Submodular function maximization via the multilinear relaxation and contention resolution schemes

Chandra Chekuri, Jan Vondrák, Rico Zenklusen

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We consider the problem of maximizing a non-negative submodular set function f: 2N → ℝ+ over a ground set N subject to a variety of packing type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that allows us to derive a number of new results, in particular when f may be a non-monotone function. Our algorithms are based on (approximately) solving the multilinear extension F of f [5] over a polytope P that represents the constraints, and then effectively rounding the fractional solution. Although this approach has been used quite successfully in some settings [6, 22, 24, 13, 3], it has been limited in some important ways. We overcome these limitations as follows. First, we give constant factor approximation algorithms to maximize F over an arbitrary down-closed polytope P that has an efficient separation oracle. Previously this was known only for monotone functions [36]. For non-monotone functions, a constant factor was known only when the polytope was either the intersection of a fixed number of knapsack constraints [24] or a matroid polytope [37,30]. Second, we show that contention resolution schemes are an effective way to round a fractional solution, even when f is non-monotone. In particular, contention resolution schemes for different polytopes can be combined to handle the intersection of different constraints. Via LP duality we show that a contention resolution scheme for a constraint is related to the correlation gap [1] of weighted rank functions of the constraint. This leads to an optimal contention resolution scheme for the matroid polytope. Our results provide a broadly applicable framework for maximizing linear and submodular functions subject to independence constraints. We give several illustrative examples. Contention resolution schemes may find other applications.

Original languageEnglish (US)
Title of host publicationSTOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing
Pages783-792
Number of pages10
DOIs
StatePublished - 2011
Event43rd ACM Symposium on Theory of Computing, STOC'11 - San Jose, CA, United States
Duration: Jun 6 2011Jun 8 2011

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Other

Other43rd ACM Symposium on Theory of Computing, STOC'11
CountryUnited States
CitySan Jose, CA
Period6/6/116/8/11

Keywords

  • approximation algorithm
  • contention resolution scheme
  • independence constraint
  • matroid
  • submodular function maximization

ASJC Scopus subject areas

  • Software

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  • Cite this

    Chekuri, C., Vondrák, J., & Zenklusen, R. (2011). Submodular function maximization via the multilinear relaxation and contention resolution schemes. In STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing (pp. 783-792). (Proceedings of the Annual ACM Symposium on Theory of Computing). https://doi.org/10.1145/1993636.1993740