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

Gruia Calinescu, Chandra Chekuri, Martin Pál, Jan Vondrák

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.