Maximizing Submodular Set Functions Subject to Multiple Linear Constraints

The concept of submodularity plays a vital role in combinatorial optimization. In particular, many important optimization problems can be cast as submodular maximization problems, including maximum coverage, maximum facility location and max cut in directed/undirected graphs. In this paper we present the first known approximation algorithms for the problem of maximizing a non-decreasing submodular set function subject to multiple linear constraints. Given a d-dimensional budget vector (L) over bar, for some d >= 1, and an oracle for a non-decreasing submodular set function f over a universe U, where each element e is an element of U is associated with a d-dimensional cost vector, we seek a subset of elements S subset of U whose total cost is at most (L) over bar, such that f(S) is maximized. We develop a framework for maximizing submodular functions subject to d linear constraints that yields a (1 - epsilon)(1 - e(-1))-approximation to the optimum for any epsilon > 0, where d > 1 is some constant. Our study is motivated by a variant of the classical maximum coverage problem that we call maximum coverage with multiple packing constraints. We use our framework to obtain the same approximation ratio for this problem. To the best of our knowledge, this is the first time the theoretical bound of 1 - e(-1) is (almost) matched for both of these problems.