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Maximizing a Submodular Set Function Subject to a Matroid Constraint (Extended Abstract)

  • Gruia Calinescu
  • Chandra Chekuri
  • Martin Pál
  • Jan Vondrák
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4513)

Abstract

Let \(f:2^{N} \rightarrow \cal R^{+}\) be a non-decreasing submodular set function, and let \((N,\cal I)\) be a matroid. We consider the problem \(\max_{S \in \cal I} f(S)\). It is known that the greedy algorithm yields a 1/2-approximation [9] for this problem. It is also known, via a reduction from the max-k-cover problem, that there is no (1 − 1/e + ε)-approximation for any constant ε> 0, unless P = NP [6]. In this paper, we improve the 1/2-approximation to a (1 − 1/e)-approximation, when f is a sum of weighted rank functions of matroids. This class of functions captures a number of interesting problems including set coverage type problems. Our main tools are the pipage rounding technique of Ageev and Sviridenko [1] and a probabilistic lemma on monotone submodular functions that might be of independent interest.

We show that the generalized assignment problem (GAP) is a special case of our problem; although the reduction requires |N| to be exponential in the original problem size, we are able to interpret the recent (1 − 1/e)-approximation for GAP by Fleischer et al. [10] in our framework. This enables us to obtain a (1 − 1/e)-approximation for variants of GAP with more complex constraints.

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Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Gruia Calinescu
    • 1
  • Chandra Chekuri
    • 2
  • Martin Pál
    • 3
  • Jan Vondrák
    • 4
  1. 1.Computer Science Dept., Illinois Institute of Technology, Chicago, IL 
  2. 2.Dept. of Computer Science, University of Illinois, Urbana, IL 61801 
  3. 3.Google Inc., 1440 Broadway, New York, NY 10018 
  4. 4.Dept. of Mathematics, Princeton University, Princeton, NJ 08544 

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