w
(\347)3117 1990 w
(\347)3792 2400 w
(\347)3792 2390 w
(\347)3792 2290 w
(\347)3792 2190 w
(\347)3792 2090 w
(\347)3792 1990 w
10 B f
(Table 2.)1 345 1 1736 2640 t
10 R f
(Minimum-cost minimum-cardinality problems.)2 1892 1 2131 2640 t
10 S f
(_ _____________________________________________________________________________________)1 4257 1 751 2950 t
(_ _____________________________________________________________________________________)1 4257 1 751 2970 t
10 R f
(P)764 3080 w
8 R f
(ROBLEM)820 3080 w
10 R f
(C)2066 3080 w
8 R f
(OST)2133 3080 w
10 R f
(P)3254 3080 w
8 R f
(ROBLEM)3310 3080 w
10 R f
(C)4328 3080 w
8 R f
(OST)4395 3080 w
10 S f
(_ _____________________________________________________________________________________)1 4257 1 751 3100 t
10 I f
(C)751 3220 w
7 R f
(3)823 3239 w
(\( 0 \))2 91 1 823 3180 t
10 R f
(,)938 3220 w
10 I f
(F)1004 3220 w
7 R f
(3)1070 3239 w
(\( 0 \))2 91 1 1070 3180 t
10 R f
(min)1319 3220 w
10 I f
({)1516 3220 w
10 R f
(\()1564 3220 w
10 I f
(m)1605 3220 w
10 S f
(+)1701 3220 w
10 I f
(n)1772 3220 w
10 R f
(log)1830 3220 w
10 I f
(n)1966 3220 w
10 R f
(\))2024 3220 w
10 I f
(L)2073 3220 w
10 R f
(,)2137 3220 w
10 I f
(m)2203 3220 w
10 R f
(\()2283 3220 w
10 I f
(m)2324 3220 w
10 S f
(+)2420 3220 w
10 I f
(n)2491 3220 w
10 R f
(log)2549 3220 w
10 I f
(n)2685 3220 w
10 R f
(\) log)1 177 1 2743 3220 t
10 I f
( CP)1 278(L })1 104 2 2928 3220 t
7 R f
(3)3315 3239 w
(\( 0 \))2 91 1 3315 3180 t
10 R f
(,)3430 3220 w
10 I f
(PT)3496 3220 w
7 R f
(3)3618 3239 w
(\( 0 \))2 91 1 3618 3180 t
10 R f
(NP-complete)4174 3220 w
10 I f
(C)751 3340 w
7 R f
(3)823 3359 w
(\( 1 \))2 91 1 823 3300 t
10 R f
(,)938 3340 w
10 I f
(F)1004 3340 w
7 R f
(3)1070 3359 w
(\( 1 \))2 91 1 1070 3300 t
10 I f
(m)1889 3340 w
10 R f
(\()1969 3340 w
10 I f
(m)2010 3340 w
10 S f
(+)2106 3340 w
10 I f
(n)2177 3340 w
10 R f
(log)2235 3340 w
10 I f
(n)2371 3340 w
10 R f
(\))2429 3340 w
10 I f
(CP)3182 3340 w
7 R f
(3)3315 3359 w
(\( 1 \))2 91 1 3315 3300 t
10 R f
(,)3430 3340 w
10 I f
(PT)3496 3340 w
7 R f
(3)3618 3359 w
(\( 1 \))2 91 1 3618 3300 t
10 I f
(m)4151 3340 w
10 R f
(\()4231 3340 w
10 I f
(m)4272 3340 w
10 S f
(+)4368 3340 w
10 I f
(n)4439 3340 w
10 R f
(log)4497 3340 w
10 I f
(n)4633 3340 w
10 R f
(\))4691 3340 w
10 I f
(C)751 3460 w
7 R f
(3)823 3479 w
(\( 2 \))2 91 1 823 3420 t
10 R f
(,)938 3460 w
10 I f
(F)1004 3460 w
7 R f
(3)1070 3479 w
(\( 2 \))2 91 1 1070 3420 t
10 I f
(n)1605 3460 w
10 R f
(\()1663 3460 w
10 I f
(m)1704 3460 w
10 S f
(+)1825 3460 w
10 I f
(n)1929 3460 w
10 R f
(log)2020 3460 w
10 I f
(n)2189 3460 w
10 R f
( \()1 74(\) log)1 210 2 2247 3460 t
10 I f
(m / n)2 166 1 2539 3460 t
10 R f
(\))2713 3460 w
10 I f
(CP)3182 3460 w
7 R f
(3)3315 3479 w
(\( 2 \))2 91 1 3315 3420 t
10 R f
(,)3430 3460 w
10 I f
(PT)3496 3460 w
7 R f
(3)3618 3479 w
(\( 2 \))2 91 1 3618 3420 t
10 I f
(n)3867 3460 w
10 R f
(\()3925 3460 w
10 I f
(m)3966 3460 w
10 S f
(+)4087 3460 w
10 I f
(n)4191 3460 w
10 R f
(log)4282 3460 w
10 I f
(n)4451 3460 w
10 R f
( \()1 74(\) log)1 210 2 4509 3460 t
10 I f
(m / n)2 166 1 4801 3460 t
10 R f
(\))4975 3460 w
10 S f
( \347)1 -3764(_ _____________________________________________________________________________________)1 4257 2 751 3480 t
(\347)1244 3470 w
(\347)1244 3370 w
(\347)1244 3270 w
(\347)1244 3170 w
(\347)1244 3070 w
(\347)3097 3480 w
(\347)3097 3470 w
(\347)3097 3370 w
(\347)3097 3270 w
(\347)3097 3170 w
(\347)3097 3070 w
(\347)3117 3480 w
(\347)3117 3470 w
(\347)3117 3370 w
(\347)3117 3270 w
(\347)3117 3170 w
(\347)3117 3070 w
(\347)3792 3480 w
(\347)3792 3470 w
(\347)3792 3370 w
(\347)3792 3270 w
(\347)3792 3170 w
(\347)3792 3070 w
10 B f
(Table 3.)1 345 1 2174 3720 t
10 R f
(Minimum-cost problems.)1 1017 1 2569 3720 t
10 B f
(Acknowledgment)720 3960 w
10 R f
(We are indebted to Mihalis Yannakakis for many valuable comments and stimulating discussions.)12 3921 1 970 4116 t
10 B f
(References)720 4416 w
10 R f
(Aho, A. V., Hopcroft, J. E., and Ullman, J. D. [1974].)10 2207 1 720 4632 t
10 I f
( Algorithms)1 477(The Design and Analysis of Computer)5 1555 2 2983 4632 t
10 R f
(,)5015 4632 w
(Addison-Wesley, Reading, Mass.)2 1340 1 1080 4752 t
( Postman,)1 407( Euler tours and the Chinese)5 1184( Matching,)1 469(Edmonds, J., and Johnson, E. L. [1973].)6 1668 4 720 4932 t
10 I f
(Mathematical)4485 4932 w
(Programming)1080 5052 w
10 B f
(5)1666 5052 w
10 R f
(, 88-124.)1 358 1 1716 5052 t
( improvements in algorithmic efficiency for network)6 2198( Theoretical)1 520( and Karp, R. M. [1972].)5 1070(Edmonds, J.,)1 532 4 720 5232 t
(flow problems,)1 605 1 1080 5352 t
10 I f
(J. ACM)1 305 1 1710 5352 t
10 B f
(19)2040 5352 w
10 R f
(, 2, 248-264.)2 508 1 2140 5352 t
(Even, S. [1979].)2 652 1 720 5532 t
10 I f
(Graph Algorithms)1 731 1 1422 5532 t
10 R f
(, Computer Science Press.)3 1046 1 2153 5532 t
(Ford, L. R. Jr and Fulkerson, D. R. [1962].)8 1713 1 720 5712 t
10 I f
(Flows in Networks)2 751 1 2483 5712 t
10 R f
(, Princeton Univ. Press. Princeton, NJ.)5 1538 1 3234 5712 t
( improved network optimization)3 1294( heaps and their uses in)5 929( Fibonacci)1 444(Fredman, M. L, and Tarjan, R. E. [1984].)7 1653 4 720 5892 t
(algorithms,)1080 6012 w
10 I f
(Proc. 25th Annual Symp. on Found. of Comp. Sci.)8 2002 1 1558 6012 t
10 R f
(, pp. 338-346.)2 558 1 3560 6012 t
( two problems in the generation of pro-)7 1602( On)1 177(Gabow, H. N., Maheshwari, S. N., and Osterweil, L. J. [1976].)10 2541 3 720 6192 t
(gram test paths,)2 630 1 1080 6312 t
10 I f
(IEEE Trans. Software Engineering)3 1400 1 1735 6312 t
10 B f
(SE-2)3160 6312 w
10 R f
(, 227-231.)1 408 1 3366 6312 t
( N. and Tarjan, R. E. [1987].)6 1235(Gabow, H.)1 449 2 720 6492 t
10 I f
(Faster scaling algorithms for network problems)5 1995 1 2469 6492 t
10 R f
(, Unpublished)1 576 1 4464 6492 t
(manuscript.)1080 6612 w
( An)1 190(Galil, Z. and Tardos, E. [1986].)5 1344 2 720 6792 t
10 I f
(O)2297 6792 w
10 R f
(\()2377 6792 w
10 I f
(n)2418 6792 w
7 R f
(2)2479 6752 w
10 R f
(log)2563 6792 w
10 I f
(n)2732 6792 w
10 R f
(\()2823 6792 w
10 I f
(m)2864 6792 w
10 S f
(+)2985 6792 w
10 I f
(n)3089 6792 w
10 R f
(log)3180 6792 w
10 I f
(n)3349 6792 w
10 R f
( min-cost flow algorithm,)3 1076(\) \))1 74 2 3407 6792 t
10 I f
(Proc. 27th)1 440 1 4600 6792 t
(IEEE Symp. of Foundations of Computer Science)6 1973 1 1080 6912 t
10 R f
(, 1-9.)1 208 1 3053 6912 t
(Gibbons, A. [1985].)2 802 1 720 7092 t
10 I f
(Algorithmic Graph Theory,)2 1097 1 1572 7092 t
10 R f
(Cambridge University Press, Cambridge.)3 1646 1 2694 7092 t
(Knuth, D. E. [1984].)3 824 1 720 7272 t
10 I f
(A torture test for TEX)4 873 1 1594 7272 t
10 R f
( University, CS Technical Report.)4 1360(. Stanford)1 419 2 2467 7272 t
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10 R f
(- 15 -)2 216 1 2772 480 t
( software test planning through auto-)5 1523( Optimal)1 387(Krause, K. A., Smith, R. W., and Goodwin, M. A. [1973].)10 2410 3 720 840 t
(mated network analysis,)2 984 1 1080 960 t
10 I f
(Proc. 1973 IEEE Symposium on Computer Software Reliability)7 2594 1 2097 960 t
10 R f
(, pp. 18-)2 349 1 4691 960 t
(22.)1080 1080 w
( points,)1 302( programming using odd or even)5 1337( Graphic)1 377(Kwan, M. \(Guan, Meigu\) [1960].)4 1360 4 720 1260 t
10 I f
(Chinese Math.)1 590 1 4128 1260 t
10 B f
(1)4750 1260 w
10 R f
(, 273-)1 240 1 4800 1260 t
(277.)1080 1380 w
(Lawler, E. [1976].)2 734 1 720 1560 t
10 I f
(Combinatorial Optimization: Networks and Matroids)4 2141 1 1504 1560 t
10 R f
(, Holt, Rinehart and Winston.)4 1180 1 3645 1560 t
( general routing problems,)3 1054( On)1 172(Lenstra, J. K. and Rinnooy Kan, A. H. G. [1976].)9 1972 3 720 1740 t
10 I f
(Networks)3943 1740 w
10 B f
(6)4346 1740 w
10 R f
(, 273-280.)1 408 1 4396 1740 t
( path cover problems in digraphs and applications to program-)9 2542( On)1 177(Ntafos, S. C. and Hakimi, S. L. [1979].)7 1601 3 720 1920 t
(ming testing,)1 523 1 1080 2040 t
10 I f
(IEEE Trans. on Software Engineering)4 1525 1 1628 2040 t
10 B f
(SE-5)3178 2040 w
10 R f
(:5, 520-529.)1 486 1 3384 2040 t
( the complexity of edge traversing,)5 1392( On)1 172(Papadimitriou, C. H. [1976].)3 1147 3 720 2220 t
10 I f
(J. ACM)1 305 1 3456 2220 t
10 B f
(23)3786 2220 w
10 R f
(:3, 544-554.)1 486 1 3886 2220 t
( chip testing costs,)3 739( Cutting)1 351(Seth, S. C., and Agrawal, V. D. [1985].)7 1573 3 720 2400 t
10 I f
(IEEE Spectrum)1 618 1 3408 2400 t
10 B f
(22)4051 2400 w
10 R f
(, 38-45.)1 308 1 4151 2400 t
( cost circulation algorithm,)3 1110( strongly polynomial minimum)3 1279( A)1 132(Tardos, E. [1985].)2 749 4 720 2580 t
10 I f
(Combinatorica)4026 2580 w
10 B f
(5)4668 2580 w
10 R f
(:3, 247-)1 322 1 4718 2580 t
(255.)1080 2700 w
( [1983].)1 321(Tarjan, R. E.)2 521 2 720 2880 t
10 I f
(Data Structures and Network Algorithms)4 1665 1 1617 2880 t
10 R f
(, Society for Industrial and Applied Mathe-)6 1758 1 3282 2880 t
(matics, Philadelphia, PA.)2 1014 1 1080 3000 t
( test sequence generation for protocols: the Chinese post-)8 2317( Optimal)1 382( Dahbura, A. T. [1986].)4 958(Uyar, M. U, and)3 663 4 720 3180 t
(man algorithm applied to Q.931,)4 1305 1 1080 3300 t
10 I f
(Proc. IEEE Global Telecommunications Conference)4 2100 1 2410 3300 t
10 R f
(, 1986.)1 275 1 4510 3300 t
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