1898:
31:
1531:
681:
of the curve is even and the subset for which the winding number is odd; these two subsets form a cut that includes all of the edges whose duals appear an odd number of times in the tour. The route inspection problem may be solved in polynomial time, and this duality allows the maximum cut problem to
786:
because the algorithm improves the cut by at least one edge at each step. When the algorithm terminates, at least half of the edges incident to every vertex belong to the cut, for otherwise moving the vertex would improve the cut. Therefore, the cut includes at least
1893:{\displaystyle {\begin{aligned}H&=-\sum _{ij\in E(V^{+})}J_{ij}-\sum _{ij\in E(V^{-})}J_{ij}+\sum _{ij\in \delta (V^{+})}J_{ij}\\&=-\sum _{ij\in E(G)}J_{ij}+2\sum _{ij\in \delta (V^{+})}J_{ij}\\&=C+2\sum _{ij\in \delta (V^{+})}J_{ij}\end{aligned}}}
224:
948:
282:
Bound (b) is often called the
Edwards-Erdős bound as Erdős conjectured it. Edwards proved the Edwards-Erdős bound using probabilistic method; Crowston et al. proved the bound using linear algebra and analysis of pseudo-boolean functions.
2725:
2510:
Crowston, R.; Fellows, M.; Gutin, G.; Jones, M.; Kim, E. J.; Rosamond, F.; Ruzsa, I. Z.; Thomassé, S.; Yeo, A. (2014), "Satisfying more than half of a system of linear equations over GF(2): A multivariate approach",
957:
is true, this is the best possible approximation ratio for maximum cut. Without such unproven assumptions, it has been proven to be NP-hard to approximate the max-cut value with an approximation ratio better than
755:
and repeatedly moves one vertex at a time from one side of the partition to the other, improving the solution at each step, until no more improvements of this type can be made. The number of iterations is at most
694:, meaning that there is no polynomial-time approximation scheme (PTAS), arbitrarily close to the optimal solution, for it, unless P = NP. Thus, every known polynomial-time approximation algorithm achieves an
486:
1358:
529:. Recently, Gutin and Yeo obtained a number of lower bounds for weighted Max-Cut extending the Poljak-Turzik bound for arbitrary weighted graphs and bounds for special classes of weighted graphs.
1536:
277:
991:
717:; therefore there is a simple deterministic polynomial-time 0.5-approximation algorithm as well. One such algorithm starts with an arbitrary partition of the vertices of the given graph
149:
863:
2383:
Barahona, Francisco; Grötschel, Martin; Jünger, Michael; Reinelt, Gerhard (1988). "An
Application of Combinatorial Optimization to Statistical Physics and Circuit Layout Design".
871:
1953:
114:
and its complement rather than the number of the edges. The weighted max-cut problem allowing both positive and negative weights can be trivially transformed into a weighted
1523:
1064:
1401:
1175:
665:(the problem of finding a shortest tour that visits each edge of a graph at least once), in the sense that the edges that do not belong to a maximum cut-set of a graph
1136:
1100:
753:
1487:
3208:
1457:
823:
1430:
1204:
1102:. Crowston et al. extended the fixed-parameter tractability result to the Balanced Subgraph Problem (BSP, see Lower bounds above) and improved the kernel size to
784:
709:: for each vertex flip a coin to decide to which half of the partition to assign it. In expectation, half of the edges are cut edges. This algorithm can be
1223:
and its edges as distances, the max cut algorithm divides a graph in two well-separated subsets. In other words, it can be naturally applied to perform
2929:
691:
2946:
3041:
2735:
828:
The polynomial-time approximation algorithm for Max-Cut with the best known approximation ratio is a method by
Goemans and Williamson using
677:. The optimal inspection tour forms a self-intersecting curve that separates the plane into two subsets, the subset of points for which the
581:
answer is easy to prove by presenting a large enough cut. The NP-completeness of the problem can be shown, for example, by a reduction from
3081:
Poljak, S.; Turzik, Z. (1986), "A polynomial time heuristic for certain subgraph optimization problems with guaranteed worst case bound",
2602:
Dunning, Iain; Gupta, Swati; Silberholz, John (2018), "What works best when? A systematic evaluation of heuristics for Max-Cut and QUBO",
3198:
714:
1227:. Compared to more common classification algorithms, it does not require a feature space, only the distances between elements within.
3167:
2485:
Bylka, S.; Idzik, A.; Tuza, I. (1999), "Maximum cuts: Improvements and local algorithmic analogues of the
Edwards-Erd6s inequality",
2354:
590:
2892:; Lingas, Andrzej; Seidel, Eike (2005), "Polynomial Time Approximation Schemes for MAX-BISECTION on Planar and Geometric Graphs",
413:
2536:
Crowston, R.; Gutin, G.; Jones, M.; Muciaccia, G. (2013), "Maximum balanced subgraph problem parameterized above lower bound",
2464:
Ausiello, Giorgio; Crescenzi, Pierluigi; Gambosi, Giorgio; Kann, Viggo; Marchetti-Spaccamela, Alberto; Protasi, Marco (2003),
586:
1267:
3203:
3193:
3171:
629:
2780:(1995), "Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming",
1964:
542:
39:
231:
961:
1220:
682:
also be solved in polynomial time for planar graphs. The
Maximum-Bisection problem is known however to be NP-hard.
2955:
2894:
1009:
829:
662:
582:
400:. Bound (a) was improved for special classes of graphs: triangle-free graphs, graphs of given maximum degree,
219:{\displaystyle \left\lceil {\frac {m}{2}}+{\sqrt {{\frac {m}{8}}+{\frac {1}{64}}}}-{\frac {1}{8}}\right\rceil }
107:
2984:
Khuller, Samir; Raghavachari, Balaji; Young, Neal E. (2007), "Greedy methods", in
Gonzalez, Teofilo F. (ed.),
3121:; Sorkin, Gregory; Sudan, Madhu; Williamson, David (2000), "Gadgets, Approximation, and Linear Programming",
2435:
954:
706:
59:
996:
In there is an extended analysis of 10 heuristics for this problem, including open-source implementation.
839:
3051:
2903:
2664:
Edwards, C. S. (1975), "An improved lower bound for the number of edges in a largest bipartite subgraph",
1012:(FPT), it is much harder to show fixed-parameter tractability for the problem of deciding whether a graph
943:{\displaystyle \alpha ={\frac {2}{\pi }}\min _{0\leq \theta \leq \pi }{\frac {\theta }{1-\cos \theta }}.}
2513:
1986:
1250:
1224:
55:
2466:
Complexity and
Approximation: Combinatorial Optimization Problems and Their Approximability Properties
1138:(holds also for BSP). Etscheid and Mnich improved the fixed-parameter tractability result for BSP to
1016:
has a cut of size at least the
Edwards-Erdős lower bound (see Lower bounds above) plus (the parameter)
3163:
2994:
1995:
1906:
702:
601:
2674:
Etscheid, M.; Mnich, M. (2018), "Linear
Kernels and Linear-Time Algorithms for Finding Large Cuts",
1492:
2908:
2777:
2760:
2538:
1242:
1023:
833:
695:
2571:
Crowston, R.; Jones, M.; Mnich, M. (2015), "Max-cut parameterized above the
Edwards–Erdős bound",
1370:
3055:
2972:
2876:
2859:
2817:
2801:
2782:
2705:
2652:
2619:
2590:
2547:
2452:
2408:
2360:
1246:
47:
2335:"Interactive graph cuts for optimal boundary & region segmentation of objects in N-D images"
1004:
While it is trivial to prove that the problem of finding a cut of size at least (the parameter)
1903:
Minimizing this energy is equivalent to the min-cut problem or by setting the graph weights as
1141:
3037:
2731:
2400:
2350:
1261:. For the Ising model on a graph G and only nearest-neighbor interactions, the Hamiltonian is
1105:
1069:
594:
720:
3144:
3090:
3067:
3029:
2964:
2913:
2868:
2842:
2791:
2721:
2717:
2695:
2685:
2642:
2611:
2582:
2557:
2522:
2496:
2444:
2392:
2342:
1990:
1462:
538:
1435:
790:
2889:
2833:
1406:
1180:
710:
574:
96:
51:
2854:
759:
624:
The optimization variant is known to be NP-Hard. The opposite problem, that of finding a
3011:
2773:
2487:
678:
2501:
3187:
3118:
3095:
3072:
2950:
2813:
2656:
2633:
2456:
1981:
295:
2805:
2709:
2594:
2976:
2942:
2925:
2880:
2751:
Maximum bipartite subgraph (decision version) is the problem GT25 in Appendix A1.2.
2676:
2573:
2364:
1525:
the set of edges that connect the two sets. We can then rewrite the Hamiltonian as
658:
3033:
2831:
Hadlock, F. (1975), "Finding a Maximum Cut of a Planar Graph in Polynomial Time",
2623:
669:
are the duals of the edges that are doubled in an optimal inspection tour of the
2477:
Maximum cut (optimisation version) is the problem ND14 in Appendix B (page 399).
1976:
1258:
1236:
625:
570:
286:
The proof of Crowston et al. allows us to extend the Edwards-Erdős bound to the
115:
95:
and the complementary subset is as large as possible. Equivalently, one wants a
2527:
370:
are different subsets. BSP aims at finding a partition with the maximum number
3148:
2968:
2917:
2759:
Gaur, Daya Ram; Krishnamurti, Ramesh (2007), "LP rounding and extensions", in
2690:
2586:
2562:
2448:
2339:
Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001
2334:
1254:
670:
2404:
2346:
2998:
3003:
Probability and Computing: Randomized Algorithms and Probabilistic Analysis
2647:
2615:
2951:"Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?"
2872:
2796:
2631:
Edwards, C. S. (1973), "Some extremal properties of bipartite subgraphs",
2396:
3123:
Proceedings of the 37th IEEE Symposium on Foundations of Computer Science
3106:
Surveys in Combinatorics, London Mathematical Society Lecture Note Series
649:, no polynomial-time algorithms for Max-Cut in general graphs are known.
407:
Poljak and Turzik extended the Edwards-Erdős bound to weighted Max-Cut:
110:, and the objective is to maximize the total weight of the edges between
646:
17:
3178:
3162:
Pierluigi Crescenzi, Viggo Kann, MagnĂşs HalldĂłrsson, Marek Karpinski,
2727:
Computers and Intractability: A Guide to the Theory of NP-Completeness
2700:
2412:
126:
Edwards obtained the following two lower bound for Max-Cut on a graph
2177:
316:, i.e. graphs where each edge is assigned + or –. For a partition of
85:
2846:
2748:
Maximum cut (decision version) is the problem ND16 in Appendix A2.2.
2822:
50:
whose size is at least the size of any other cut. That is, it is a
30:
2552:
29:
1020:. Crowston et al. proved that the problem can be solved in time
3104:
Scott, A. (2005), "Judicious partitions and related problems",
3058:(1991), "Optimization, approximation, and complexity classes",
2260:
2283:
3024:
Newman, Alantha (2008), "Max cut", in Kao, Ming-Yang (ed.),
2249:
77:
is as large as possible. Finding such a cut is known as the
2816:; Yeo, A. (2021), "Lower Bounds for Maximum Weighted Cut",
2161:
589:). The weighted version of the decision problem was one of
2429:
Alon, N.; Krivelevich, M.; Sudakov, B. (2005), "Maxcut in
593:; Karp showed the NP-completeness by a reduction from the
481:{\displaystyle {\frac {w(G)}{2}}+{\frac {w(T_{min})}{4}},}
2053:
84:
The problem can be stated simply as follows. One wants a
2225:
106:, where each edge is associated with a real number, its
91:
of the vertex set such that the number of edges between
2986:
Handbook of Approximation Algorithms and Metaheuristics
2765:
Handbook of Approximation Algorithms and Metaheuristics
2173:
1367:
of the graph is a spin site that can take a spin value
2294:
2065:
1249:, the Max Cut problem is equivalent to minimizing the
966:
604:
of the above decision problem is usually known as the
102:
There is a more general version of the problem called
99:
subgraph of the graph with as many edges as possible.
1909:
1534:
1495:
1465:
1438:
1409:
1373:
1353:{\displaystyle H=-\sum _{ij\in E(G)}J_{ij}s_{i}s_{j}}
1270:
1183:
1144:
1108:
1072:
1026:
964:
874:
842:
793:
762:
723:
416:
234:
152:
557:, determine whether there is a cut of size at least
2341:. Vol. 1. IEEE Comput. Soc. pp. 105–112.
541:related to maximum cuts has been studied widely in
3131:Zeng, Q.; Hou, J. (2017), "Bipartite Subgraphs of
2857:(2001), "Some optimal inapproximability results",
2308:
1947:
1892:
1517:
1481:
1451:
1424:
1395:
1352:
1198:
1169:
1130:
1094:
1058:
985:
942:
857:
817:
778:
747:
480:
271:
218:
2213:
2189:
1989:, equivalent to asking for the largest bipartite
2304:
2302:
2237:
892:
272:{\displaystyle {\frac {m}{2}}+{\frac {n-1}{4}}.}
2201:
2049:
2047:
2038:
986:{\displaystyle {\tfrac {16}{17}}\approx 0.941}
2320:
118:problem by flipping the sign in all weights.
8:
2932:", in Miller, R. E.; Thacher, J. W. (eds.),
2125:
2101:
628:is known to be efficiently solvable via the
142:is arbitrary, but in (b) it is connected):
573:. It is easy to see that the problem is in
385:. The Edwards-Erdős gives a lower bound on
27:Problem of finding a maximum cut in a graph
3172:"A compendium of NP optimization problems"
1000:Parameterized algorithms and kernelization
3177:Andrea Casini, Nicola Rebagliati (2012),
3094:
3071:
2930:Reducibility among combinatorial problems
2907:
2821:
2795:
2699:
2689:
2646:
2561:
2551:
2526:
2500:
1933:
1914:
1908:
1877:
1862:
1842:
1810:
1795:
1775:
1756:
1728:
1702:
1687:
1667:
1651:
1636:
1616:
1600:
1585:
1565:
1535:
1533:
1506:
1494:
1470:
1464:
1443:
1437:
1408:
1378:
1372:
1344:
1334:
1321:
1293:
1269:
1182:
1149:
1143:
1119:
1107:
1083:
1071:
1047:
1031:
1025:
965:
963:
913:
895:
881:
873:
841:
807:
802:
794:
792:
771:
763:
761:
722:
661:, the Maximum-Cut Problem is dual to the
454:
441:
417:
415:
248:
235:
233:
201:
186:
173:
171:
158:
151:
2226:Khuller, Raghavachari & Young (2007)
2113:
1963:The max cut problem has applications in
69:, such that the number of edges between
3060:Journal of Computer and System Sciences
2149:
2089:
2026:
2014:
2007:
1998:, a related concept for infinite graphs
3209:Computational problems in graph theory
3179:"A Python library for solving Max Cut"
2295:Dunning, Gupta & Silberholz (2018)
2272:
2066:Alon, Krivelevich & Sudakov (2005)
2378:
2376:
2374:
2174:Papadimitriou & Yannakakis (1991)
2077:
858:{\displaystyle \alpha \approx 0.878,}
836:that achieves an approximation ratio
520:and its minimum weight spanning tree
7:
2137:
2333:Boykov, Y.Y.; Jolly, M.-P. (2001).
715:method of conditional probabilities
2945:; Kindler, Guy; Mossel, Elchanan;
2934:Complexity of Computer Computation
2309:Crowston, Jones & Mnich (2015)
1432:into two sets, those with spin up
25:
396:for every connected signed graph
1403:A spin configuration partitions
2936:, Plenum Press, pp. 85–103
2666:Recent Advances in Graph Theory
2214:Mitzenmacher & Upfal (2005)
2190:Mitzenmacher & Upfal (2005)
1948:{\displaystyle w_{ij}=-J_{ij},}
3028:, Springer, pp. 489–492,
3014:; Raghavan, Prabhakar (1995),
2238:Gaur & Krishnamurti (2007)
2039:Bylka, Idzik & Tuza (1999)
1868:
1855:
1801:
1788:
1747:
1741:
1693:
1680:
1642:
1629:
1591:
1578:
1548:
1542:
1518:{\displaystyle \delta (V^{+})}
1512:
1499:
1419:
1413:
1312:
1306:
1280:
1274:
1193:
1187:
1177:and the kernel-size result to
1164:
1158:
1125:
1112:
1089:
1076:
1053:
1040:
803:
795:
772:
764:
742:
730:
591:Karp's 21 NP-complete problems
587:maximum satisfiability problem
466:
447:
429:
423:
1:
3034:10.1007/978-0-387-30162-4_219
2502:10.1016/S0012-365X(98)00115-0
2202:Motwani & Raghavan (1995)
1066:and admits a kernel of size
1059:{\displaystyle 8^{k}O(n^{4})}
404:-free graphs, etc., see e.g.
3096:10.1016/0012-365X(86)90192-5
3073:10.1016/0022-0000(91)90023-X
2604:INFORMS Journal on Computing
1396:{\displaystyle s_{i}=\pm 1.}
569:This problem is known to be
543:theoretical computer science
2321:Etscheid & Mnich (2018)
351:are in the same subset, or
34:An example of a maximum cut
3225:
3199:Combinatorial optimization
3052:Papadimitriou, Christos H.
3026:Encyclopedia of Algorithms
2528:10.1016/j.jcss.2013.10.002
2126:Garey & Johnson (1979)
2102:Poljak & Turzik (1986)
1234:
645:As the Max-Cut Problem is
641:Polynomial-time algorithms
3149:10.1017/S0004972716001295
2969:10.1137/S0097539705447372
2956:SIAM Journal on Computing
2918:10.1137/s009753970139567x
2895:SIAM Journal on Computing
2691:10.1007/s00453-017-0388-z
2587:10.1007/s00453-014-9870-z
2563:10.1016/j.tcs.2013.10.026
2449:10.1017/S0963548305007017
1459:and those with spin down
1170:{\displaystyle 8^{k}O(m)}
1010:fixed-parameter tractable
288:Balanced Subgraph Problem
2988:, Chapman & Hall/CRC
2767:, Chapman & Hall/CRC
2347:10.1109/iccv.2001.937505
1131:{\displaystyle O(k^{3})}
1095:{\displaystyle O(k^{5})}
830:semidefinite programming
698:strictly less than one.
686:Approximation algorithms
663:route inspection problem
630:Ford–Fulkerson algorithm
583:maximum 2-satisfiability
533:Computational complexity
2436:Combin. Probab. Comput.
1257:model, most simply the
955:unique games conjecture
748:{\displaystyle G=(V,E)}
707:approximation algorithm
690:The Max-Cut Problem is
3137:Bull. Aust. Math. Soc.
2648:10.4153/CJM-1973-048-x
2616:10.1287/ijoc.2017.0798
2284:Trevisan et al. (2000)
2250:Ausiello et al. (2003)
2114:Gutin & Yeo (2021)
2054:Crowston et al. (2014)
1949:
1894:
1519:
1483:
1482:{\displaystyle V^{-}.}
1453:
1426:
1397:
1354:
1219:Treating its nodes as
1200:
1171:
1132:
1096:
1060:
987:
944:
859:
819:
780:
749:
585:(a restriction of the
482:
332:is balanced if either
273:
220:
35:
3016:Randomized Algorithms
2995:Mitzenmacher, Michael
2873:10.1145/502090.502098
2797:10.1145/227683.227684
2514:J. Comput. Syst. Sci.
2397:10.1287/opre.36.3.493
2090:Zeng & Hou (2017)
1987:Odd cycle transversal
1955:the max-cut problem.
1950:
1895:
1520:
1484:
1454:
1452:{\displaystyle V^{+}}
1427:
1398:
1355:
1225:binary classification
1201:
1172:
1133:
1097:
1061:
988:
945:
860:
820:
818:{\displaystyle |E|/2}
781:
750:
620:, find a maximum cut.
483:
381:of balanced edges in
274:
221:
33:
3204:NP-complete problems
3194:Graph theory objects
2778:Williamson, David P.
2761:Gonzalez, Teofilo F.
2162:Jansen et al. (2005)
1996:Unfriendly partition
1907:
1532:
1493:
1463:
1436:
1425:{\displaystyle V(G)}
1407:
1371:
1268:
1199:{\displaystyle O(k)}
1181:
1142:
1106:
1070:
1024:
962:
872:
840:
791:
760:
721:
602:optimization variant
414:
232:
150:
3056:Yannakakis, Mihalis
2539:Theor. Comput. Sci.
2385:Operations Research
1243:statistical physics
1231:Theoretical physics
834:randomized rounding
779:{\displaystyle |E|}
696:approximation ratio
612:and is defined as:
606:Maximum-Cut Problem
516:are the weights of
2860:Journal of the ACM
2783:Journal of the ACM
2774:Goemans, Michel X.
2668:, pp. 167–181
2261:Khot et al. (2007)
1945:
1890:
1888:
1872:
1805:
1751:
1697:
1646:
1595:
1515:
1479:
1449:
1422:
1393:
1350:
1316:
1247:disordered systems
1196:
1167:
1128:
1092:
1056:
983:
975:
940:
912:
855:
815:
776:
745:
701:There is a simple
478:
269:
216:
60:complementary sets
36:
3164:Gerhard Woeginger
3043:978-0-387-30770-1
2737:978-0-7167-1045-5
2722:Johnson, David S.
2718:Garey, Michael R.
1838:
1771:
1724:
1663:
1612:
1561:
1363:Here each vertex
1289:
974:
935:
891:
889:
595:partition problem
473:
436:
264:
243:
209:
196:
194:
181:
166:
16:(Redirected from
3216:
3151:
3126:
3113:
3099:
3098:
3076:
3075:
3046:
3019:
3006:
2989:
2979:
2937:
2926:Karp, Richard M.
2920:
2911:
2890:Karpinski, Marek
2883:
2849:
2826:
2825:
2808:
2799:
2790:(6): 1115–1145,
2768:
2740:
2730:, W.H. Freeman,
2712:
2703:
2693:
2684:(9): 2574–2615,
2669:
2659:
2650:
2626:
2597:
2566:
2565:
2555:
2531:
2530:
2505:
2504:
2469:
2459:
2417:
2416:
2380:
2369:
2368:
2330:
2324:
2318:
2312:
2306:
2297:
2292:
2286:
2281:
2275:
2270:
2264:
2258:
2252:
2247:
2241:
2235:
2229:
2223:
2217:
2211:
2205:
2199:
2193:
2187:
2181:
2171:
2165:
2159:
2153:
2147:
2141:
2135:
2129:
2123:
2117:
2111:
2105:
2099:
2093:
2087:
2081:
2075:
2069:
2063:
2057:
2051:
2042:
2036:
2030:
2024:
2018:
2012:
1991:induced subgraph
1954:
1952:
1951:
1946:
1941:
1940:
1922:
1921:
1899:
1897:
1896:
1891:
1889:
1885:
1884:
1871:
1867:
1866:
1822:
1818:
1817:
1804:
1800:
1799:
1764:
1763:
1750:
1714:
1710:
1709:
1696:
1692:
1691:
1659:
1658:
1645:
1641:
1640:
1608:
1607:
1594:
1590:
1589:
1524:
1522:
1521:
1516:
1511:
1510:
1488:
1486:
1485:
1480:
1475:
1474:
1458:
1456:
1455:
1450:
1448:
1447:
1431:
1429:
1428:
1423:
1402:
1400:
1399:
1394:
1383:
1382:
1359:
1357:
1356:
1351:
1349:
1348:
1339:
1338:
1329:
1328:
1315:
1215:Machine learning
1205:
1203:
1202:
1197:
1176:
1174:
1173:
1168:
1154:
1153:
1137:
1135:
1134:
1129:
1124:
1123:
1101:
1099:
1098:
1093:
1088:
1087:
1065:
1063:
1062:
1057:
1052:
1051:
1036:
1035:
992:
990:
989:
984:
976:
967:
949:
947:
946:
941:
936:
934:
914:
911:
890:
882:
864:
862:
861:
856:
824:
822:
821:
816:
811:
806:
798:
785:
783:
782:
777:
775:
767:
754:
752:
751:
746:
539:decision problem
528:
519:
515:
501:
487:
485:
484:
479:
474:
469:
465:
464:
442:
437:
432:
418:
403:
399:
395:
384:
380:
369:
365:
361:
350:
346:
342:
331:
327:
323:
319:
315:
278:
276:
275:
270:
265:
260:
249:
244:
236:
225:
223:
222:
217:
215:
211:
210:
202:
197:
195:
187:
182:
174:
172:
167:
159:
141:
137:
133:
129:
113:
104:weighted max-cut
94:
90:
76:
72:
68:
64:
21:
3224:
3223:
3219:
3218:
3217:
3215:
3214:
3213:
3184:
3183:
3159:
3135:-free Graphs",
3130:
3117:
3103:
3080:
3050:
3044:
3023:
3012:Motwani, Rajeev
3010:
2993:
2983:
2947:O'Donnell, Ryan
2941:
2924:
2888:Jansen, Klaus;
2887:
2853:
2847:10.1137/0204019
2834:SIAM J. Comput.
2830:
2812:
2772:
2758:
2738:
2716:
2673:
2663:
2630:
2601:
2570:
2535:
2509:
2484:
2463:
2433:-free graphs",
2428:
2425:
2420:
2382:
2381:
2372:
2357:
2332:
2331:
2327:
2319:
2315:
2307:
2300:
2293:
2289:
2282:
2278:
2271:
2267:
2259:
2255:
2248:
2244:
2236:
2232:
2224:
2220:
2212:
2208:
2200:
2196:
2188:
2184:
2172:
2168:
2160:
2156:
2148:
2144:
2136:
2132:
2124:
2120:
2112:
2108:
2100:
2096:
2088:
2084:
2076:
2072:
2064:
2060:
2052:
2045:
2037:
2033:
2025:
2021:
2013:
2009:
2005:
1973:
1961:
1929:
1910:
1905:
1904:
1887:
1886:
1873:
1858:
1820:
1819:
1806:
1791:
1752:
1712:
1711:
1698:
1683:
1647:
1632:
1596:
1581:
1551:
1530:
1529:
1502:
1491:
1490:
1489:We denote with
1466:
1461:
1460:
1439:
1434:
1433:
1405:
1404:
1374:
1369:
1368:
1340:
1330:
1317:
1266:
1265:
1239:
1233:
1217:
1212:
1179:
1178:
1145:
1140:
1139:
1115:
1104:
1103:
1079:
1068:
1067:
1043:
1027:
1022:
1021:
1002:
960:
959:
918:
870:
869:
838:
837:
789:
788:
758:
757:
719:
718:
688:
655:
643:
638:
553:and an integer
535:
527:
521:
517:
513:
503:
492:
450:
443:
419:
412:
411:
401:
397:
386:
382:
371:
367:
363:
352:
348:
344:
333:
329:
325:
321:
317:
298:
250:
230:
229:
157:
153:
148:
147:
139:
135:
131:
127:
124:
111:
92:
88:
79:max-cut problem
74:
70:
66:
62:
54:of the graph's
28:
23:
22:
15:
12:
11:
5:
3222:
3220:
3212:
3211:
3206:
3201:
3196:
3186:
3185:
3182:
3181:
3175:
3158:
3157:External links
3155:
3154:
3153:
3128:
3119:Trevisan, Luca
3115:
3101:
3083:Discrete Math.
3078:
3066:(3): 425–440,
3048:
3042:
3021:
3008:
2991:
2981:
2963:(1): 319–357,
2939:
2922:
2909:10.1.1.62.5082
2902:(1): 110–119,
2885:
2867:(4): 798–859,
2851:
2841:(3): 221–225,
2828:
2810:
2770:
2755:
2754:
2753:
2752:
2749:
2743:
2742:
2736:
2714:
2671:
2661:
2641:(3): 475–485,
2628:
2610:(3): 608–624,
2599:
2581:(3): 734–757,
2568:
2533:
2521:(4): 687–696,
2507:
2495:(1–3): 39–58,
2488:Discrete Math.
2481:
2480:
2479:
2478:
2472:
2471:
2461:
2424:
2421:
2419:
2418:
2391:(3): 493–513.
2370:
2355:
2325:
2313:
2298:
2287:
2276:
2265:
2253:
2242:
2230:
2218:
2206:
2194:
2182:
2180:-completeness.
2166:
2154:
2150:Hadlock (1975)
2142:
2130:
2118:
2106:
2094:
2082:
2070:
2058:
2043:
2031:
2027:Edwards (1975)
2019:
2015:Edwards (1973)
2006:
2004:
2001:
2000:
1999:
1993:
1984:
1979:
1972:
1969:
1960:
1959:Circuit design
1957:
1944:
1939:
1936:
1932:
1928:
1925:
1920:
1917:
1913:
1901:
1900:
1883:
1880:
1876:
1870:
1865:
1861:
1857:
1854:
1851:
1848:
1845:
1841:
1837:
1834:
1831:
1828:
1825:
1823:
1821:
1816:
1813:
1809:
1803:
1798:
1794:
1790:
1787:
1784:
1781:
1778:
1774:
1770:
1767:
1762:
1759:
1755:
1749:
1746:
1743:
1740:
1737:
1734:
1731:
1727:
1723:
1720:
1717:
1715:
1713:
1708:
1705:
1701:
1695:
1690:
1686:
1682:
1679:
1676:
1673:
1670:
1666:
1662:
1657:
1654:
1650:
1644:
1639:
1635:
1631:
1628:
1625:
1622:
1619:
1615:
1611:
1606:
1603:
1599:
1593:
1588:
1584:
1580:
1577:
1574:
1571:
1568:
1564:
1560:
1557:
1554:
1552:
1550:
1547:
1544:
1541:
1538:
1537:
1514:
1509:
1505:
1501:
1498:
1478:
1473:
1469:
1446:
1442:
1421:
1418:
1415:
1412:
1392:
1389:
1386:
1381:
1377:
1361:
1360:
1347:
1343:
1337:
1333:
1327:
1324:
1320:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1292:
1288:
1285:
1282:
1279:
1276:
1273:
1235:Main article:
1232:
1229:
1216:
1213:
1211:
1208:
1195:
1192:
1189:
1186:
1166:
1163:
1160:
1157:
1152:
1148:
1127:
1122:
1118:
1114:
1111:
1091:
1086:
1082:
1078:
1075:
1055:
1050:
1046:
1042:
1039:
1034:
1030:
1001:
998:
982:
979:
973:
970:
951:
950:
939:
933:
930:
927:
924:
921:
917:
910:
907:
904:
901:
898:
894:
888:
885:
880:
877:
854:
851:
848:
845:
814:
810:
805:
801:
797:
774:
770:
766:
744:
741:
738:
735:
732:
729:
726:
687:
684:
679:winding number
654:
651:
642:
639:
637:
634:
622:
621:
616:Given a graph
600:The canonical
567:
566:
549:Given a graph
537:The following
534:
531:
525:
511:
489:
488:
477:
472:
468:
463:
460:
457:
453:
449:
446:
440:
435:
431:
428:
425:
422:
280:
279:
268:
263:
259:
256:
253:
247:
242:
239:
226:
214:
208:
205:
200:
193:
190:
185:
180:
177:
170:
165:
162:
156:
138:edges (in (a)
123:
120:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3221:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3192:
3191:
3189:
3180:
3176:
3173:
3169:
3168:"Maximum Cut"
3165:
3161:
3160:
3156:
3150:
3146:
3142:
3138:
3134:
3129:
3124:
3120:
3116:
3111:
3107:
3102:
3097:
3092:
3089:(1): 99–104,
3088:
3084:
3079:
3074:
3069:
3065:
3061:
3057:
3053:
3049:
3045:
3039:
3035:
3031:
3027:
3022:
3017:
3013:
3009:
3004:
3000:
2996:
2992:
2987:
2982:
2978:
2974:
2970:
2966:
2962:
2958:
2957:
2952:
2948:
2944:
2943:Khot, Subhash
2940:
2935:
2931:
2927:
2923:
2919:
2915:
2910:
2905:
2901:
2897:
2896:
2891:
2886:
2882:
2878:
2874:
2870:
2866:
2862:
2861:
2856:
2855:HĂĄstad, Johan
2852:
2848:
2844:
2840:
2836:
2835:
2829:
2824:
2819:
2815:
2811:
2807:
2803:
2798:
2793:
2789:
2785:
2784:
2779:
2775:
2771:
2766:
2762:
2757:
2756:
2750:
2747:
2746:
2745:
2744:
2739:
2733:
2729:
2728:
2723:
2719:
2715:
2711:
2707:
2702:
2697:
2692:
2687:
2683:
2679:
2678:
2672:
2667:
2662:
2658:
2654:
2649:
2644:
2640:
2636:
2635:
2634:Can. J. Math.
2629:
2625:
2621:
2617:
2613:
2609:
2605:
2600:
2596:
2592:
2588:
2584:
2580:
2576:
2575:
2569:
2564:
2559:
2554:
2549:
2545:
2541:
2540:
2534:
2529:
2524:
2520:
2516:
2515:
2508:
2503:
2498:
2494:
2490:
2489:
2483:
2482:
2476:
2475:
2474:
2473:
2467:
2462:
2458:
2454:
2450:
2446:
2442:
2438:
2437:
2432:
2427:
2426:
2422:
2414:
2410:
2406:
2402:
2398:
2394:
2390:
2386:
2379:
2377:
2375:
2371:
2366:
2362:
2358:
2356:0-7695-1143-0
2352:
2348:
2344:
2340:
2336:
2329:
2326:
2322:
2317:
2314:
2310:
2305:
2303:
2299:
2296:
2291:
2288:
2285:
2280:
2277:
2274:
2273:HĂĄstad (2001)
2269:
2266:
2262:
2257:
2254:
2251:
2246:
2243:
2239:
2234:
2231:
2227:
2222:
2219:
2215:
2210:
2207:
2203:
2198:
2195:
2191:
2186:
2183:
2179:
2175:
2170:
2167:
2163:
2158:
2155:
2151:
2146:
2143:
2139:
2134:
2131:
2127:
2122:
2119:
2115:
2110:
2107:
2103:
2098:
2095:
2091:
2086:
2083:
2079:
2074:
2071:
2067:
2062:
2059:
2055:
2050:
2048:
2044:
2040:
2035:
2032:
2028:
2023:
2020:
2016:
2011:
2008:
2002:
1997:
1994:
1992:
1988:
1985:
1983:
1982:Minimum k-cut
1980:
1978:
1975:
1974:
1970:
1968:
1966:
1958:
1956:
1942:
1937:
1934:
1930:
1926:
1923:
1918:
1915:
1911:
1881:
1878:
1874:
1863:
1859:
1852:
1849:
1846:
1843:
1839:
1835:
1832:
1829:
1826:
1824:
1814:
1811:
1807:
1796:
1792:
1785:
1782:
1779:
1776:
1772:
1768:
1765:
1760:
1757:
1753:
1744:
1738:
1735:
1732:
1729:
1725:
1721:
1718:
1716:
1706:
1703:
1699:
1688:
1684:
1677:
1674:
1671:
1668:
1664:
1660:
1655:
1652:
1648:
1637:
1633:
1626:
1623:
1620:
1617:
1613:
1609:
1604:
1601:
1597:
1586:
1582:
1575:
1572:
1569:
1566:
1562:
1558:
1555:
1553:
1545:
1539:
1528:
1527:
1526:
1507:
1503:
1496:
1476:
1471:
1467:
1444:
1440:
1416:
1410:
1390:
1387:
1384:
1379:
1375:
1366:
1345:
1341:
1335:
1331:
1325:
1322:
1318:
1309:
1303:
1300:
1297:
1294:
1290:
1286:
1283:
1277:
1271:
1264:
1263:
1262:
1260:
1256:
1252:
1248:
1244:
1238:
1230:
1228:
1226:
1222:
1214:
1209:
1207:
1190:
1184:
1161:
1155:
1150:
1146:
1120:
1116:
1109:
1084:
1080:
1073:
1048:
1044:
1037:
1032:
1028:
1019:
1015:
1011:
1007:
999:
997:
994:
980:
977:
971:
968:
956:
937:
931:
928:
925:
922:
919:
915:
908:
905:
902:
899:
896:
886:
883:
878:
875:
868:
867:
866:
852:
849:
846:
843:
835:
831:
826:
812:
808:
799:
768:
739:
736:
733:
727:
724:
716:
712:
708:
704:
699:
697:
693:
685:
683:
680:
676:
672:
668:
664:
660:
659:planar graphs
653:Planar graphs
652:
650:
648:
640:
635:
633:
631:
627:
619:
615:
614:
613:
611:
607:
603:
598:
596:
592:
588:
584:
580:
576:
572:
564:
560:
556:
552:
548:
547:
546:
544:
540:
532:
530:
524:
510:
506:
499:
495:
475:
470:
461:
458:
455:
451:
444:
438:
433:
426:
420:
410:
409:
408:
405:
393:
389:
378:
374:
359:
355:
340:
336:
320:into subsets
313:
309:
305:
301:
297:
296:signed graphs
293:
289:
284:
266:
261:
257:
254:
251:
245:
240:
237:
227:
212:
206:
203:
198:
191:
188:
183:
178:
175:
168:
163:
160:
154:
145:
144:
143:
134:vertices and
121:
119:
117:
109:
105:
100:
98:
87:
82:
80:
61:
57:
53:
49:
45:
41:
32:
19:
3140:
3136:
3132:
3122:
3109:
3105:
3086:
3082:
3063:
3059:
3025:
3015:
3002:
2985:
2960:
2954:
2933:
2899:
2893:
2864:
2858:
2838:
2832:
2787:
2781:
2764:
2726:
2681:
2677:Algorithmica
2675:
2665:
2638:
2632:
2607:
2603:
2578:
2574:Algorithmica
2572:
2543:
2537:
2518:
2512:
2492:
2486:
2465:
2440:
2434:
2430:
2388:
2384:
2338:
2328:
2316:
2290:
2279:
2268:
2256:
2245:
2233:
2221:
2216:, Sect. 6.3.
2209:
2204:, Sect. 5.1.
2197:
2192:, Sect. 6.2.
2185:
2169:
2157:
2145:
2133:
2121:
2109:
2097:
2085:
2078:Scott (2005)
2073:
2061:
2034:
2022:
2010:
1962:
1902:
1364:
1362:
1240:
1218:
1210:Applications
1017:
1013:
1005:
1003:
995:
952:
827:
711:derandomized
700:
689:
674:
666:
657:However, in
656:
644:
623:
617:
609:
605:
599:
578:
568:
562:
558:
554:
550:
536:
522:
508:
504:
497:
493:
490:
406:
391:
387:
376:
372:
357:
353:
338:
334:
311:
307:
303:
299:
291:
287:
285:
281:
125:
122:Lower bounds
103:
101:
83:
78:
43:
37:
3143:(3): 1–13,
3018:, Cambridge
3005:, Cambridge
2443:: 629–647,
2138:Karp (1972)
1977:Minimum cut
1965:VLSI design
1259:Ising model
1251:Hamiltonian
1237:Ising model
626:minimum cut
571:NP-complete
116:minimum cut
44:maximum cut
3188:Categories
2999:Upfal, Eli
2823:2104.05536
2701:11420/4693
2468:, Springer
2423:References
1255:spin glass
1206:vertices.
703:randomized
671:dual graph
636:Algorithms
328:, an edge
3125:: 617–626
2928:(1972), "
2904:CiteSeerX
2814:Gutin, G.
2657:121925638
2553:1212.6848
2546:: 53–64,
2457:123485000
2405:0030-364X
1927:−
1853:δ
1850:∈
1840:∑
1786:δ
1783:∈
1773:∑
1736:∈
1726:∑
1722:−
1678:δ
1675:∈
1665:∑
1638:−
1624:∈
1614:∑
1610:−
1573:∈
1563:∑
1559:−
1497:δ
1472:−
1388:±
1301:∈
1291:∑
1287:−
978:≈
932:θ
929:
923:−
916:θ
909:π
906:≤
903:θ
900:≤
887:π
876:α
847:≈
844:α
713:with the
255:−
199:−
97:bipartite
58:into two
52:partition
3166:(2000),
3112:: 95–117
3001:(2005),
2949:(2007),
2806:15794408
2724:(1979),
2710:16301072
2595:14973734
1971:See also
1221:features
692:APX-hard
213:⌉
155:⌈
56:vertices
2977:2090495
2881:5120748
2763:(ed.),
2365:2245438
953:If the
825:edges.
647:NP-hard
610:Max-Cut
294:) on
18:Max cut
3040:
2975:
2906:
2879:
2804:
2734:
2708:
2655:
2624:485706
2622:
2593:
2455:
2413:170992
2411:
2403:
2363:
2353:
2178:MaxSNP
2176:prove
865:where
491:where
108:weight
86:subset
3170:, in
2973:S2CID
2877:S2CID
2818:arXiv
2802:S2CID
2706:S2CID
2653:S2CID
2620:S2CID
2591:S2CID
2548:arXiv
2453:S2CID
2409:JSTOR
2361:S2CID
2003:Notes
1253:of a
981:0.941
850:0.878
360:) = –
341:) = +
130:with
46:is a
40:graph
38:In a
3038:ISBN
2732:ISBN
2401:ISSN
2351:ISBN
1245:and
832:and
705:0.5-
577:: a
502:and
366:and
362:and
347:and
343:and
324:and
228:(b)
146:(a)
73:and
65:and
42:, a
3145:doi
3110:327
3091:doi
3068:doi
3030:doi
2965:doi
2914:doi
2869:doi
2843:doi
2792:doi
2696:hdl
2686:doi
2643:doi
2612:doi
2583:doi
2558:doi
2544:513
2523:doi
2497:doi
2493:194
2445:doi
2393:doi
2343:doi
1241:In
1008:is
926:cos
893:min
673:of
608:or
579:yes
561:in
526:min
512:min
302:= (
292:BSP
48:cut
3190::
3141:30
3139:,
3108:,
3087:58
3085:,
3064:43
3062:,
3054:;
3036:,
2997:;
2971:,
2961:37
2959:,
2953:,
2912:,
2900:35
2898:,
2875:,
2865:48
2863:,
2837:,
2800:,
2788:42
2786:,
2776:;
2720:;
2704:,
2694:,
2682:80
2680:,
2651:,
2639:25
2637:,
2618:,
2608:30
2606:,
2589:,
2579:72
2577:,
2556:,
2542:,
2519:80
2517:,
2491:,
2451:,
2441:14
2439:,
2407:.
2399:.
2389:36
2387:.
2373:^
2359:.
2349:.
2337:.
2301:^
2046:^
1967:.
1391:1.
993:.
972:17
969:16
632:.
597:.
575:NP
545::
358:xy
339:xy
330:xy
310:,
306:,
192:64
81:.
3174:.
3152:.
3147::
3133:H
3127:.
3114:.
3100:.
3093::
3077:.
3070::
3047:.
3032::
3020:.
3007:.
2990:.
2980:.
2967::
2938:.
2921:.
2916::
2884:.
2871::
2850:.
2845::
2839:4
2827:.
2820::
2809:.
2794::
2769:.
2741:.
2713:.
2698::
2688::
2670:.
2660:.
2645::
2627:.
2614::
2598:.
2585::
2567:.
2560::
2550::
2532:.
2525::
2506:.
2499::
2470:.
2460:.
2447::
2431:H
2415:.
2395::
2367:.
2345::
2323:.
2311:.
2263:.
2240:.
2228:.
2164:.
2152:.
2140:.
2128:.
2116:.
2104:.
2092:.
2080:.
2068:.
2056:.
2041:.
2029:.
2017:.
1943:,
1938:j
1935:i
1931:J
1924:=
1919:j
1916:i
1912:w
1882:j
1879:i
1875:J
1869:)
1864:+
1860:V
1856:(
1847:j
1844:i
1836:2
1833:+
1830:C
1827:=
1815:j
1812:i
1808:J
1802:)
1797:+
1793:V
1789:(
1780:j
1777:i
1769:2
1766:+
1761:j
1758:i
1754:J
1748:)
1745:G
1742:(
1739:E
1733:j
1730:i
1719:=
1707:j
1704:i
1700:J
1694:)
1689:+
1685:V
1681:(
1672:j
1669:i
1661:+
1656:j
1653:i
1649:J
1643:)
1634:V
1630:(
1627:E
1621:j
1618:i
1605:j
1602:i
1598:J
1592:)
1587:+
1583:V
1579:(
1576:E
1570:j
1567:i
1556:=
1549:]
1546:s
1543:[
1540:H
1513:)
1508:+
1504:V
1500:(
1477:.
1468:V
1445:+
1441:V
1420:)
1417:G
1414:(
1411:V
1385:=
1380:i
1376:s
1365:i
1346:j
1342:s
1336:i
1332:s
1326:j
1323:i
1319:J
1313:)
1310:G
1307:(
1304:E
1298:j
1295:i
1284:=
1281:]
1278:s
1275:[
1272:H
1194:)
1191:k
1188:(
1185:O
1165:)
1162:m
1159:(
1156:O
1151:k
1147:8
1126:)
1121:3
1117:k
1113:(
1110:O
1090:)
1085:5
1081:k
1077:(
1074:O
1054:)
1049:4
1045:n
1041:(
1038:O
1033:k
1029:8
1018:k
1014:G
1006:k
938:.
920:1
897:0
884:2
879:=
853:,
813:2
809:/
804:|
800:E
796:|
773:|
769:E
765:|
743:)
740:E
737:,
734:V
731:(
728:=
725:G
675:G
667:G
618:G
565:.
563:G
559:k
555:k
551:G
523:T
518:G
514:)
509:T
507:(
505:w
500:)
498:G
496:(
494:w
476:,
471:4
467:)
462:n
459:i
456:m
452:T
448:(
445:w
439:+
434:2
430:)
427:G
424:(
421:w
402:H
398:G
394:)
392:G
390:(
388:b
383:G
379:)
377:G
375:(
373:b
368:y
364:x
356:(
354:s
349:y
345:x
337:(
335:s
326:W
322:U
318:V
314:)
312:s
308:E
304:V
300:G
290:(
267:.
262:4
258:1
252:n
246:+
241:2
238:m
207:8
204:1
189:1
184:+
179:8
176:m
169:+
164:2
161:m
140:G
136:m
132:n
128:G
112:S
93:S
89:S
75:T
71:S
67:T
63:S
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
