Maximizing a Monotone Submodular Function Subject to a Matroid Constraint

Abstract

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.

Comparing Apples and Oranges: Query Trade-off in Submodular Maximization

“…Feige [15] proved the latter holds even when the objective function is restricted to being a coverage function. Calinescu et al [8] presented the continuous greedy algorithm, which enabled one to achieve the same tight (1 − 1 /e) guarantee for the more general matroid constraint.…”

“…An approximation of 0.309 was given by Vondrák [41] for the general matroid independence constraint, which was later improved to 0.325 by Oveis Gharan and Vondrák [22] using a simulated annealing technique. Extending the continuous greedy algorithm of [8] to general non-negative submodular objectives, Feldman et al [19] obtained an improved approximation of 1 /e − o(1) for the same problem.…”

“…We denote by ∂ u F (x) the derivative of F at point x with respect to the coordinate corresponding to u. The multilinear extension has been extensively used for maximizing submodular functions subject to a matroid constrained, starting with the continuous greedy algorithm of Calinescu et al [8]. All algorithms based on the multilinear extension approximate its value at various points using sampling.…”

“…Elegant algorithmic techniques were developed in the course of this line of research which achieved provable, and in some cases even tight, approximation guarantees. A prime example for the latter is the continuous greedy algorithm of [8] for maximizing a monotone submodular function subject to a matroid independence constraint. Unfortunately, most of these techniques result in algorithms which are efficient in theory but are not practical.…”

Approximation Algorithms for the Connected Sensor Cover Problem

“…Submodular maximization under cardinality constraint, which generalizes the maximum coverage problem, is a classical combinatorial optimization problem and it is known the optimal approximation is 1−1/e [30,14]. Submodular maximization under various more general combinatorial constraints (in particular, downward monotone set systems) is a vibrant research area in theoretical computer science and there have been a number of exciting new developments in the past few years (see e.g., [3,33] and the references therein). The connectivity constraint has also been considered in some previous work [38,26,24], some of which we mentioned before.…”