Constrained Non-monotone Submodular Maximization: Offline and Secretary Algorithms

  • Anupam Gupta
  • Aaron Roth
  • Grant Schoenebeck
  • Kunal Talwar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6484)

Abstract

Constrained submodular maximization problems have long been studied, most recently in the context of auctions and computational advertising, with near-optimal results known under a variety of constraints when the submodular function is monotone. In this paper, we give constant approximation algorithms for the non-monotone case that work for p-independence systems (which generalize constraints given by the intersection of p matroids that had been studied previously), where the running time is \(\text{poly}(n,p)\). Our algorithms and analyses are simple, and essentially reduce non-monotone maximization to multiple runs of the greedy algorithm previously used in the monotone case.

We extend these ideas to give a simple greedy-based constant factor algorithms for non-monotone submodular maximization subject to a knapsack constraint, and for (online) secretary setting (where elements arrive one at a time in random order and the algorithm must make irrevocable decisions) subject to uniform matroid or a partition matroid constraint. Finally, we give an O(logk) approximation in the secretary setting subject to a general matroid constraint of rank k.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Anupam Gupta
    • 1
  • Aaron Roth
    • 2
  • Grant Schoenebeck
    • 3
  • Kunal Talwar
    • 4
  1. 1.Carnegie Mellon UniversityPittsburgh
  2. 2.Microsoft Research New EnglandCambridge
  3. 3.Princeton UniversityPrinceton
  4. 4.Microsoft ResearchMountain View

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