5769:. Owing to the diminishing returns property, submodular functions naturally model costs of items, since there is often a larger discount, with an increase in the items one buys. Submodular functions model notions of complexity, similarity and cooperation when they appear in minimization problems. In maximization problems, on the other hand, they model notions of diversity, information and coverage.
4348:
4011:
5552:
3686:
4459:
786:
4178:
3841:
5736:
Minimization/maximization of a submodular function subject to a submodular level set constraint (also known as submodular optimization subject to submodular cover or submodular knapsack constraint) admits bounded approximation
3362:
5135:
1286:
1525:
4614:
400:
3521:
4134:
3797:
3230:
4716:
2691:
2261:
1630:
1422:
157:
5226:
2475:
2027:
1828:
521:
602:
3026:
5393:
4789:
4555:
1156:
2645:
sets. For example: we know that the value of receiving house A and house B is V, and we want to know the value of receiving 40% of house A and 60% of house B. To this end, we need a
302:
5740:
Partitioning data based on a submodular function to maximize the average welfare is known as the submodular welfare problem, which also admits bounded approximation guarantees (see
5388:
1198:
3565:
2844:
435:
244:
2347:
2792:
2505:
2291:
2057:
1899:
1660:
1551:
550:
4851:
4173:
3836:
3560:
3269:
1949:
5864:
3091:
6270:
N. Buchbinder, M. Feldman, J. Naor and R. Schwartz, A tight linear time (1/2)-approximation for unconstrained submodular maximization, Proc. of 53rd FOCS (2012), pp. 649-658.
642:
2630:
2388:
2182:
3142:
1450:
1091:
5575:
which concerns optimizing a convex or concave function can also be described as the problem of maximizing or minimizing a submodular function subject to some constraints.
5155:
4944:
1047:
270:
2946:
2882:
2766:
2727:
1319:
184:
4361:
3384:
5007:
2572:
2124:
1345:
814:
208:
112:
2908:
2598:
2150:
924:
869:
5715:
5670:
5334:
5307:
5280:
5253:
4053:
2534:
2086:
1718:
1689:
5175:
5047:
5027:
4987:
4964:
4891:
4871:
4739:
4634:
4479:
3440:
3046:
2973:
1869:
1738:
1017:
984:
964:
944:
889:
834:
3420:
4343:{\displaystyle f^{+}(\mathbf {x} )=\max \left(\sum _{S}\alpha _{S}f(S):\sum _{S}\alpha _{S}1_{S}=\mathbf {x} ,\sum _{S}\alpha _{S}=1,\alpha _{S}\geq 0\right)}
4006:{\displaystyle f^{-}(\mathbf {x} )=\min \left(\sum _{S}\alpha _{S}f(S):\sum _{S}\alpha _{S}1_{S}=\mathbf {x} ,\sum _{S}\alpha _{S}=1,\alpha _{S}\geq 0\right)}
647:
6335:
G. Calinescu, C. Chekuri, M. Pál and J. Vondrák, Maximizing a submodular set function subject to a matroid constraint, SIAM J. Comp. 40:6 (2011), 1740-1766.
6353:
Y. Filmus, J. Ward, A tight combinatorial algorithm for submodular maximization subject to a matroid constraint, Proc. of 53rd FOCS (2012), pp. 659-668.
42:
that, informally, describes the relationship between a set of inputs and an output, where adding more of one input has a decreasing additional benefit (
3276:
5619:
even in the unconstrained setting. Thus, most of the works in this field are concerned with polynomial-time approximation algorithms, including
5792:
6362:
M. Narasimhan and J. Bilmes, A submodular-supermodular procedure with applications to discriminative structure learning, In Proc. UAI (2005).
5991:
5946:
5729:
Apart from submodular minimization and maximization, there are several other natural optimization problems related to submodular functions.
6380:
R. Iyer and J. Bilmes, Submodular
Optimization Subject to Submodular Cover and Submodular Knapsack Constraints, In Advances of NIPS (2013).
6550:
5819:
S. Tschiatschek, R. Iyer, H. Wei and J. Bilmes, Learning
Mixtures of Submodular Functions for Image Collection Summarization, NIPS-2014.
5052:
83:
6233:
Z. Svitkina and L. Fleischer, Submodular approximation: Sampling-based algorithms and lower bounds, SIAM Journal on
Computing (2011).
6163:
Iwata, S.; Fleischer, L.; Fujishige, S. (2001). "A combinatorial strongly polynomial algorithm for minimizing submodular functions".
1211:
6555:
6505:
6483:
6465:
6443:
6418:
5884:
3387:
6371:
R. Iyer and J. Bilmes, Algorithms for
Approximate Minimization of the Difference between Submodular Functions, In Proc. UAI (2012).
6344:
M. Feldman, J. Naor and R. Schwartz, A unified continuous greedy algorithm for submodular maximization, Proc. of 52nd FOCS (2011).
1455:
4560:
307:
3453:
4066:
3729:
3162:
4639:
2655:
2195:
1564:
1356:
121:
5180:
2409:
1961:
1762:
440:
2030:
1556:
555:
75:
5624:
5547:{\displaystyle \mathbb {E} \geq \sum _{R\subseteq }\Pi _{i\in R}p_{i}\Pi _{i\notin R}(1-p_{i})f(\cup _{i\in R}A_{i})}
2978:
6435:
6261:, V. Mirrokni and J. Vondrák, Maximizing non-monotone submodular functions, Proc. of 48th FOCS (2007), pp. 461–471.
5869:
Handbook of
Approximation Algorithms and Metaheuristics, Second Edition: Methodologies and Traditional Applications
2641:
Often, given a submodular set function that describes the values of various sets, we need to compute the values of
6283:; Wolsey, L. A.; Fisher, M. L. (1978). "An analysis of approximations for maximizing submodular set functions I".
5631:
The problem of maximizing a non-negative submodular function admits a 1/2 approximation algorithm. Computing the
2951:
1203:
4744:
1830:
be the ground set on which a matroid is defined. Then the rank function of the matroid is a submodular function.
5673:
4501:
793:
51:
1103:
3681:{\displaystyle F(\mathbf {x} )=\sum _{S\subseteq \Omega }f(S)\prod _{i\in S}x_{i}\prod _{i\notin S}(1-x_{i})}
3147:
Several kinds of continuous extensions of submodular functions are commonly used, which are described below.
275:
6560:
6497:
6427:
5339:
71:
67:
1161:
2797:
408:
217:
6076:
5926:
3156:
2296:
5778:
5733:
Minimizing the difference between two submodular functions is not only NP hard, but also inapproximable.
4491:
2771:
2484:
2270:
2036:
1878:
1639:
1530:
529:
4797:
4139:
3802:
3526:
3235:
1904:
5638:
The problem of maximizing a monotone submodular function subject to a cardinality constraint admits a
3051:
1553:
is called a coverage function. This can be generalized by adding non-negative weights to the elements.
5840:
A. Krause and C. Guestrin, Beyond
Convexity: Submodularity in Machine Learning, Tutorial at ICML-2008
5828:
A. Krause and C. Guestrin, Near-optimal nonmyopic value of information in graphical models, UAI-2005.
5741:
5572:
607:
5583:
The hardness of minimizing a submodular set function depends on constraints imposed on the problem.
2603:
2352:
2155:
6410:
6402:
6198:
6080:
6072:
5900:
5592:
4454:{\displaystyle f^{+}(\mathbf {x} )\geq F(\mathbf {x} )\geq f^{-}(\mathbf {x} )=f^{L}(\mathbf {x} )}
3096:
1052:
47:
43:
5721:
Many of these algorithms can be unified within a semi-differential based framework of algorithms.
5140:
4896:
1026:
249:
6300:
6180:
6145:
6110:
5997:
5952:
4495:
2913:
2849:
2732:
2696:
2187:
1291:
162:
59:
816:
is not assumed finite, then the above conditions are not equivalent. In particular a function
6249:
and J. Bilmes, Fast
Semidifferential based submodular function optimization, Proc. ICML (2013).
6501:
6479:
6461:
6439:
6414:
6036:
5987:
5942:
5880:
3369:
1430:
115:
79:
6319:
4992:
2539:
2091:
1324:
799:
193:
97:
6511:
6292:
6280:
6246:
6214:
6172:
6137:
6100:
6092:
6028:
5979:
5934:
5872:
5762:
5620:
5568:
2887:
2577:
2129:
1633:
894:
839:
63:
5810:
H. Lin and J. Bilmes, A Class of
Submodular Functions for Document Summarization, ACL-2011.
5686:
5641:
5312:
5285:
5258:
5231:
62:. Recently, submodular functions have also found utility in several real world problems in
6515:
5766:
5588:
5564:
4019:
2510:
2264:
2062:
1745:
1741:
1694:
1665:
6055:
6203:"A combinatorial algorithm minimizing submodular functions in strongly polynomial time"
5160:
5032:
5012:
4972:
4949:
4876:
4856:
4724:
4619:
4464:
3425:
3031:
2958:
2478:
1854:
1723:
1002:
969:
949:
929:
874:
819:
781:{\displaystyle f(X\cup \{x_{1}\})+f(X\cup \{x_{2}\})\geq f(X\cup \{x_{1},x_{2}\})+f(X)}
3393:
946:
is infinite satisfies the first condition above, but the second condition fails when
6544:
6528:
6304:
5956:
6149:
6114:
5978:. STOC '08. New York, NY, USA: Association for Computing Machinery. pp. 67–74.
5972:"Optimal approximation for the submodular welfare problem in the value oracle model"
6083:(1981). "The ellipsoid method and its consequences in combinatorial optimization".
5157:
is the empty set. More generally consider the following random process where a set
1754:
39:
6184:
6001:
5602:
The problem of minimizing a submodular function with a cardinality lower bound is
3706:, the two inner products represent the probability that the chosen set is exactly
6389:
J. Bilmes, Submodularity in
Machine Learning Applications, Tutorial at AAAI-2015.
5938:
6258:
5787:
5758:
5632:
5596:
2399:
A non-monotone submodular function which is not symmetric is called asymmetric.
55:
5587:
The unconstrained problem of minimizing a submodular function is computable in
2632:. This can be generalized by adding non-negative weights to the directed edges.
6016:
6015:
Calinescu, Gruia; Chekuri, Chandra; Pál, Martin; Vondrák, Jan (January 2011).
6040:
5615:
Unlike the case of minimization, maximizing a generic submodular function is
3357:{\displaystyle f^{L}(\mathbf {x} )=\mathbb {E} (f(\{i|x_{i}\geq \lambda \}))}
5983:
5754:
5753:
Submodular functions naturally occur in several real world applications, in
187:
6219:
6017:"Maximizing a Monotone Submodular Function Subject to a Matroid Constraint"
86:, sensor placement, image collection summarization and many other domains.
6176:
5971:
5876:
17:
6457:
6105:
6296:
6141:
6096:
5976:
Proceedings of the fortieth annual ACM symposium on Theory of computing
5783:
5680:
5616:
5603:
1751:
1744:. Further inequalities for the entropy function are known to hold, see
1425:
6032:
5679:
The problem of maximizing a monotone submodular function subject to a
2184:. This can be generalized by adding non-negative weights to the edges.
50:
property which makes them suitable for many applications, including
1951:. Examples of symmetric non-monotone submodular functions include:
5606:, with polynomial factor lower bounds on the approximation factor.
6128:
Cunningham, W. H. (1985). "On submodular function minimization".
6056:"Polyhedral techniques in combinatorial optimization: Lecture 17"
5130:{\displaystyle \mathbb {E} \geq pf(\Omega )+(1-p)f(\varnothing )}
3718:
at random with probability xi, independently of the other items.
796:
function, but a subadditive function need not be submodular. If
5563:
Submodular functions have properties which are very similar to
1281:{\displaystyle f(S)=\min \left\{B,~\sum _{i\in S}w_{i}\right\}}
210:, which satisfies one of the following equivalent conditions.
6202:
6534:
1520:{\displaystyle f(S)=\left|\bigcup _{E_{i}\in S}E_{i}\right|}
588:
6320:"Bridging Continuous and Discrete Optimization: Lecture 23"
4609:{\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{k}}
3422:. The Lovász extension is a convex function if and only if
395:{\displaystyle f(X\cup \{x\})-f(X)\geq f(Y\cup \{x\})-f(Y)}
5599:
in a graph is a special case of this minimization problem.
3516:{\displaystyle \mathbf {x} =\{x_{1},x_{2},\ldots ,x_{n}\}}
5683:
constraint (which subsumes the case above) also admits a
4358:
For the extensions discussed above, it can be shown that
4129:{\displaystyle \mathbf {x} =\{x_{1},x_{2},\dots ,x_{n}\}}
3792:{\displaystyle \mathbf {x} =\{x_{1},x_{2},\dots ,x_{n}\}}
3225:{\displaystyle \mathbf {x} =\{x_{1},x_{2},\dots ,x_{n}\}}
4946:
is submodular, for any non decreasing concave function
4711:{\displaystyle g(S)=\sum _{i=1}^{k}\alpha _{i}f_{i}(S)}
4016:
The convex closure of any set function is convex over
3710:. Therefore, the sum represents the expected value of
5689:
5644:
5396:
5342:
5315:
5288:
5261:
5234:
5183:
5163:
5143:
5055:
5035:
5015:
4995:
4975:
4952:
4899:
4879:
4859:
4800:
4747:
4727:
4642:
4622:
4563:
4504:
4467:
4364:
4181:
4142:
4069:
4022:
3844:
3805:
3732:
3568:
3529:
3456:
3428:
3396:
3372:
3279:
3238:
3165:
3099:
3054:
3034:
2981:
2961:
2916:
2890:
2852:
2800:
2774:
2735:
2699:
2686:{\displaystyle f:2^{\Omega }\rightarrow \mathbb {R} }
2658:
2606:
2580:
2542:
2513:
2487:
2412:
2355:
2299:
2273:
2256:{\displaystyle \Omega =\{X_{1},X_{2},\ldots ,X_{n}\}}
2198:
2158:
2132:
2094:
2065:
2039:
1964:
1907:
1881:
1857:
1839:
A submodular function that is not monotone is called
1765:
1726:
1697:
1668:
1642:
1625:{\displaystyle \Omega =\{X_{1},X_{2},\ldots ,X_{n}\}}
1567:
1533:
1458:
1433:
1417:{\displaystyle \Omega =\{E_{1},E_{2},\ldots ,E_{n}\}}
1359:
1327:
1294:
1214:
1164:
1106:
1093:. Examples of monotone submodular functions include:
1055:
1029:
1005:
972:
952:
932:
897:
877:
842:
822:
802:
650:
610:
558:
532:
443:
411:
310:
278:
252:
220:
196:
165:
152:{\displaystyle f:2^{\Omega }\rightarrow \mathbb {R} }
124:
100:
5221:{\displaystyle 1\leq i\leq l,A_{i}\subseteq \Omega }
2470:{\displaystyle \Omega =\{v_{1},v_{2},\dots ,v_{n}\}}
2022:{\displaystyle \Omega =\{v_{1},v_{2},\dots ,v_{n}\}}
1823:{\displaystyle \Omega =\{e_{1},e_{2},\dots ,e_{n}\}}
516:{\displaystyle f(S)+f(T)\geq f(S\cup T)+f(S\cap T)}
6529:http://www.cs.berkeley.edu/~stefje/references.html
5709:
5664:
5546:
5382:
5328:
5301:
5274:
5247:
5220:
5169:
5149:
5129:
5041:
5021:
5001:
4981:
4958:
4938:
4885:
4865:
4845:
4783:
4733:
4710:
4628:
4608:
4549:
4473:
4453:
4342:
4167:
4128:
4047:
4005:
3830:
3791:
3680:
3554:
3515:
3434:
3414:
3378:
3356:
3263:
3224:
3136:
3085:
3040:
3020:
2967:
2940:
2902:
2876:
2838:
2786:
2760:
2721:
2685:
2624:
2592:
2566:
2528:
2499:
2469:
2382:
2341:
2285:
2255:
2176:
2144:
2118:
2080:
2051:
2021:
1943:
1893:
1863:
1822:
1732:
1712:
1683:
1654:
1624:
1545:
1519:
1444:
1416:
1339:
1313:
1280:
1192:
1150:
1085:
1041:
1011:
978:
958:
938:
918:
883:
863:
828:
808:
780:
636:
597:{\displaystyle x_{1},x_{2}\in \Omega \backslash X}
596:
544:
515:
429:
394:
296:
264:
238:
202:
178:
151:
106:
2637:Continuous extensions of submodular set functions
4816:
4206:
3869:
1230:
3562:. Then the multilinear extension is defined as
5929:(1983). "Submodular functions and convexity".
3021:{\displaystyle F:^{n}\rightarrow \mathbb {R} }
1720:is the entropy of the set of random variables
5931:Mathematical Programming the State of the Art
5635:of a graph is a special case of this problem.
1158:is called a linear function. Additionally if
58:(as functions modeling user preferences) and
8:
6476:Submodular Functions and Electrical Networks
6432:A First Course in Combinatorial Optimization
5865:"Submodular Functions Maximization Problems"
4123:
4078:
3786:
3741:
3510:
3465:
3345:
3318:
3219:
3174:
3155:This extension is named after mathematician
3074:
3061:
2827:
2814:
2749:
2736:
2464:
2419:
2250:
2205:
2016:
1971:
1817:
1772:
1619:
1574:
1411:
1366:
986:are infinite sets with finite intersection.
792:A nonnegative submodular function is also a
757:
731:
710:
697:
676:
663:
371:
365:
329:
323:
4784:{\displaystyle g(S)=f(\Omega \setminus S)}
3271:. Then the Lovász extension is defined as
990:Types and examples of submodular functions
118:, a submodular function is a set function
6218:
6104:
5850:
5699:
5688:
5654:
5643:
5535:
5519:
5500:
5475:
5465:
5449:
5427:
5398:
5397:
5395:
5374:
5364:
5353:
5341:
5320:
5314:
5293:
5287:
5266:
5260:
5239:
5233:
5206:
5182:
5162:
5142:
5057:
5056:
5054:
5034:
5014:
4994:
4974:
4951:
4898:
4878:
4873:is a real number, is submodular whenever
4858:
4799:
4746:
4726:
4693:
4683:
4673:
4662:
4641:
4621:
4600:
4581:
4568:
4562:
4550:{\displaystyle f_{1},f_{2},\ldots ,f_{k}}
4541:
4522:
4509:
4503:
4466:
4443:
4434:
4419:
4410:
4395:
4378:
4369:
4363:
4323:
4304:
4294:
4282:
4273:
4263:
4253:
4228:
4218:
4195:
4186:
4180:
4175:. Then the concave closure is defined as
4153:
4141:
4117:
4098:
4085:
4070:
4068:
4039:
4021:
3986:
3967:
3957:
3945:
3936:
3926:
3916:
3891:
3881:
3858:
3849:
3843:
3816:
3804:
3780:
3761:
3748:
3733:
3731:
3714:for the set formed by choosing each item
3669:
3644:
3634:
3618:
3590:
3575:
3567:
3540:
3528:
3504:
3485:
3472:
3457:
3455:
3427:
3395:
3371:
3333:
3324:
3305:
3304:
3293:
3284:
3278:
3249:
3237:
3213:
3194:
3181:
3166:
3164:
3110:
3098:
3077:
3053:
3033:
3014:
3013:
3004:
2980:
2960:
2926:
2921:
2915:
2889:
2862:
2857:
2851:
2830:
2805:
2799:
2773:
2752:
2734:
2708:
2700:
2698:
2679:
2678:
2669:
2657:
2605:
2579:
2541:
2512:
2486:
2458:
2439:
2426:
2411:
2354:
2298:
2272:
2244:
2225:
2212:
2197:
2157:
2131:
2093:
2064:
2038:
2010:
1991:
1978:
1963:
1906:
1880:
1856:
1811:
1792:
1779:
1764:
1725:
1696:
1667:
1641:
1613:
1594:
1581:
1566:
1532:
1506:
1488:
1483:
1457:
1432:
1405:
1386:
1373:
1358:
1326:
1299:
1293:
1267:
1251:
1213:
1178:
1163:
1142:
1126:
1105:
1054:
1028:
1004:
971:
951:
931:
896:
876:
841:
821:
801:
751:
738:
704:
670:
649:
628:
615:
609:
576:
563:
557:
531:
442:
410:
309:
277:
251:
219:
195:
170:
164:
145:
144:
135:
123:
99:
6537:includes further material on the subject
5863:Buchbinder, Niv; Feldman, Moran (2018).
5390:. Then the following inequality is true
5049:. Then the following inequality is true
4893:is monotone submodular. More generally,
3838:. Then the convex closure is defined as
1151:{\displaystyle f(S)=\sum _{i\in S}w_{i}}
27:Set-to-real map with diminishing returns
6496:, Oxford Science Publications, Oxford:
5803:
5177:is constructed as follows. For each of
5144:
5121:
4772:
4354:Relations between continuous extensions
297:{\displaystyle x\in \Omega \setminus Y}
288:
6241:
6239:
5793:Utility functions on indivisible goods
5383:{\displaystyle S=\cup _{i=1}^{l}S_{i}}
4969:Consider a random process where a set
6454:Submodular Functions and Optimization
5901:"Information Processing and Learning"
4490:The class of submodular functions is
3702:is chosen for the set. For every set
3698:represents the probability that item
1193:{\displaystyle \forall i,w_{i}\geq 0}
7:
5836:
5834:
5611:Submodular set function maximization
5579:Submodular set function minimization
2839:{\displaystyle x^{S}\in \{0,1\}^{n}}
2729:can be represented as a function on
430:{\displaystyle S,T\subseteq \Omega }
239:{\displaystyle X,Y\subseteq \Omega }
4498:. Consider any submodular function
2342:{\displaystyle f(S)=I(S;\Omega -S)}
1851:A non-monotone submodular function
1424:be a collection of subsets of some
5676:is a special case of this problem.
5472:
5446:
5215:
5091:
4996:
4769:
3597:
2787:{\displaystyle S\subseteq \Omega }
2781:
2705:
2670:
2613:
2500:{\displaystyle S\subseteq \Omega }
2494:
2413:
2368:
2327:
2286:{\displaystyle S\subseteq \Omega }
2280:
2199:
2165:
2052:{\displaystyle S\subseteq \Omega }
2046:
1965:
1929:
1894:{\displaystyle S\subseteq \Omega }
1888:
1766:
1655:{\displaystyle S\subseteq \Omega }
1649:
1568:
1546:{\displaystyle S\subseteq \Omega }
1540:
1435:
1360:
1165:
803:
585:
545:{\displaystyle X\subseteq \Omega }
539:
424:
285:
233:
197:
171:
136:
101:
25:
5867:. In Gonzalez, Teofilo F. (ed.).
4846:{\displaystyle g(S)=\min(f(S),c)}
4168:{\displaystyle 0\leq x_{i}\leq 1}
3831:{\displaystyle 0\leq x_{i}\leq 1}
3555:{\displaystyle 0\leq x_{i}\leq 1}
3264:{\displaystyle 0\leq x_{i}\leq 1}
1944:{\displaystyle f(S)=f(\Omega -S)}
4444:
4420:
4396:
4379:
4283:
4196:
4071:
3946:
3859:
3734:
3576:
3458:
3294:
3167:
3086:{\displaystyle x\in \{0,1\}^{n}}
2649:of the submodular set function.
2349:is a submodular function, where
1691:is a submodular function, where
5029:independently with probability
4989:is chosen with each element in
637:{\displaystyle x_{1}\neq x_{2}}
5541:
5512:
5506:
5487:
5440:
5434:
5417:
5414:
5408:
5402:
5124:
5118:
5112:
5100:
5094:
5088:
5076:
5073:
5067:
5061:
4933:
4930:
4924:
4918:
4909:
4903:
4840:
4831:
4825:
4819:
4810:
4804:
4778:
4766:
4757:
4751:
4705:
4699:
4652:
4646:
4448:
4440:
4424:
4416:
4400:
4392:
4383:
4375:
4243:
4237:
4200:
4192:
4036:
4023:
3906:
3900:
3863:
3855:
3675:
3656:
3611:
3605:
3580:
3572:
3409:
3397:
3366:where the expectation is over
3351:
3348:
3325:
3315:
3309:
3298:
3290:
3131:
3125:
3116:
3103:
3010:
3001:
2988:
2709:
2701:
2675:
2625:{\displaystyle v\in \Omega -S}
2561:
2549:
2523:
2517:
2383:{\displaystyle I(S;\Omega -S)}
2377:
2359:
2336:
2318:
2309:
2303:
2177:{\displaystyle v\in \Omega -S}
2113:
2101:
2075:
2069:
1938:
1926:
1917:
1911:
1707:
1701:
1678:
1672:
1468:
1462:
1224:
1218:
1116:
1110:
1080:
1074:
1065:
1059:
907:
901:
852:
846:
775:
769:
760:
722:
713:
688:
679:
654:
510:
498:
489:
477:
468:
462:
453:
447:
389:
383:
374:
356:
347:
341:
332:
314:
141:
1:
5725:Related optimization problems
5672:approximation algorithm. The
5255:by including each element in
3137:{\displaystyle F(x^{S})=f(S)}
1086:{\displaystyle f(T)\leq f(S)}
5939:10.1007/978-3-642-68874-4_10
5150:{\displaystyle \varnothing }
4939:{\displaystyle g(S)=h(f(S))}
4721:For any submodular function
3028:, that matches the value of
1042:{\displaystyle T\subseteq S}
265:{\displaystyle X\subseteq Y}
76:multi-document summarization
5970:Vondrak, Jan (2008-05-17).
2941:{\displaystyle x_{i}^{S}=0}
2877:{\displaystyle x_{i}^{S}=1}
2761:{\displaystyle \{0,1\}^{n}}
2722:{\displaystyle |\Omega |=n}
2536:denote the number of edges
2088:denote the number of edges
1314:{\displaystyle w_{i}\geq 0}
179:{\displaystyle 2^{\Omega }}
6577:
6551:Combinatorial optimization
6452:Fujishige, Satoru (2005),
6436:Cambridge University Press
6407:Combinatorial Optimization
4741:, the function defined by
3442:is a submodular function.
2481:. For any set of vertices
2390:is the mutual information.
2033:. For any set of vertices
1347:is called budget additive.
1097:Linear (Modular) functions
6535:http://submodularity.org/
6531:has a longer bibliography
6021:SIAM Journal on Computing
4557:and non-negative numbers
2975:is a continuous function
2652:Formally, a set function
1208:Any function of the form
1204:Budget-additive functions
1100:Any function of the form
6556:Approximation algorithms
6492:Oxley, James G. (1992),
6285:Mathematical Programming
6207:J. Combin. Theory Ser. B
5871:. Chapman and Hall/CRC.
5717:approximation algorithm.
5674:maximum coverage problem
3379:{\displaystyle \lambda }
1445:{\displaystyle \Omega '}
52:approximation algorithms
6498:Oxford University Press
5984:10.1145/1374376.1374389
5625:local search algorithms
5002:{\displaystyle \Omega }
2567:{\displaystyle e=(u,v)}
2119:{\displaystyle e=(u,v)}
1340:{\displaystyle B\geq 0}
809:{\displaystyle \Omega }
203:{\displaystyle \Omega }
107:{\displaystyle \Omega }
72:automatic summarization
68:artificial intelligence
32:submodular set function
6474:Narayanan, H. (1997),
6220:10.1006/jctb.2000.1989
5711:
5666:
5571:. For this reason, an
5548:
5384:
5330:
5303:
5276:
5249:
5222:
5171:
5151:
5131:
5043:
5023:
5003:
4983:
4960:
4940:
4887:
4867:
4847:
4785:
4735:
4712:
4678:
4630:
4610:
4551:
4475:
4455:
4344:
4169:
4130:
4049:
4007:
3832:
3793:
3682:
3556:
3517:
3436:
3416:
3380:
3358:
3265:
3226:
3159:. Consider any vector
3138:
3087:
3042:
3022:
2969:
2942:
2904:
2903:{\displaystyle i\in S}
2878:
2840:
2788:
2768:, by associating each
2762:
2723:
2687:
2626:
2594:
2593:{\displaystyle u\in S}
2568:
2530:
2501:
2471:
2384:
2343:
2287:
2257:
2178:
2146:
2145:{\displaystyle u\in S}
2120:
2082:
2053:
2023:
1945:
1895:
1865:
1824:
1734:
1714:
1685:
1656:
1626:
1547:
1521:
1446:
1418:
1341:
1315:
1282:
1194:
1152:
1087:
1043:
1013:
980:
960:
940:
920:
919:{\displaystyle f(S)=0}
885:
865:
864:{\displaystyle f(S)=1}
830:
810:
782:
638:
598:
546:
517:
431:
396:
298:
266:
240:
204:
180:
153:
108:
6318:Williamson, David P.
6177:10.1145/502090.502096
5916:Fujishige (2005) p.22
5877:10.1201/9781351236423
5779:Supermodular function
5712:
5710:{\displaystyle 1-1/e}
5667:
5665:{\displaystyle 1-1/e}
5559:Optimization problems
5549:
5385:
5331:
5329:{\displaystyle p_{i}}
5304:
5302:{\displaystyle S_{i}}
5277:
5275:{\displaystyle A_{i}}
5250:
5248:{\displaystyle S_{i}}
5223:
5172:
5152:
5132:
5044:
5024:
5004:
4984:
4961:
4941:
4888:
4868:
4848:
4786:
4736:
4713:
4658:
4631:
4611:
4552:
4476:
4456:
4345:
4170:
4131:
4050:
4008:
3833:
3794:
3683:
3557:
3518:
3446:Multilinear extension
3437:
3417:
3381:
3359:
3266:
3227:
3139:
3088:
3043:
3023:
2970:
2943:
2905:
2879:
2841:
2794:with a binary vector
2789:
2763:
2724:
2688:
2627:
2595:
2569:
2531:
2502:
2477:be the vertices of a
2472:
2385:
2344:
2288:
2258:
2179:
2147:
2121:
2083:
2054:
2029:be the vertices of a
2024:
1946:
1896:
1866:
1825:
1735:
1715:
1686:
1657:
1627:
1548:
1522:
1447:
1419:
1342:
1316:
1283:
1195:
1153:
1088:
1044:
1014:
981:
961:
941:
921:
886:
866:
831:
811:
783:
639:
599:
547:
518:
432:
397:
299:
267:
241:
205:
181:
154:
109:
6403:Schrijver, Alexander
5933:. pp. 235–257.
5853:, §44, p. 766)
5742:welfare maximization
5687:
5642:
5595:time. Computing the
5573:optimization problem
5394:
5340:
5313:
5286:
5259:
5232:
5181:
5161:
5141:
5053:
5033:
5013:
4993:
4973:
4950:
4897:
4877:
4857:
4798:
4745:
4725:
4640:
4620:
4616:. Then the function
4561:
4502:
4465:
4362:
4179:
4140:
4067:
4063:Consider any vector
4048:{\displaystyle ^{n}}
4020:
3842:
3803:
3730:
3726:Consider any vector
3566:
3527:
3454:
3450:Consider any vector
3426:
3394:
3388:uniform distribution
3370:
3277:
3236:
3163:
3097:
3052:
3032:
2979:
2959:
2914:
2888:
2850:
2798:
2772:
2733:
2697:
2656:
2647:continuous extension
2604:
2578:
2540:
2529:{\displaystyle f(S)}
2511:
2485:
2410:
2353:
2297:
2271:
2196:
2156:
2130:
2092:
2081:{\displaystyle f(S)}
2063:
2037:
1962:
1905:
1879:
1855:
1763:
1742:Shannon's inequality
1724:
1713:{\displaystyle H(S)}
1695:
1684:{\displaystyle H(S)}
1666:
1640:
1565:
1531:
1456:
1431:
1357:
1325:
1292:
1212:
1162:
1104:
1053:
1027:
1003:
970:
950:
930:
895:
875:
840:
820:
800:
648:
608:
556:
530:
441:
409:
308:
276:
250:
218:
194:
163:
122:
98:
5593:strongly-polynomial
5369:
5282:independently into
4496:linear combinations
4494:under non-negative
2931:
2867:
1200:then f is monotone.
60:electrical networks
48:diminishing returns
44:diminishing returns
36:submodular function
6297:10.1007/BF01588971
6142:10.1007/BF02579361
6097:10.1007/BF02579273
5707:
5662:
5544:
5444:
5380:
5349:
5336:. Furthermore let
5326:
5299:
5272:
5245:
5218:
5167:
5147:
5127:
5039:
5019:
5009:being included in
4999:
4979:
4956:
4936:
4883:
4863:
4843:
4781:
4731:
4708:
4626:
4606:
4547:
4471:
4451:
4340:
4299:
4258:
4223:
4165:
4126:
4045:
4003:
3962:
3921:
3886:
3828:
3789:
3678:
3655:
3629:
3601:
3552:
3513:
3432:
3412:
3376:
3354:
3261:
3222:
3134:
3083:
3038:
3018:
2965:
2938:
2917:
2900:
2874:
2853:
2836:
2784:
2758:
2719:
2683:
2622:
2590:
2564:
2526:
2497:
2467:
2380:
2339:
2283:
2253:
2188:Mutual information
2174:
2142:
2116:
2078:
2049:
2019:
1941:
1891:
1861:
1820:
1740:, a fact known as
1730:
1710:
1681:
1652:
1622:
1543:
1517:
1501:
1442:
1414:
1350:Coverage functions
1337:
1311:
1278:
1262:
1190:
1148:
1137:
1083:
1039:
1009:
976:
956:
936:
916:
881:
861:
826:
806:
778:
634:
594:
542:
513:
427:
392:
294:
262:
236:
200:
176:
149:
104:
30:In mathematics, a
6281:Nemhauser, George
6033:10.1137/080733991
5993:978-1-60558-047-0
5948:978-3-642-68876-8
5621:greedy algorithms
5569:concave functions
5423:
5309:with probability
5170:{\displaystyle S}
5042:{\displaystyle p}
5022:{\displaystyle T}
4982:{\displaystyle T}
4959:{\displaystyle h}
4886:{\displaystyle f}
4866:{\displaystyle c}
4734:{\displaystyle f}
4629:{\displaystyle g}
4474:{\displaystyle f}
4290:
4249:
4214:
3953:
3912:
3877:
3640:
3614:
3586:
3435:{\displaystyle f}
3041:{\displaystyle f}
2968:{\displaystyle f}
1864:{\displaystyle f}
1733:{\displaystyle S}
1479:
1247:
1246:
1122:
1012:{\displaystyle f}
979:{\displaystyle T}
959:{\displaystyle S}
939:{\displaystyle S}
884:{\displaystyle S}
829:{\displaystyle f}
80:feature selection
34:(also known as a
16:(Redirected from
6568:
6518:
6488:
6470:
6448:
6423:
6390:
6387:
6381:
6378:
6372:
6369:
6363:
6360:
6354:
6351:
6345:
6342:
6336:
6333:
6327:
6326:
6324:
6315:
6309:
6308:
6277:
6271:
6268:
6262:
6256:
6250:
6243:
6234:
6231:
6225:
6224:
6222:
6195:
6189:
6188:
6160:
6154:
6153:
6125:
6119:
6118:
6108:
6069:
6063:
6062:
6060:
6051:
6045:
6044:
6027:(6): 1740–1766.
6012:
6006:
6005:
5967:
5961:
5960:
5923:
5917:
5914:
5908:
5907:
5905:
5897:
5891:
5890:
5860:
5854:
5849:(Schrijver
5847:
5841:
5838:
5829:
5826:
5820:
5817:
5811:
5808:
5763:machine learning
5716:
5714:
5713:
5708:
5703:
5671:
5669:
5668:
5663:
5658:
5553:
5551:
5550:
5545:
5540:
5539:
5530:
5529:
5505:
5504:
5486:
5485:
5470:
5469:
5460:
5459:
5443:
5401:
5389:
5387:
5386:
5381:
5379:
5378:
5368:
5363:
5335:
5333:
5332:
5327:
5325:
5324:
5308:
5306:
5305:
5300:
5298:
5297:
5281:
5279:
5278:
5273:
5271:
5270:
5254:
5252:
5251:
5246:
5244:
5243:
5227:
5225:
5224:
5219:
5211:
5210:
5176:
5174:
5173:
5168:
5156:
5154:
5153:
5148:
5136:
5134:
5133:
5128:
5060:
5048:
5046:
5045:
5040:
5028:
5026:
5025:
5020:
5008:
5006:
5005:
5000:
4988:
4986:
4985:
4980:
4965:
4963:
4962:
4957:
4945:
4943:
4942:
4937:
4892:
4890:
4889:
4884:
4872:
4870:
4869:
4864:
4852:
4850:
4849:
4844:
4790:
4788:
4787:
4782:
4740:
4738:
4737:
4732:
4717:
4715:
4714:
4709:
4698:
4697:
4688:
4687:
4677:
4672:
4635:
4633:
4632:
4627:
4615:
4613:
4612:
4607:
4605:
4604:
4586:
4585:
4573:
4572:
4556:
4554:
4553:
4548:
4546:
4545:
4527:
4526:
4514:
4513:
4480:
4478:
4477:
4472:
4460:
4458:
4457:
4452:
4447:
4439:
4438:
4423:
4415:
4414:
4399:
4382:
4374:
4373:
4349:
4347:
4346:
4341:
4339:
4335:
4328:
4327:
4309:
4308:
4298:
4286:
4278:
4277:
4268:
4267:
4257:
4233:
4232:
4222:
4199:
4191:
4190:
4174:
4172:
4171:
4166:
4158:
4157:
4135:
4133:
4132:
4127:
4122:
4121:
4103:
4102:
4090:
4089:
4074:
4054:
4052:
4051:
4046:
4044:
4043:
4012:
4010:
4009:
4004:
4002:
3998:
3991:
3990:
3972:
3971:
3961:
3949:
3941:
3940:
3931:
3930:
3920:
3896:
3895:
3885:
3862:
3854:
3853:
3837:
3835:
3834:
3829:
3821:
3820:
3798:
3796:
3795:
3790:
3785:
3784:
3766:
3765:
3753:
3752:
3737:
3687:
3685:
3684:
3679:
3674:
3673:
3654:
3639:
3638:
3628:
3600:
3579:
3561:
3559:
3558:
3553:
3545:
3544:
3522:
3520:
3519:
3514:
3509:
3508:
3490:
3489:
3477:
3476:
3461:
3441:
3439:
3438:
3433:
3421:
3419:
3418:
3415:{\displaystyle }
3413:
3390:on the interval
3386:chosen from the
3385:
3383:
3382:
3377:
3363:
3361:
3360:
3355:
3338:
3337:
3328:
3308:
3297:
3289:
3288:
3270:
3268:
3267:
3262:
3254:
3253:
3231:
3229:
3228:
3223:
3218:
3217:
3199:
3198:
3186:
3185:
3170:
3151:Lovász extension
3143:
3141:
3140:
3135:
3115:
3114:
3092:
3090:
3089:
3084:
3082:
3081:
3047:
3045:
3044:
3039:
3027:
3025:
3024:
3019:
3017:
3009:
3008:
2974:
2972:
2971:
2966:
2947:
2945:
2944:
2939:
2930:
2925:
2909:
2907:
2906:
2901:
2883:
2881:
2880:
2875:
2866:
2861:
2845:
2843:
2842:
2837:
2835:
2834:
2810:
2809:
2793:
2791:
2790:
2785:
2767:
2765:
2764:
2759:
2757:
2756:
2728:
2726:
2725:
2720:
2712:
2704:
2692:
2690:
2689:
2684:
2682:
2674:
2673:
2631:
2629:
2628:
2623:
2599:
2597:
2596:
2591:
2573:
2571:
2570:
2565:
2535:
2533:
2532:
2527:
2506:
2504:
2503:
2498:
2476:
2474:
2473:
2468:
2463:
2462:
2444:
2443:
2431:
2430:
2389:
2387:
2386:
2381:
2348:
2346:
2345:
2340:
2292:
2290:
2289:
2284:
2265:random variables
2262:
2260:
2259:
2254:
2249:
2248:
2230:
2229:
2217:
2216:
2183:
2181:
2180:
2175:
2151:
2149:
2148:
2143:
2125:
2123:
2122:
2117:
2087:
2085:
2084:
2079:
2058:
2056:
2055:
2050:
2028:
2026:
2025:
2020:
2015:
2014:
1996:
1995:
1983:
1982:
1950:
1948:
1947:
1942:
1900:
1898:
1897:
1892:
1870:
1868:
1867:
1862:
1829:
1827:
1826:
1821:
1816:
1815:
1797:
1796:
1784:
1783:
1739:
1737:
1736:
1731:
1719:
1717:
1716:
1711:
1690:
1688:
1687:
1682:
1661:
1659:
1658:
1653:
1634:random variables
1631:
1629:
1628:
1623:
1618:
1617:
1599:
1598:
1586:
1585:
1552:
1550:
1549:
1544:
1526:
1524:
1523:
1518:
1516:
1512:
1511:
1510:
1500:
1493:
1492:
1451:
1449:
1448:
1443:
1441:
1423:
1421:
1420:
1415:
1410:
1409:
1391:
1390:
1378:
1377:
1346:
1344:
1343:
1338:
1320:
1318:
1317:
1312:
1304:
1303:
1287:
1285:
1284:
1279:
1277:
1273:
1272:
1271:
1261:
1244:
1199:
1197:
1196:
1191:
1183:
1182:
1157:
1155:
1154:
1149:
1147:
1146:
1136:
1092:
1090:
1089:
1084:
1048:
1046:
1045:
1040:
1018:
1016:
1015:
1010:
985:
983:
982:
977:
965:
963:
962:
957:
945:
943:
942:
937:
925:
923:
922:
917:
890:
888:
887:
882:
870:
868:
867:
862:
835:
833:
832:
827:
815:
813:
812:
807:
787:
785:
784:
779:
756:
755:
743:
742:
709:
708:
675:
674:
643:
641:
640:
635:
633:
632:
620:
619:
603:
601:
600:
595:
581:
580:
568:
567:
551:
549:
548:
543:
522:
520:
519:
514:
436:
434:
433:
428:
401:
399:
398:
393:
303:
301:
300:
295:
271:
269:
268:
263:
245:
243:
242:
237:
209:
207:
206:
201:
185:
183:
182:
177:
175:
174:
158:
156:
155:
150:
148:
140:
139:
113:
111:
110:
105:
64:machine learning
21:
6576:
6575:
6571:
6570:
6569:
6567:
6566:
6565:
6541:
6540:
6525:
6508:
6491:
6486:
6473:
6468:
6451:
6446:
6426:
6421:
6401:
6398:
6393:
6388:
6384:
6379:
6375:
6370:
6366:
6361:
6357:
6352:
6348:
6343:
6339:
6334:
6330:
6322:
6317:
6316:
6312:
6291:(14): 265–294.
6279:
6278:
6274:
6269:
6265:
6257:
6253:
6244:
6237:
6232:
6228:
6197:
6196:
6192:
6162:
6161:
6157:
6127:
6126:
6122:
6071:
6070:
6066:
6058:
6053:
6052:
6048:
6014:
6013:
6009:
5994:
5969:
5968:
5964:
5949:
5925:
5924:
5920:
5915:
5911:
5903:
5899:
5898:
5894:
5887:
5862:
5861:
5857:
5848:
5844:
5839:
5832:
5827:
5823:
5818:
5814:
5809:
5805:
5801:
5775:
5767:computer vision
5751:
5727:
5685:
5684:
5640:
5639:
5613:
5589:polynomial time
5581:
5561:
5531:
5515:
5496:
5471:
5461:
5445:
5392:
5391:
5370:
5338:
5337:
5316:
5311:
5310:
5289:
5284:
5283:
5262:
5257:
5256:
5235:
5230:
5229:
5202:
5179:
5178:
5159:
5158:
5139:
5138:
5051:
5050:
5031:
5030:
5011:
5010:
4991:
4990:
4971:
4970:
4948:
4947:
4895:
4894:
4875:
4874:
4855:
4854:
4796:
4795:
4743:
4742:
4723:
4722:
4689:
4679:
4638:
4637:
4618:
4617:
4596:
4577:
4564:
4559:
4558:
4537:
4518:
4505:
4500:
4499:
4487:
4481:is submodular.
4463:
4462:
4430:
4406:
4365:
4360:
4359:
4356:
4319:
4300:
4269:
4259:
4224:
4213:
4209:
4182:
4177:
4176:
4149:
4138:
4137:
4136:such that each
4113:
4094:
4081:
4065:
4064:
4061:
4059:Concave closure
4035:
4018:
4017:
3982:
3963:
3932:
3922:
3887:
3876:
3872:
3845:
3840:
3839:
3812:
3801:
3800:
3799:such that each
3776:
3757:
3744:
3728:
3727:
3724:
3696:
3665:
3630:
3564:
3563:
3536:
3525:
3524:
3523:such that each
3500:
3481:
3468:
3452:
3451:
3448:
3424:
3423:
3392:
3391:
3368:
3367:
3329:
3280:
3275:
3274:
3245:
3234:
3233:
3232:such that each
3209:
3190:
3177:
3161:
3160:
3153:
3106:
3095:
3094:
3073:
3050:
3049:
3030:
3029:
3000:
2977:
2976:
2957:
2956:
2912:
2911:
2886:
2885:
2848:
2847:
2826:
2801:
2796:
2795:
2770:
2769:
2748:
2731:
2730:
2695:
2694:
2665:
2654:
2653:
2639:
2602:
2601:
2576:
2575:
2538:
2537:
2509:
2508:
2483:
2482:
2454:
2435:
2422:
2408:
2407:
2397:
2351:
2350:
2295:
2294:
2269:
2268:
2267:. Then for any
2240:
2221:
2208:
2194:
2193:
2154:
2153:
2128:
2127:
2090:
2089:
2061:
2060:
2035:
2034:
2006:
1987:
1974:
1960:
1959:
1903:
1902:
1877:
1876:
1853:
1852:
1849:
1837:
1807:
1788:
1775:
1761:
1760:
1746:entropic vector
1722:
1721:
1693:
1692:
1664:
1663:
1638:
1637:
1636:. Then for any
1609:
1590:
1577:
1563:
1562:
1529:
1528:
1502:
1484:
1478:
1474:
1454:
1453:
1452:. The function
1434:
1429:
1428:
1401:
1382:
1369:
1355:
1354:
1323:
1322:
1295:
1290:
1289:
1263:
1237:
1233:
1210:
1209:
1174:
1160:
1159:
1138:
1102:
1101:
1051:
1050:
1025:
1024:
1001:
1000:
999:A set function
997:
992:
968:
967:
948:
947:
928:
927:
893:
892:
873:
872:
838:
837:
818:
817:
798:
797:
747:
734:
700:
666:
646:
645:
624:
611:
606:
605:
572:
559:
554:
553:
528:
527:
439:
438:
407:
406:
306:
305:
274:
273:
248:
247:
216:
215:
192:
191:
166:
161:
160:
131:
120:
119:
96:
95:
92:
84:active learning
46:). The natural
28:
23:
22:
15:
12:
11:
5:
6574:
6572:
6564:
6563:
6561:Matroid theory
6558:
6553:
6543:
6542:
6539:
6538:
6532:
6524:
6523:External links
6521:
6520:
6519:
6506:
6494:Matroid theory
6489:
6484:
6471:
6466:
6449:
6444:
6424:
6419:
6397:
6394:
6392:
6391:
6382:
6373:
6364:
6355:
6346:
6337:
6328:
6310:
6272:
6263:
6251:
6235:
6226:
6213:(2): 346–355.
6190:
6171:(4): 761–777.
6155:
6136:(3): 185–192.
6120:
6091:(2): 169–197.
6064:
6054:Vondrák, Jan.
6046:
6007:
5992:
5962:
5947:
5918:
5909:
5892:
5885:
5855:
5842:
5830:
5821:
5812:
5802:
5800:
5797:
5796:
5795:
5790:
5781:
5774:
5771:
5750:
5747:
5746:
5745:
5738:
5734:
5726:
5723:
5719:
5718:
5706:
5702:
5698:
5695:
5692:
5677:
5661:
5657:
5653:
5650:
5647:
5636:
5612:
5609:
5608:
5607:
5600:
5591:, and even in
5580:
5577:
5560:
5557:
5556:
5555:
5543:
5538:
5534:
5528:
5525:
5522:
5518:
5514:
5511:
5508:
5503:
5499:
5495:
5492:
5489:
5484:
5481:
5478:
5474:
5468:
5464:
5458:
5455:
5452:
5448:
5442:
5439:
5436:
5433:
5430:
5426:
5422:
5419:
5416:
5413:
5410:
5407:
5404:
5400:
5377:
5373:
5367:
5362:
5359:
5356:
5352:
5348:
5345:
5323:
5319:
5296:
5292:
5269:
5265:
5242:
5238:
5217:
5214:
5209:
5205:
5201:
5198:
5195:
5192:
5189:
5186:
5166:
5146:
5126:
5123:
5120:
5117:
5114:
5111:
5108:
5105:
5102:
5099:
5096:
5093:
5090:
5087:
5084:
5081:
5078:
5075:
5072:
5069:
5066:
5063:
5059:
5038:
5018:
4998:
4978:
4967:
4955:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4911:
4908:
4905:
4902:
4882:
4862:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4806:
4803:
4792:
4791:is submodular.
4780:
4777:
4774:
4771:
4768:
4765:
4762:
4759:
4756:
4753:
4750:
4730:
4719:
4718:is submodular.
4707:
4704:
4701:
4696:
4692:
4686:
4682:
4676:
4671:
4668:
4665:
4661:
4657:
4654:
4651:
4648:
4645:
4625:
4603:
4599:
4595:
4592:
4589:
4584:
4580:
4576:
4571:
4567:
4544:
4540:
4536:
4533:
4530:
4525:
4521:
4517:
4512:
4508:
4486:
4483:
4470:
4450:
4446:
4442:
4437:
4433:
4429:
4426:
4422:
4418:
4413:
4409:
4405:
4402:
4398:
4394:
4391:
4388:
4385:
4381:
4377:
4372:
4368:
4355:
4352:
4338:
4334:
4331:
4326:
4322:
4318:
4315:
4312:
4307:
4303:
4297:
4293:
4289:
4285:
4281:
4276:
4272:
4266:
4262:
4256:
4252:
4248:
4245:
4242:
4239:
4236:
4231:
4227:
4221:
4217:
4212:
4208:
4205:
4202:
4198:
4194:
4189:
4185:
4164:
4161:
4156:
4152:
4148:
4145:
4125:
4120:
4116:
4112:
4109:
4106:
4101:
4097:
4093:
4088:
4084:
4080:
4077:
4073:
4060:
4057:
4042:
4038:
4034:
4031:
4028:
4025:
4001:
3997:
3994:
3989:
3985:
3981:
3978:
3975:
3970:
3966:
3960:
3956:
3952:
3948:
3944:
3939:
3935:
3929:
3925:
3919:
3915:
3911:
3908:
3905:
3902:
3899:
3894:
3890:
3884:
3880:
3875:
3871:
3868:
3865:
3861:
3857:
3852:
3848:
3827:
3824:
3819:
3815:
3811:
3808:
3788:
3783:
3779:
3775:
3772:
3769:
3764:
3760:
3756:
3751:
3747:
3743:
3740:
3736:
3723:
3722:Convex closure
3720:
3694:
3677:
3672:
3668:
3664:
3661:
3658:
3653:
3650:
3647:
3643:
3637:
3633:
3627:
3624:
3621:
3617:
3613:
3610:
3607:
3604:
3599:
3596:
3593:
3589:
3585:
3582:
3578:
3574:
3571:
3551:
3548:
3543:
3539:
3535:
3532:
3512:
3507:
3503:
3499:
3496:
3493:
3488:
3484:
3480:
3475:
3471:
3467:
3464:
3460:
3447:
3444:
3431:
3411:
3408:
3405:
3402:
3399:
3375:
3353:
3350:
3347:
3344:
3341:
3336:
3332:
3327:
3323:
3320:
3317:
3314:
3311:
3307:
3303:
3300:
3296:
3292:
3287:
3283:
3260:
3257:
3252:
3248:
3244:
3241:
3221:
3216:
3212:
3208:
3205:
3202:
3197:
3193:
3189:
3184:
3180:
3176:
3173:
3169:
3152:
3149:
3133:
3130:
3127:
3124:
3121:
3118:
3113:
3109:
3105:
3102:
3080:
3076:
3072:
3069:
3066:
3063:
3060:
3057:
3037:
3016:
3012:
3007:
3003:
2999:
2996:
2993:
2990:
2987:
2984:
2964:
2937:
2934:
2929:
2924:
2920:
2899:
2896:
2893:
2873:
2870:
2865:
2860:
2856:
2833:
2829:
2825:
2822:
2819:
2816:
2813:
2808:
2804:
2783:
2780:
2777:
2755:
2751:
2747:
2744:
2741:
2738:
2718:
2715:
2711:
2707:
2703:
2681:
2677:
2672:
2668:
2664:
2661:
2638:
2635:
2634:
2633:
2621:
2618:
2615:
2612:
2609:
2589:
2586:
2583:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2525:
2522:
2519:
2516:
2496:
2493:
2490:
2479:directed graph
2466:
2461:
2457:
2453:
2450:
2447:
2442:
2438:
2434:
2429:
2425:
2421:
2418:
2415:
2404:
2396:
2393:
2392:
2391:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2282:
2279:
2276:
2252:
2247:
2243:
2239:
2236:
2233:
2228:
2224:
2220:
2215:
2211:
2207:
2204:
2201:
2190:
2185:
2173:
2170:
2167:
2164:
2161:
2141:
2138:
2135:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2077:
2074:
2071:
2068:
2048:
2045:
2042:
2018:
2013:
2009:
2005:
2002:
1999:
1994:
1990:
1986:
1981:
1977:
1973:
1970:
1967:
1956:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1890:
1887:
1884:
1860:
1848:
1845:
1836:
1833:
1832:
1831:
1819:
1814:
1810:
1806:
1803:
1800:
1795:
1791:
1787:
1782:
1778:
1774:
1771:
1768:
1757:
1755:rank functions
1749:
1729:
1709:
1706:
1703:
1700:
1680:
1677:
1674:
1671:
1651:
1648:
1645:
1621:
1616:
1612:
1608:
1605:
1602:
1597:
1593:
1589:
1584:
1580:
1576:
1573:
1570:
1559:
1554:
1542:
1539:
1536:
1515:
1509:
1505:
1499:
1496:
1491:
1487:
1482:
1477:
1473:
1470:
1467:
1464:
1461:
1440:
1437:
1413:
1408:
1404:
1400:
1397:
1394:
1389:
1385:
1381:
1376:
1372:
1368:
1365:
1362:
1351:
1348:
1336:
1333:
1330:
1310:
1307:
1302:
1298:
1276:
1270:
1266:
1260:
1257:
1254:
1250:
1243:
1240:
1236:
1232:
1229:
1226:
1223:
1220:
1217:
1206:
1201:
1189:
1186:
1181:
1177:
1173:
1170:
1167:
1145:
1141:
1135:
1132:
1129:
1125:
1121:
1118:
1115:
1112:
1109:
1098:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1038:
1035:
1032:
1008:
996:
993:
991:
988:
975:
955:
935:
915:
912:
909:
906:
903:
900:
891:is finite and
880:
860:
857:
854:
851:
848:
845:
825:
805:
790:
789:
777:
774:
771:
768:
765:
762:
759:
754:
750:
746:
741:
737:
733:
730:
727:
724:
721:
718:
715:
712:
707:
703:
699:
696:
693:
690:
687:
684:
681:
678:
673:
669:
665:
662:
659:
656:
653:
631:
627:
623:
618:
614:
593:
590:
587:
584:
579:
575:
571:
566:
562:
541:
538:
535:
524:
512:
509:
506:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
473:
470:
467:
464:
461:
458:
455:
452:
449:
446:
426:
423:
420:
417:
414:
403:
391:
388:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
293:
290:
287:
284:
281:
261:
258:
255:
235:
232:
229:
226:
223:
199:
173:
169:
147:
143:
138:
134:
130:
127:
103:
91:
88:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6573:
6562:
6559:
6557:
6554:
6552:
6549:
6548:
6546:
6536:
6533:
6530:
6527:
6526:
6522:
6517:
6513:
6509:
6507:0-19-853563-5
6503:
6499:
6495:
6490:
6487:
6485:0-444-82523-1
6481:
6477:
6472:
6469:
6467:0-444-52086-4
6463:
6459:
6455:
6450:
6447:
6445:0-521-01012-8
6441:
6437:
6433:
6429:
6425:
6422:
6420:3-540-44389-4
6416:
6412:
6408:
6404:
6400:
6399:
6395:
6386:
6383:
6377:
6374:
6368:
6365:
6359:
6356:
6350:
6347:
6341:
6338:
6332:
6329:
6321:
6314:
6311:
6306:
6302:
6298:
6294:
6290:
6286:
6282:
6276:
6273:
6267:
6264:
6260:
6255:
6252:
6248:
6242:
6240:
6236:
6230:
6227:
6221:
6216:
6212:
6208:
6204:
6200:
6199:Schrijver, A.
6194:
6191:
6186:
6182:
6178:
6174:
6170:
6166:
6159:
6156:
6151:
6147:
6143:
6139:
6135:
6131:
6130:Combinatorica
6124:
6121:
6116:
6112:
6107:
6102:
6098:
6094:
6090:
6086:
6085:Combinatorica
6082:
6081:Schrijver, A.
6078:
6074:
6073:Grötschel, M.
6068:
6065:
6057:
6050:
6047:
6042:
6038:
6034:
6030:
6026:
6022:
6018:
6011:
6008:
6003:
5999:
5995:
5989:
5985:
5981:
5977:
5973:
5966:
5963:
5958:
5954:
5950:
5944:
5940:
5936:
5932:
5928:
5922:
5919:
5913:
5910:
5902:
5896:
5893:
5888:
5886:9781351236423
5882:
5878:
5874:
5870:
5866:
5859:
5856:
5852:
5846:
5843:
5837:
5835:
5831:
5825:
5822:
5816:
5813:
5807:
5804:
5798:
5794:
5791:
5789:
5785:
5782:
5780:
5777:
5776:
5772:
5770:
5768:
5764:
5760:
5756:
5748:
5743:
5739:
5735:
5732:
5731:
5730:
5724:
5722:
5704:
5700:
5696:
5693:
5690:
5682:
5678:
5675:
5659:
5655:
5651:
5648:
5645:
5637:
5634:
5630:
5629:
5628:
5626:
5622:
5618:
5610:
5605:
5601:
5598:
5594:
5590:
5586:
5585:
5584:
5578:
5576:
5574:
5570:
5566:
5558:
5536:
5532:
5526:
5523:
5520:
5516:
5509:
5501:
5497:
5493:
5490:
5482:
5479:
5476:
5466:
5462:
5456:
5453:
5450:
5437:
5431:
5428:
5424:
5420:
5411:
5405:
5375:
5371:
5365:
5360:
5357:
5354:
5350:
5346:
5343:
5321:
5317:
5294:
5290:
5267:
5263:
5240:
5236:
5212:
5207:
5203:
5199:
5196:
5193:
5190:
5187:
5184:
5164:
5115:
5109:
5106:
5103:
5097:
5085:
5082:
5079:
5070:
5064:
5036:
5016:
4976:
4968:
4953:
4927:
4921:
4915:
4912:
4906:
4900:
4880:
4860:
4837:
4834:
4828:
4822:
4813:
4807:
4801:
4794:The function
4793:
4775:
4763:
4760:
4754:
4748:
4728:
4720:
4702:
4694:
4690:
4684:
4680:
4674:
4669:
4666:
4663:
4659:
4655:
4649:
4643:
4623:
4601:
4597:
4593:
4590:
4587:
4582:
4578:
4574:
4569:
4565:
4542:
4538:
4534:
4531:
4528:
4523:
4519:
4515:
4510:
4506:
4497:
4493:
4489:
4488:
4484:
4482:
4468:
4435:
4431:
4427:
4411:
4407:
4403:
4389:
4386:
4370:
4366:
4353:
4351:
4336:
4332:
4329:
4324:
4320:
4316:
4313:
4310:
4305:
4301:
4295:
4291:
4287:
4279:
4274:
4270:
4264:
4260:
4254:
4250:
4246:
4240:
4234:
4229:
4225:
4219:
4215:
4210:
4203:
4187:
4183:
4162:
4159:
4154:
4150:
4146:
4143:
4118:
4114:
4110:
4107:
4104:
4099:
4095:
4091:
4086:
4082:
4075:
4058:
4056:
4040:
4032:
4029:
4026:
4014:
3999:
3995:
3992:
3987:
3983:
3979:
3976:
3973:
3968:
3964:
3958:
3954:
3950:
3942:
3937:
3933:
3927:
3923:
3917:
3913:
3909:
3903:
3897:
3892:
3888:
3882:
3878:
3873:
3866:
3850:
3846:
3825:
3822:
3817:
3813:
3809:
3806:
3781:
3777:
3773:
3770:
3767:
3762:
3758:
3754:
3749:
3745:
3738:
3721:
3719:
3717:
3713:
3709:
3705:
3701:
3697:
3691:Intuitively,
3689:
3670:
3666:
3662:
3659:
3651:
3648:
3645:
3641:
3635:
3631:
3625:
3622:
3619:
3615:
3608:
3602:
3594:
3591:
3587:
3583:
3569:
3549:
3546:
3541:
3537:
3533:
3530:
3505:
3501:
3497:
3494:
3491:
3486:
3482:
3478:
3473:
3469:
3462:
3445:
3443:
3429:
3406:
3403:
3400:
3389:
3373:
3364:
3342:
3339:
3334:
3330:
3321:
3312:
3301:
3285:
3281:
3272:
3258:
3255:
3250:
3246:
3242:
3239:
3214:
3210:
3206:
3203:
3200:
3195:
3191:
3187:
3182:
3178:
3171:
3158:
3157:László Lovász
3150:
3148:
3145:
3128:
3122:
3119:
3111:
3107:
3100:
3078:
3070:
3067:
3064:
3058:
3055:
3035:
3005:
2997:
2994:
2991:
2985:
2982:
2962:
2954:
2953:
2948:otherwise. A
2935:
2932:
2927:
2922:
2918:
2897:
2894:
2891:
2871:
2868:
2863:
2858:
2854:
2831:
2823:
2820:
2817:
2811:
2806:
2802:
2778:
2775:
2753:
2745:
2742:
2739:
2716:
2713:
2666:
2662:
2659:
2650:
2648:
2644:
2636:
2619:
2616:
2610:
2607:
2587:
2584:
2581:
2558:
2555:
2552:
2546:
2543:
2520:
2514:
2491:
2488:
2480:
2459:
2455:
2451:
2448:
2445:
2440:
2436:
2432:
2427:
2423:
2416:
2405:
2403:Directed cuts
2402:
2401:
2400:
2394:
2374:
2371:
2365:
2362:
2356:
2333:
2330:
2324:
2321:
2315:
2312:
2306:
2300:
2293:we have that
2277:
2274:
2266:
2245:
2241:
2237:
2234:
2231:
2226:
2222:
2218:
2213:
2209:
2202:
2191:
2189:
2186:
2171:
2168:
2162:
2159:
2139:
2136:
2133:
2110:
2107:
2104:
2098:
2095:
2072:
2066:
2043:
2040:
2032:
2011:
2007:
2003:
2000:
1997:
1992:
1988:
1984:
1979:
1975:
1968:
1957:
1954:
1953:
1952:
1935:
1932:
1923:
1920:
1914:
1908:
1901:we have that
1885:
1882:
1875:if for every
1874:
1858:
1846:
1844:
1842:
1834:
1812:
1808:
1804:
1801:
1798:
1793:
1789:
1785:
1780:
1776:
1769:
1758:
1756:
1753:
1750:
1747:
1743:
1727:
1704:
1698:
1675:
1669:
1662:we have that
1646:
1643:
1635:
1614:
1610:
1606:
1603:
1600:
1595:
1591:
1587:
1582:
1578:
1571:
1560:
1558:
1555:
1537:
1534:
1513:
1507:
1503:
1497:
1494:
1489:
1485:
1480:
1475:
1471:
1465:
1459:
1438:
1427:
1406:
1402:
1398:
1395:
1392:
1387:
1383:
1379:
1374:
1370:
1363:
1352:
1349:
1334:
1331:
1328:
1308:
1305:
1300:
1296:
1274:
1268:
1264:
1258:
1255:
1252:
1248:
1241:
1238:
1234:
1227:
1221:
1215:
1207:
1205:
1202:
1187:
1184:
1179:
1175:
1171:
1168:
1143:
1139:
1133:
1130:
1127:
1123:
1119:
1113:
1107:
1099:
1096:
1095:
1094:
1077:
1071:
1068:
1062:
1056:
1049:we have that
1036:
1033:
1030:
1023:if for every
1022:
1006:
994:
989:
987:
973:
953:
933:
913:
910:
904:
898:
878:
858:
855:
849:
843:
823:
795:
772:
766:
763:
752:
748:
744:
739:
735:
728:
725:
719:
716:
705:
701:
694:
691:
685:
682:
671:
667:
660:
657:
651:
644:we have that
629:
625:
621:
616:
612:
591:
582:
577:
573:
569:
564:
560:
536:
533:
525:
507:
504:
501:
495:
492:
486:
483:
480:
474:
471:
465:
459:
456:
450:
444:
437:we have that
421:
418:
415:
412:
404:
386:
380:
377:
368:
362:
359:
353:
350:
344:
338:
335:
326:
320:
317:
311:
304:we have that
291:
282:
279:
259:
256:
253:
230:
227:
224:
221:
213:
212:
211:
189:
167:
132:
128:
125:
117:
89:
87:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
6493:
6478:, Elsevier,
6475:
6453:
6431:
6406:
6385:
6376:
6367:
6358:
6349:
6340:
6331:
6313:
6288:
6284:
6275:
6266:
6254:
6229:
6210:
6206:
6193:
6168:
6164:
6158:
6133:
6129:
6123:
6106:10068/182482
6088:
6084:
6067:
6049:
6024:
6020:
6010:
5975:
5965:
5930:
5921:
5912:
5895:
5868:
5858:
5845:
5824:
5815:
5806:
5752:
5749:Applications
5728:
5720:
5614:
5582:
5562:
4357:
4062:
4015:
3725:
3715:
3711:
3707:
3703:
3699:
3692:
3690:
3449:
3365:
3273:
3154:
3146:
2949:
2651:
2646:
2642:
2640:
2398:
2263:be a set of
1872:
1850:
1841:non-monotone
1840:
1838:
1835:Non-monotone
1632:be a set of
1020:
998:
791:
186:denotes the
114:is a finite
93:
70:, including
40:set function
35:
31:
29:
5788:Polymatroid
5759:game theory
5737:guarantees.
5633:maximum cut
5597:minimum cut
4636:defined by
2950:continuous
836:defined by
794:subadditive
56:game theory
6545:Categories
6516:0784.05002
6396:References
6247:S. Jegelka
6077:Lovasz, L.
5927:Lovász, L.
5228:construct
4485:Properties
2846:such that
2643:fractional
2574:such that
2395:Asymmetric
2126:such that
1955:Graph cuts
1871:is called
1426:ground set
604:such that
526:For every
405:For every
272:and every
214:For every
90:Definition
18:Submodular
6305:206800425
6245:R. Iyer,
6041:0097-5397
5957:117358746
5799:Citations
5755:economics
5694:−
5649:−
5524:∈
5517:∪
5494:−
5480:∉
5473:Π
5454:∈
5447:Π
5432:⊆
5425:∑
5421:≥
5351:∪
5216:Ω
5213:⊆
5194:≤
5188:≤
5145:∅
5122:∅
5107:−
5092:Ω
5080:≥
4997:Ω
4773:∖
4770:Ω
4681:α
4660:∑
4598:α
4591:…
4579:α
4566:α
4532:…
4412:−
4404:≥
4387:≥
4330:≥
4321:α
4302:α
4292:∑
4261:α
4251:∑
4226:α
4216:∑
4160:≤
4147:≤
4108:…
3993:≥
3984:α
3965:α
3955:∑
3924:α
3914:∑
3889:α
3879:∑
3851:−
3823:≤
3810:≤
3771:…
3663:−
3649:∉
3642:∏
3623:∈
3616:∏
3598:Ω
3595:⊆
3588:∑
3547:≤
3534:≤
3495:…
3374:λ
3343:λ
3340:≥
3256:≤
3243:≤
3204:…
3059:∈
3011:→
2952:extension
2895:∈
2812:∈
2782:Ω
2779:⊆
2706:Ω
2676:→
2671:Ω
2617:−
2614:Ω
2611:∈
2585:∈
2495:Ω
2492:⊆
2449:…
2414:Ω
2372:−
2369:Ω
2331:−
2328:Ω
2281:Ω
2278:⊆
2235:…
2200:Ω
2169:−
2166:Ω
2163:∈
2137:∈
2047:Ω
2044:⊆
2001:…
1966:Ω
1933:−
1930:Ω
1889:Ω
1886:⊆
1873:symmetric
1847:Symmetric
1802:…
1767:Ω
1650:Ω
1647:⊆
1604:…
1569:Ω
1541:Ω
1538:⊆
1495:∈
1481:⋃
1436:Ω
1396:…
1361:Ω
1332:≥
1306:≥
1288:for each
1256:∈
1249:∑
1185:≥
1166:∀
1131:∈
1124:∑
1069:≤
1034:⊆
804:Ω
729:∪
717:≥
695:∪
661:∪
622:≠
589:∖
586:Ω
583:∈
540:Ω
537:⊆
505:∩
484:∪
472:≥
425:Ω
422:⊆
378:−
363:∪
351:≥
336:−
321:∪
289:∖
286:Ω
283:∈
257:⊆
234:Ω
231:⊆
198:Ω
188:power set
172:Ω
142:→
137:Ω
102:Ω
6458:Elsevier
6430:(2004),
6428:Lee, Jon
6411:Springer
6405:(2003),
6259:U. Feige
6201:(2000).
6150:33192360
6115:43787103
5773:See also
4853:, where
1439:′
1021:monotone
995:Monotone
159:, where
5784:Matroid
5681:matroid
5617:NP-hard
5604:NP-hard
3093:, i.e.
1752:Matroid
1557:Entropy
38:) is a
6514:
6504:
6482:
6464:
6442:
6417:
6303:
6185:888513
6183:
6165:J. ACM
6148:
6113:
6039:
6002:170510
6000:
5990:
5955:
5945:
5906:. cmu.
5883:
5565:convex
5137:where
4492:closed
2910:, and
1245:
6323:(PDF)
6301:S2CID
6181:S2CID
6146:S2CID
6111:S2CID
6059:(PDF)
5998:S2CID
5953:S2CID
5904:(PDF)
4461:when
2884:when
2693:with
2031:graph
246:with
6502:ISBN
6480:ISBN
6462:ISBN
6440:ISBN
6415:ISBN
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