Let f : 2 X → R + be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem max S∈I f (S). It is known that the greedy algorithm yields a 1/2approximation [17] for this problem. For certain special cases, e.g. max |S|≤k f (S), the greedy algorithm yields a (1 − 1/e)-approximation. It is known that this is optimal both in the value oracle model (where the only access to f is through a black box returning f (S) for a given set S) [37], and also for explicitly posed instances assuming P = N P [13]. In this paper, we provide a randomized (1 − 1/e)-approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [1], and a continuous greedy process that may be of independent interest. As a special case, our algorithm implies an optimal approximation for the Submodular Welfare Problem in the value oracle model [42]. As a second application, we show that the Generalized Assignment Problem (GAP) is also a special case; although the reduction requires |X| to be exponential in the original problem size, we are able to achieve a (1 − 1/e − o(1))approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.

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We consider the problem of maximizing a non-negative submodular set function f : 2 N → R+ over a ground set N subject to a variety of packing type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that allows us to derive a number of new results, in particular when f may be a non-monotone function. Our algorithms are based on (approximately) solving the multilinear extension F of f [5] over a polytope P that represents the constraints, and then effectively rounding the fractional solution. Although this approach has been used quite successfully in some settings [6,22,24,13,3], it has been limited in some important ways. We overcome these limitations as follows.First, we give constant factor approximation algorithms to maximize F over an arbitrary down-closed polytope P that has an efficient separation oracle. Previously this was known only for monotone functions [36]. For non-monotone functions, a constant factor was known only when the polytope was either the intersection of a fixed number of knapsack constraints [24] or a matroid polytope [37,30]. Second, we show that contention resolution schemes are an effective way to round a fractional solution, even when f is non-monotone. In particular, contention resolution schemes for different polytopes can be combined to handle the intersection of different constraints. Via LP duality we show that a contention resolution scheme for a constraint is related to the correlation gap [1] of weighted rank functions of the constraint. This leads to an optimal contention resolution scheme for the matroid polytope.Our results provide a broadly applicable framework for maximizing linear and submodular functions subject to independence constraints. We give several illustrative examples. Contention resolution schemes may find other applications.

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