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Optimal approximation for the submodular welfare problem in the value oracle model

Author:

Jan VondrakAuthors Info & Claims

STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing

Pages 67 - 74

https://doi.org/10.1145/1374376.1374389

Published: 17 May 2008 Publication History

Abstract

In the Submodular Welfare Problem, m items are to be distributed among n players with utility functions w_i: 2^[m] → R₊. The utility functions are assumed to be monotone and submodular. Assuming that player i receives a set of items S_i, we wish to maximize the total utility ∑_i=1ⁿ w_i(S_i). In this paper, we work in the value oracle model where the only access to the utility functions is through a black box returning w_i(S) for a given set S. Submodular Welfare is in fact a special case of the more general problem of submodular maximization subject to a matroid constraint: max{f(S): S ∈ I}, where f is monotone submodular and I is the collection of independent sets in some matroid.

For both problems, a greedy algorithm is known to yield a 1/2-approximation [21, 16]. In special cases where the matroid is uniform (I = S: |S| ≤ k) [20] or the submodular function is of a special type [4, 2], a (1-1/e)-approximation has been achieved and this is optimal for these problems in the value oracle model [22, 6, 15]. A (1-1/e)-approximation for the general Submodular Welfare Problem has been known only in a stronger demand oracle model [4], where in fact 1-1/e can be improved [9].

In this paper, we develop a randomized continuous greedy algorithm which achieves a (1-1/e)-approximation for the Submodular Welfare Problem in the value oracle model. We also show that the special case of n equal players is approximation resistant, in the sense that the optimal (1-1/e)-approximation is achieved by a uniformly random solution. Using the pipage rounding technique [1, 2], we obtain a (1-1/e)-approximation for submodular maximization subject to any matroid constraint. The continuous greedy algorithm has a potential of wider applicability, which we demonstrate on the examples of the Generalized Assignment Problem and the AdWords Assignment Problem.

References

[1]

A. Ageev and M. Sviridenko. Pipage rounding: a new method of constructing algorithms with proven performance guarantee, J. of Combinatorial Optimization 8 (2004), 307--328.

[2]

G. Calinescu, C. Chekuri, M. Pal and J. Vondrak. Maximizing a submodular set function subject to a matroid constraint, Proc. of 12th IPCO (2007), 182--196.

Digital Library

[3]

C. Chekuri and A. Kumar. Maximum coverage problem with group budget constraints and applications, Proc. of APPROX} 2004, Lecture Notes in Computer Science 3122, 72--83.

[4]

S. Dobzinski and M. Schapira. An improved approximation algorithm for combinatorial auctions with submodular bidders, Proc. of 17th SODA (2006), 1064--1073.

Digital Library

[5]

J. Edmonds. Matroids, submodular functions and certain polyhedra, Combinatorial Structures and Their Applications} (1970), 69--87.

[6]

U. Feige. A threshold of ln n for approximating Set Cover, Journal of the ACM 45 (1998), 634--652.

Digital Library

[7]

U. Feige. Maximizing social welfare when utility functions are subadditive, Proc. of 38th STOC (2006), 41--50.

Digital Library

[8]

U. Feige, V. Mirrokni and J. Vondrak. Maximizing non-monotone submodular functions, Proc. of 48th FOCS (2007), 461--471.

Digital Library

[9]

U. Feige and J. Vondrak. Approximation algorithms for combinatorial allocation problems: Improving the factor of 1-1/e, Proc. of 47th FOCS (2006), 667--676.

Digital Library

[10]

L. Fleischer, S. Fujishige and S. Iwata. A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions, Journal of the ACM 48:4 (2001), 761--777.

Digital Library

[11]

L. Fleischer, M. X. Goemans, V. Mirrokni and M. Sviridenko. Tight approximation algorithms for maximum general assignment problems, Proc. of 17th ACM-SIAM SODA (2006), 611--620.

Digital Library

[12]

P. Goundan and A. Schulz. Revisiting the greedy approach to submodular set function maximization, manuscript (2007).

[13]

J. Hastad. Clique is hard to approximate within n^1-ε, Acta Mathematica 182 (1999), 105--142.

[14]

K. Jansen and G. Zhang. On rectangle packing: maximizing benefits, Proc. of 15th ACM-SIAM SODA (2004), 204--213.

Digital Library

[15]

S. Khot, R. Lipton, E. Markakis and A. Mehta. Inapproximability results for combinatorial auctions with submodular utility functions, Proc. of WINE 2005, Lecture Notes in Computer Science 3828, 92--101.

Digital Library

[16]

B. Lehmann, D. J. Lehmann and N. Nisan. Combinatorial auctions with decreasing marginal utilities, ACM Conference on El. Commerce 2001, 18--28.

Digital Library

[17]

B. Lehmann, D. J. Lehmann and N. Nisan. Combinatorial auctions with decreasing marginal utilities (journal version), Games and Economic Behavior 55 (2006), 270--296.

[18]

L. Lovasz. Submodular functions and convexity. A. Bachem et al., editors, Mathematical Programmming: The State of the Art, 235--257.

[19]

V. Mirrokni, M. Schapira and J. Vondrak. Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions, manuscript (2007).

Digital Library

[20]

G. L. Nemhauser, L. A. Wolsey and M. L. Fisher. An analysis of approximations for maximizing submodular set functions I, Mathematical Programming 14 (1978), 265--294.

Digital Library

[21]

M. L. Fisher, G. L. Nemhauser and L. A. Wolsey. An analysis of approximations for maximizing submodular set functions II, Math. Programming Study 8 (1978), 73--87.

[22]

G. L. Nemhauser and L. A. Wolsey. Best algorithms for approximating the maximum of a submodular set function, Math. Operations Research 3:3 (1978), 177--188.

Digital Library

[23]

A. Schrijver. A combinatorial algorithm minimizing submodular functions in strongly polynomial time, Journal of Combinatorial Theory, Series B 80 (2000), 346--355.

Digital Library

[24]

A. Schrijver. Combinatorial optimization - polyhedra and efficiency, Springer-Verlag Berlin Heidelberg, 2003.

[25]

M. Sviridenko. A note on maximizing a submodular set function subject to knapsack constraint, Operations Research Letters 32 (2004), 41--43.

Digital Library

[26]

L. Wolsey. Maximizing real-valued submodular functions: Primal and dual heuristics for location problems, Math. of Operations Research 7 (1982), 410--425.

Digital Library

[27]

D. Zuckerman. Linear degree extractors and inapproximability, Theory of Computing 3 (2007), 103--128.

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Index Terms

Optimal approximation for the submodular welfare problem in the value oracle model
1. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing

May 2008

712 pages

ISBN:9781605580470

DOI:10.1145/1374376

General Chair:
Richard Ladner
University of Washington
,
Program Chair:
Cynthia Dwork
Microsoft Research, Silicon Valley

Copyright © 2008 ACM.

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Published: 17 May 2008

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https://doi.org/10.1360/SSM-2024-0195
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https://dl.acm.org/doi/10.1007/s11704-024-40266-4
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Rafiey ASalakhutdinov RKolter ZHeller KWeller AOliver NScarlett JBerkenkamp F(2024)Decomposable submodular maximization in federated settingProceedings of the 41st International Conference on Machine Learning10.5555/3692070.3693772(41841-41866)Online publication date: 21-Jul-2024
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