Document

# Constrained Monotone Function Maximization and the Supermodular Degree

## File

LIPIcs.APPROX-RANDOM.2014.160.pdf
• Filesize: 0.59 MB
• 16 pages

## Cite As

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)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.160

## Abstract

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.
##### Keywords
• supermodular degree
• set function
• submodular
• matroid
• extendible system

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Sushil Bikhchandani and John W. Mamer. Competitive equilibrium in an exchange economy with indivisibilities. Journal of Economic Theory, 74(2):385-413, 1997.
2. Liad Blumrosen and Noam Nisan. On the computational power of demand queries. SIAM Journal on Computing, 39:1372-1391, 2009.
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.
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.
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.
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.
7. V. Conitzer, T. Sandholm, and P. Santi. Combinatorial auctions with k-wise dependent valuations. In AAAI, pages 248-254, 2005.
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.
9. Shahar Dobzinski and Michael Schapira. An improved approximation algorithm for combinatorial auctions with submodular bidders. In SODA, pages 1064-1073, 2006.
10. Uriel Feige. On maximizing welfare when utility functions are subadditive. SIAM Journal on Computing, 39:122-142, 2009. Preliminary version in STOC'06.
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.
12. Uriel Feige and Rani Izsak. Welfare maximization and the supermodular degree. In ITCS, pages 247-256, 2013.
13. Uriel Feige and Jan Vondrák. The submodular welfare problem with demand queries. Theory of Computing, 6(1):247-290, 2010.
14. Moran Feldman, Joseph (Seffi) Naor, and Roy Schwartz. A unified continuous greedy algorithm for submodular maximization. In FOCS, 2011.
15. Moran Feldman, Joseph (Seffi) Naor, Roy Schwartz, and Justin Ward. Improved approximations for k-exchange systems. In ESA, pages 784-798, 2011.
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.
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.
18. M. Grötschel, L. Lovász, and A. Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatoria, 1(2):169-197, 1981.
19. Faruk Gul and Ennio Stacchetti. Walrasian equilibrium with gross substitutes. Journal of Economic Theory, 87(1):95-124, 1999.
20. D. Hausmann and B. Korte. K-greedy algorithms for independence systems. Oper. Res. Ser. A-B, 22(1):219-228, 1978.
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.
22. Elad Hazan, Shmuel Safra, and Oded Schwartz. On the complexity of approximating k-set packing. Computational Complexity, 15(1):20-39, May 2006.
23. Satoru Iwata and Kiyohito Nagano. Submodular function minimization under covering constraints. In FOCS, pages 671-680, 2009.
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.
25. T. Jenkyns. The efficacy of the greedy algorithm. Cong. Num., 17:341-350, 1976.
26. B. Korte and D. Hausmann. An analysis of the greedy heuristic for independence systems. Annals of Discrete Math., 2:65-74, 1978.
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.
28. Julián Mestre. Greedy in approximation algorithms. In ESA, pages 528-539, 2006.
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.
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.
31. Prasad Raghavendra and David Steurer. Graph expansion and the unique games conjecture. In STOC, pages 755-764, 2010.
32. Prasad Raghavendra, David Steurer, and Madhur Tulsiani. Reductions between expansion problems. In IEEE Conference on Computational Complexity, pages 64-73, 2012.
33. Jan Vondrák. Symmetry and approximability of submodular maximization problems. SIAM J. Comput., 42(1):265-304, 2013.
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.