Maximizing a Submodular Set Function subject to a Matroid Constraint
Abstract
Let f : 2N → R+ be a non-decreasing submodular set function, and let (N, I) be a matroid. We consider the problem maxS∈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 + )-approximation for any constant > 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.
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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