1272:
535:
31:
also being required for the source and sink (i.e. there are no special nodes). In variants of the problem, there are multiple commodities flowing through the network, and a cost on the flow.
898:
733:
379:
821:
1021:
438:
1832:
1609:
1168:
1340:
593:
1521:
1922:
1873:
1738:
1697:
1650:
1439:
1477:
264:
187:
110:
72:
1770:
1122:
1095:
1048:
296:
1547:
1398:
1378:
1292:
1068:
655:
635:
615:
227:
207:
150:
130:
1345:
1526:
2013:
1173:
453:
2008:
951:
909:
836:
664:
307:
1351:
755:
545:
In a multi-commodity circulation problem, you also need to keep track of the flow of the individual commodities:
957:
1969:
386:
444:
Finding a flow assignment satisfying the constraints gives a solution to the given circulation problem.
1878:
1655:
1775:
1552:
1357:
935:
Below are given some problems, and how to solve them with the general circulation setup given above.
1127:
1297:
555:
924:
1482:
1884:
1837:
1702:
1661:
1614:
1403:
1444:
1981:
1950:
234:
157:
80:
42:
1743:
1100:
1073:
1026:
269:
920:
1532:
1383:
1363:
1277:
1053:
640:
620:
600:
212:
192:
135:
115:
2002:
939:
Minimum cost multi-commodity circulation problem - Using all constraints given above.
908:
For the circulation problem, many polynomial algorithms have been developed (e.g.,
24:
912:, 1972; Tarjan 1987-1988). Tardos found the first strongly polynomial algorithm.
916:
1941:Éva Tardos (1985). "A strongly polynomial minimum cost circulation algorithm".
830:
The conservation constraint must be upheld individually for the commodities:
27:
problems, with the added constraint of a lower bound on edge flows, and with
1989:
1954:
1985:
1970:"On the complexity of timetable and multi-commodity flow problems"
1881:- Let all capacities be unlimited, and find a flow of 1 for
945:
Multi-commodity circulation - Solve without optimising cost.
919:
for integer flows. For fractional flows, it is solvable in
948:
Simple circulation - Just use one commodity, and no cost.
942:
Minimum cost circulation problem - Use a single commodity
1267:{\displaystyle l_{i}(t_{i},s_{i})=u(t_{i},s_{i})=d_{i}}
745:
There is also a lower bound on each flow of commodity.
1887:
1840:
1778:
1746:
1705:
1664:
1617:
1555:
1535:
1485:
1447:
1406:
1386:
1366:
1300:
1280:
1176:
1130:
1103:
1076:
1056:
1029:
960:
915:
For the case of multiple commodities, the problem is
839:
758:
667:
643:
623:
603:
558:
530:{\displaystyle \sum _{(v,w)\in E}c(v,w)\cdot f(v,w).}
456:
447:
In the minimum cost variant of the problem, minimize
389:
310:
272:
237:
215:
195:
160:
138:
118:
83:
45:
1360:- Set all costs to 0, and add an edge from the sink
1916:
1867:
1826:
1764:
1732:
1691:
1644:
1603:
1541:
1515:
1471:
1433:
1392:
1372:
1334:
1286:
1266:
1162:
1116:
1089:
1062:
1042:
1015:
892:
815:
727:
649:
629:
609:
587:
529:
432:
373:
290:
258:
221:
201:
181:
144:
124:
104:
66:
893:{\displaystyle \ \sum _{w\in V}f_{i}(u,w)=0.}
8:
1740:for all edges in the graph, and add an edge
728:{\displaystyle \,f(v,w)=\sum _{i}f_{i}(v,w)}
374:{\displaystyle l(v,w)\leq f(v,w)\leq u(v,w)}
816:{\displaystyle \,l_{i}(v,w)\leq f_{i}(v,w)}
440:(flow cannot appear or disappear in nodes).
1968:S. Even and A. Itai and A. Shamir (1976).
1906:
1886:
1839:
1777:
1745:
1704:
1663:
1616:
1554:
1534:
1484:
1446:
1405:
1385:
1365:
1346:Minimum cost multi-commodity flow problem
1305:
1299:
1279:
1258:
1242:
1229:
1207:
1194:
1181:
1175:
1151:
1138:
1129:
1108:
1102:
1081:
1075:
1055:
1034:
1028:
1004:
991:
978:
965:
959:
863:
847:
838:
792:
764:
759:
757:
704:
694:
668:
666:
642:
622:
602:
564:
559:
557:
461:
455:
394:
388:
309:
271:
236:
214:
194:
159:
137:
117:
82:
44:
23:and its variants are a generalisation of
1924:commodities, one for each pair of nodes.
1016:{\displaystyle K_{i}(s_{i},t_{i},d_{i})}
923:, as one can formulate the problem as a
1933:
16:Generalization of network flow problems
433:{\displaystyle \sum _{w\in V}f(u,w)=0}
1529:- First find the maximum flow amount
7:
1348:- As above, but minimize the cost.
14:
1527:Minimum cost maximum flow problem
189:, upper bound on flow from node
112:, lower bound on flow from node
1827:{\displaystyle l(t,s)=u(t,s)=1}
1604:{\displaystyle l(t,s)=u(t,s)=m}
1903:
1891:
1856:
1844:
1815:
1803:
1794:
1782:
1759:
1747:
1721:
1709:
1680:
1668:
1633:
1621:
1592:
1580:
1571:
1559:
1501:
1489:
1463:
1451:
1422:
1410:
1323:
1311:
1248:
1222:
1213:
1187:
1157:
1131:
1010:
971:
881:
869:
810:
798:
782:
770:
722:
710:
684:
672:
582:
570:
521:
509:
500:
488:
474:
462:
421:
409:
368:
356:
347:
335:
326:
314:
285:
273:
253:
241:
176:
164:
99:
87:
61:
49:
1:
1354:- As above, with 1 commodity.
1163:{\displaystyle (t_{i},s_{i})}
1335:{\displaystyle l_{i}(u,v)=0}
588:{\displaystyle \,f_{i}(v,w)}
266:, cost of a unit of flow on
1656:Single-source shortest path
541:Multi-commodity circulation
2030:
1974:SIAM Journal on Computing
1516:{\displaystyle c(t,s)=-1}
1352:Minimum cost flow problem
1917:{\displaystyle v(v-1)/2}
1868:{\displaystyle c(t,s)=0}
1733:{\displaystyle c(u,v)=1}
1692:{\displaystyle l(u,v)=0}
1645:{\displaystyle c(t,s)=0}
1434:{\displaystyle l(t,s)=0}
1879:All-pairs shortest path
1472:{\displaystyle u(t,s)=}
1918:
1869:
1828:
1766:
1734:
1693:
1646:
1605:
1543:
1517:
1473:
1435:
1394:
1374:
1336:
1288:
1268:
1164:
1118:
1091:
1064:
1044:
1017:
910:Edmonds–Karp algorithm
894:
817:
729:
651:
631:
611:
597:The flow of commodity
589:
531:
434:
375:
292:
260:
259:{\displaystyle c(v,w)}
223:
203:
183:
182:{\displaystyle u(v,w)}
146:
126:
106:
105:{\displaystyle l(v,w)}
68:
67:{\displaystyle G(V,E)}
2014:Mathematical problems
1919:
1870:
1829:
1767:
1765:{\displaystyle (t,s)}
1735:
1694:
1647:
1606:
1544:
1518:
1474:
1436:
1395:
1375:
1337:
1289:
1269:
1165:
1119:
1117:{\displaystyle t_{i}}
1092:
1090:{\displaystyle s_{i}}
1065:
1045:
1043:{\displaystyle d_{i}}
1018:
895:
818:
730:
652:
632:
612:
590:
532:
435:
376:
301:and the constraints:
293:
291:{\displaystyle (v,w)}
261:
224:
204:
184:
147:
127:
107:
69:
2009:Network flow problem
1980:(4). SIAM: 691–703.
1885:
1838:
1776:
1744:
1703:
1662:
1615:
1553:
1533:
1483:
1445:
1404:
1384:
1364:
1358:Maximum flow problem
1342:for all other edges.
1298:
1278:
1274:for all commodities
1174:
1128:
1101:
1074:
1054:
1027:
1023:denotes a demand of
958:
952:Multi-commodity flow
837:
756:
665:
641:
621:
601:
556:
454:
387:
308:
270:
235:
213:
193:
158:
136:
116:
81:
43:
39:Given flow network
21:circulation problem
1955:10.1007/BF02579369
1914:
1865:
1824:
1762:
1730:
1689:
1642:
1601:
1549:. Then solve with
1539:
1513:
1469:
1431:
1390:
1370:
1332:
1284:
1264:
1160:
1114:
1087:
1060:
1040:
1013:
890:
858:
813:
725:
699:
647:
627:
607:
585:
527:
484:
430:
405:
371:
288:
256:
219:
199:
179:
142:
122:
102:
64:
1542:{\displaystyle m}
1393:{\displaystyle s}
1373:{\displaystyle t}
1287:{\displaystyle i}
1124:, create an edge
1063:{\displaystyle i}
843:
842:
826:
825:
741:
740:
690:
650:{\displaystyle w}
630:{\displaystyle v}
610:{\displaystyle i}
457:
390:
222:{\displaystyle w}
202:{\displaystyle v}
145:{\displaystyle w}
125:{\displaystyle v}
29:flow conservation
2021:
1994:
1993:
1988:. Archived from
1965:
1959:
1958:
1938:
1923:
1921:
1920:
1915:
1910:
1874:
1872:
1871:
1866:
1833:
1831:
1830:
1825:
1771:
1769:
1768:
1763:
1739:
1737:
1736:
1731:
1698:
1696:
1695:
1690:
1651:
1649:
1648:
1643:
1610:
1608:
1607:
1602:
1548:
1546:
1545:
1540:
1522:
1520:
1519:
1514:
1478:
1476:
1475:
1470:
1440:
1438:
1437:
1432:
1399:
1397:
1396:
1391:
1379:
1377:
1376:
1371:
1341:
1339:
1338:
1333:
1310:
1309:
1293:
1291:
1290:
1285:
1273:
1271:
1270:
1265:
1263:
1262:
1247:
1246:
1234:
1233:
1212:
1211:
1199:
1198:
1186:
1185:
1169:
1167:
1166:
1161:
1156:
1155:
1143:
1142:
1123:
1121:
1120:
1115:
1113:
1112:
1096:
1094:
1093:
1088:
1086:
1085:
1069:
1067:
1066:
1061:
1049:
1047:
1046:
1041:
1039:
1038:
1022:
1020:
1019:
1014:
1009:
1008:
996:
995:
983:
982:
970:
969:
931:Related problems
899:
897:
896:
891:
868:
867:
857:
840:
822:
820:
819:
814:
797:
796:
769:
768:
750:
749:
737:The total flow.
734:
732:
731:
726:
709:
708:
698:
656:
654:
653:
648:
636:
634:
633:
628:
616:
614:
613:
608:
594:
592:
591:
586:
569:
568:
550:
549:
536:
534:
533:
528:
483:
439:
437:
436:
431:
404:
380:
378:
377:
372:
297:
295:
294:
289:
265:
263:
262:
257:
228:
226:
225:
220:
208:
206:
205:
200:
188:
186:
185:
180:
151:
149:
148:
143:
131:
129:
128:
123:
111:
109:
108:
103:
73:
71:
70:
65:
2029:
2028:
2024:
2023:
2022:
2020:
2019:
2018:
1999:
1998:
1997:
1986:10.1137/0205048
1967:
1966:
1962:
1940:
1939:
1935:
1931:
1883:
1882:
1836:
1835:
1774:
1773:
1742:
1741:
1701:
1700:
1660:
1659:
1613:
1612:
1551:
1550:
1531:
1530:
1481:
1480:
1443:
1442:
1402:
1401:
1382:
1381:
1362:
1361:
1301:
1296:
1295:
1276:
1275:
1254:
1238:
1225:
1203:
1190:
1177:
1172:
1171:
1147:
1134:
1126:
1125:
1104:
1099:
1098:
1077:
1072:
1071:
1052:
1051:
1030:
1025:
1024:
1000:
987:
974:
961:
956:
955:
933:
921:polynomial time
906:
859:
835:
834:
788:
760:
754:
753:
700:
663:
662:
639:
638:
619:
618:
599:
598:
560:
554:
553:
543:
452:
451:
385:
384:
306:
305:
268:
267:
233:
232:
211:
210:
191:
190:
156:
155:
134:
133:
114:
113:
79:
78:
41:
40:
37:
17:
12:
11:
5:
2027:
2025:
2017:
2016:
2011:
2001:
2000:
1996:
1995:
1992:on 2013-01-12.
1960:
1949:(3): 247–255.
1932:
1930:
1927:
1926:
1925:
1913:
1909:
1905:
1902:
1899:
1896:
1893:
1890:
1876:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1761:
1758:
1755:
1752:
1749:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1653:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1538:
1524:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1389:
1380:to the source
1369:
1355:
1349:
1343:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1308:
1304:
1283:
1261:
1257:
1253:
1250:
1245:
1241:
1237:
1232:
1228:
1224:
1221:
1218:
1215:
1210:
1206:
1202:
1197:
1193:
1189:
1184:
1180:
1159:
1154:
1150:
1146:
1141:
1137:
1133:
1111:
1107:
1084:
1080:
1059:
1050:for commodity
1037:
1033:
1012:
1007:
1003:
999:
994:
990:
986:
981:
977:
973:
968:
964:
949:
946:
943:
940:
932:
929:
925:linear program
905:
902:
901:
900:
889:
886:
883:
880:
877:
874:
871:
866:
862:
856:
853:
850:
846:
828:
827:
824:
823:
812:
809:
806:
803:
800:
795:
791:
787:
784:
781:
778:
775:
772:
767:
763:
743:
742:
739:
738:
735:
724:
721:
718:
715:
712:
707:
703:
697:
693:
689:
686:
683:
680:
677:
674:
671:
659:
658:
646:
626:
606:
595:
584:
581:
578:
575:
572:
567:
563:
542:
539:
538:
537:
526:
523:
520:
517:
514:
511:
508:
505:
502:
499:
496:
493:
490:
487:
482:
479:
476:
473:
470:
467:
464:
460:
442:
441:
429:
426:
423:
420:
417:
414:
411:
408:
403:
400:
397:
393:
382:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
299:
298:
287:
284:
281:
278:
275:
255:
252:
249:
246:
243:
240:
230:
218:
198:
178:
175:
172:
169:
166:
163:
153:
141:
121:
101:
98:
95:
92:
89:
86:
63:
60:
57:
54:
51:
48:
36:
33:
15:
13:
10:
9:
6:
4:
3:
2:
2026:
2015:
2012:
2010:
2007:
2006:
2004:
1991:
1987:
1983:
1979:
1975:
1971:
1964:
1961:
1956:
1952:
1948:
1944:
1943:Combinatorica
1937:
1934:
1928:
1911:
1907:
1900:
1897:
1894:
1888:
1880:
1877:
1862:
1859:
1853:
1850:
1847:
1841:
1821:
1818:
1812:
1809:
1806:
1800:
1797:
1791:
1788:
1785:
1779:
1756:
1753:
1750:
1727:
1724:
1718:
1715:
1712:
1706:
1686:
1683:
1677:
1674:
1671:
1665:
1657:
1654:
1639:
1636:
1630:
1627:
1624:
1618:
1598:
1595:
1589:
1586:
1583:
1577:
1574:
1568:
1565:
1562:
1556:
1536:
1528:
1525:
1510:
1507:
1504:
1498:
1495:
1492:
1486:
1466:
1460:
1457:
1454:
1448:
1428:
1425:
1419:
1416:
1413:
1407:
1387:
1367:
1359:
1356:
1353:
1350:
1347:
1344:
1329:
1326:
1320:
1317:
1314:
1306:
1302:
1281:
1259:
1255:
1251:
1243:
1239:
1235:
1230:
1226:
1219:
1216:
1208:
1204:
1200:
1195:
1191:
1182:
1178:
1152:
1148:
1144:
1139:
1135:
1109:
1105:
1082:
1078:
1057:
1035:
1031:
1005:
1001:
997:
992:
988:
984:
979:
975:
966:
962:
953:
950:
947:
944:
941:
938:
937:
936:
930:
928:
926:
922:
918:
913:
911:
903:
887:
884:
878:
875:
872:
864:
860:
854:
851:
848:
844:
833:
832:
831:
807:
804:
801:
793:
789:
785:
779:
776:
773:
765:
761:
752:
751:
748:
747:
746:
736:
719:
716:
713:
705:
701:
695:
691:
687:
681:
678:
675:
669:
661:
660:
644:
624:
604:
596:
579:
576:
573:
565:
561:
552:
551:
548:
547:
546:
540:
524:
518:
515:
512:
506:
503:
497:
494:
491:
485:
480:
477:
471:
468:
465:
458:
450:
449:
448:
445:
427:
424:
418:
415:
412:
406:
401:
398:
395:
391:
383:
365:
362:
359:
353:
350:
344:
341:
338:
332:
329:
323:
320:
317:
311:
304:
303:
302:
282:
279:
276:
250:
247:
244:
238:
231:
216:
196:
173:
170:
167:
161:
154:
139:
119:
96:
93:
90:
84:
77:
76:
75:
58:
55:
52:
46:
34:
32:
30:
26:
22:
1990:the original
1977:
1973:
1963:
1946:
1942:
1936:
934:
914:
907:
829:
744:
544:
446:
443:
300:
38:
28:
25:network flow
20:
18:
917:NP-complete
2003:Categories
1929:References
35:Definition
1898:−
1508:−
852:∈
845:∑
786:≤
692:∑
504:⋅
478:∈
459:∑
399:∈
392:∑
351:≤
330:≤
904:Solution
209:to node
132:to node
1658:- Let
1479:∞ and
1294:. Let
841:
74:with:
1772:with
1400:with
1170:with
1070:from
954:- If
617:from
1834:and
1699:and
1611:and
19:The
1982:doi
1951:doi
1097:to
637:to
2005::
1976:.
1972:.
1945:.
1441:,
927:.
888:0.
657:.
1984::
1978:5
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