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

摘要

Let f : 2~N → R~+ be a non-decreasing submodular set function, and let (N, I) be a matroid. We consider the problem max s∈x 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 / 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.