Constrained Monotone Function Maximization and the Supermodular Degree

Authors Moran Feldman, Rani Izsak

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Moran Feldman
Rani Izsak

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Moran Feldman and Rani Izsak. Constrained Monotone Function Maximization and the Supermodular Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 160-175, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


The problem of maximizing a constrained monotone set function has many practical applications and generalizes many combinatorial problems such as k-Coverage, Max-SAT, Set Packing, Maximum Independent Set and Welfare Maximization. Unfortunately, it is generally not possible to maximize a monotone set function up to an acceptable approximation ratio, even subject to simple constraints. One highly studied approach to cope with this hardness is to restrict the set function, for example, by requiring it to be submodular. An outstanding disadvantage of imposing such a restriction on the set function is that no result is implied for set functions deviating from the restriction, even slightly. A more flexible approach, studied by Feige and Izsak [ITCS 2013], is to design an approximation algorithm whose approximation ratio depends on the complexity of the instance, as measured by some complexity measure. Specifically, they introduced a complexity measure called supermodular degree, measuring deviation from submodularity, and designed an algorithm for the welfare maximization problem with an approximation ratio that depends on this measure. In this work, we give the first (to the best of our knowledge) algorithm for maximizing an arbitrary monotone set function, subject to a k-extendible system. This class of constraints captures, for example, the intersection of k-matroids (note that a single matroid constraint is sufficient to capture the welfare maximization problem). Our approximation ratio deteriorates gracefully with the complexity of the set function and k. Our work can be seen as generalizing both the classic result of Fisher, Nemhauser and Wolsey [Mathematical Programming Study 1978], for maximizing a submodular set function subject to a k-extendible system, and the result of Feige and Izsak for the welfare maximization problem. Moreover, when our algorithm is applied to each one of these simpler cases, it obtains the same approximation ratio as of the respective original work. That is, the generalization does not incur any penalty. Finally, we also consider the less general problem of maximizing a monotone set function subject to a uniform matroid constraint, and give a somewhat better approximation ratio for it.
  • supermodular degree
  • set function
  • submodular
  • matroid
  • extendible system


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  1. Sushil Bikhchandani and John W. Mamer. Competitive equilibrium in an exchange economy with indivisibilities. Journal of Economic Theory, 74(2):385-413, 1997. Google Scholar
  2. Liad Blumrosen and Noam Nisan. On the computational power of demand queries. SIAM Journal on Computing, 39:1372-1391, 2009. Google Scholar
  3. Niv Buchbinder, Moran Feldman, Joseph (Seffi) Naor, and Roy Schwartz. A tight linear time (1/2)-approximation for unconstrained submodular maximization. In FOCS, pages 649-658, 2012. Google Scholar
  4. Gruia Calinescu, Chandra Chekuri, Martin Pal, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing, 40(6):1740-1766, 2011. Google Scholar
  5. Y. Chevaleyre, U. Endriss, S. Estivie, and N. Maudet. Multiagent resource allocation in k-additive domains: preference representation and complexity. Annals of Operations Research, 163:49-62, 2008. Google Scholar
  6. M. Conforti and G. Cornuèjols. Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the rado-edmonds theorem. Disc. Appl. Math., 7(3):251-274, 1984. Google Scholar
  7. V. Conitzer, T. Sandholm, and P. Santi. Combinatorial auctions with k-wise dependent valuations. In AAAI, pages 248-254, 2005. Google Scholar
  8. Shahar Dobzinski, Noam Nisan, and Michael Schapira. Approximation algorithms for combinatorial auctions with complement-free bidders. In STOC, pages 610-618, New York, NY, USA, 2005. ACM. Google Scholar
  9. Shahar Dobzinski and Michael Schapira. An improved approximation algorithm for combinatorial auctions with submodular bidders. In SODA, pages 1064-1073, 2006. Google Scholar
  10. Uriel Feige. On maximizing welfare when utility functions are subadditive. SIAM Journal on Computing, 39:122-142, 2009. Preliminary version in STOC'06. Google Scholar
  11. Uriel Feige, Michal Feldman, Nicole Immorlica, Rani Izsak, Brendan Lucier, and Vasilis Syrgkanis. A unifying hierarchy of valuations with complements and substitutes, 2014. Working paper. Google Scholar
  12. Uriel Feige and Rani Izsak. Welfare maximization and the supermodular degree. In ITCS, pages 247-256, 2013. Google Scholar
  13. Uriel Feige and Jan Vondrák. The submodular welfare problem with demand queries. Theory of Computing, 6(1):247-290, 2010. Google Scholar
  14. Moran Feldman, Joseph (Seffi) Naor, and Roy Schwartz. A unified continuous greedy algorithm for submodular maximization. In FOCS, 2011. Google Scholar
  15. Moran Feldman, Joseph (Seffi) Naor, Roy Schwartz, and Justin Ward. Improved approximations for k-exchange systems. In ESA, pages 784-798, 2011. Google Scholar
  16. M. L. Fisher, G. L. Nemhauser, and L. A. Wolsey. An analysis of approximations for maximizing submodular set functions - II. In Polyhedral Combinatorics, volume 8 of Mathematical Programming Study, pages 73-87. North-Holland Publishing Company, 1978. Google Scholar
  17. Gagan Goel, Chinmay Karande, Pushkar Tripathi, and Lei Wang. Approximability of combinatorial problems with multi-agent submodular cost functions. SIGecom Exchanges, 9(1):8, 2010. Google Scholar
  18. M. Grötschel, L. Lovász, and A. Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatoria, 1(2):169-197, 1981. Google Scholar
  19. Faruk Gul and Ennio Stacchetti. Walrasian equilibrium with gross substitutes. Journal of Economic Theory, 87(1):95-124, 1999. Google Scholar
  20. D. Hausmann and B. Korte. K-greedy algorithms for independence systems. Oper. Res. Ser. A-B, 22(1):219-228, 1978. Google Scholar
  21. D. Hausmann, B. Korte, and T. Jenkyns. Worst case analysis of greedy type algorithms for independence systems. Math. Prog. Study, 12:120-131, 1980. Google Scholar
  22. Elad Hazan, Shmuel Safra, and Oded Schwartz. On the complexity of approximating k-set packing. Computational Complexity, 15(1):20-39, May 2006. Google Scholar
  23. Satoru Iwata and Kiyohito Nagano. Submodular function minimization under covering constraints. In FOCS, pages 671-680, 2009. Google Scholar
  24. Satoru Iwata and James B. Orlin. A simple combinatorial algorithm for submodular function minimization. In SODA, pages 1230-1237, Philadelphia, PA, USA, 2009. Society for Industrial and Applied Mathematics. Google Scholar
  25. T. Jenkyns. The efficacy of the greedy algorithm. Cong. Num., 17:341-350, 1976. Google Scholar
  26. B. Korte and D. Hausmann. An analysis of the greedy heuristic for independence systems. Annals of Discrete Math., 2:65-74, 1978. Google Scholar
  27. Jon Lee, Maxim Sviridenko, and Jan Vondrák. Submodular maximization over multiple matroids via generalized exchange properties. Math. Oper. Res., 35(4):795-806, 2010. Google Scholar
  28. Julián Mestre. Greedy in approximation algorithms. In ESA, pages 528-539, 2006. Google Scholar
  29. G. Nemhauser and L. Wolsey. Best algorithms for approximating the maximum of a submodular set function. Math. Oper. Res., 3(3):177-188, 1978. Google Scholar
  30. G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher. An analysis of approximations for maximizing submodular set functions - I. Mathematical Programming, 14:265-294, 1978. Google Scholar
  31. Prasad Raghavendra and David Steurer. Graph expansion and the unique games conjecture. In STOC, pages 755-764, 2010. Google Scholar
  32. Prasad Raghavendra, David Steurer, and Madhur Tulsiani. Reductions between expansion problems. In IEEE Conference on Computational Complexity, pages 64-73, 2012. Google Scholar
  33. Jan Vondrák. Symmetry and approximability of submodular maximization problems. SIAM J. Comput., 42(1):265-304, 2013. Google Scholar
  34. Justin Ward. A (k+3)/2-approximation algorithm for monotone submodular k-set packing and general k-exchange systems. In STACS, pages 42-53, 2012. Google Scholar
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