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

• Authors: Gruia Calinescu;Chandra Chekuri;Martin Pál;Jan Vondrák
• Venue: IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
• Year: 2007

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 茂戮驴 1/e+ 茂戮驴)-approximation for any constant 茂戮驴 0, unless P= NP[6]. In this paper, we improve the 1/2-approximation to a (1 茂戮驴 1/e)-approximation, when fis 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 茂戮驴 1/e)-approximation for GAP by Fleischer et al.[10] in our framework. This enables us to obtain a (1 茂戮驴 1/e)-approximation for variants of GAP with more complex constraints.