Optimal approximation for the submodular welfare problem in the value oracle model

Abstract

In the Submodular Welfare Problem, m items are to be distributed among n players with utility functions wi: 2[m] → R+. The utility functions are assumed to be monotone and submodular. Assuming that player i receives a set of items Si, we wish to maximize the total utility ∑i=1n wi(Si). In this paper, we work in the value oracle model where the only access to the utility functions is through a black box returning wi(S) for a given set S. Submodular Welfare is in fact a special case of the more general problem of submodular maximization subject to a matroid constraint: max{f(S): S ∈ I}, where f is monotone submodular and I is the collection of independent sets in some matroid. For both problems, a greedy algorithm is known to yield a 1/2-approximation [21, 16]. In special cases where the matroid is uniform (I = S: |S| ≤ k) [20] or the submodular function is of a special type [4, 2], a (1-1/e)-approximation has been achieved and this is optimal for these problems in the value oracle model [22, 6, 15]. A (1-1/e)-approximation for the general Submodular Welfare Problem has been known only in a stronger demand oracle model [4], where in fact 1-1/e can be improved [9]. In this paper, we develop a randomized continuous greedy algorithm which achieves a (1-1/e)-approximation for the Submodular Welfare Problem in the value oracle model. We also show that the special case of n equal players is approximation resistant, in the sense that the optimal (1-1/e)-approximation is achieved by a uniformly random solution. Using the pipage rounding technique [1, 2], we obtain a (1-1/e)-approximation for submodular maximization subject to any matroid constraint. The continuous greedy algorithm has a potential of wider applicability, which we demonstrate on the examples of the Generalized Assignment Problem and the AdWords Assignment Problem.