Approximations for monotone and nonmonotone submodular maximization with knapsack constraints

Submodular maximization generalizes many fundamental problems in discrete optimization, including Max-Cut in directed/undirected graphs, maximum coverage, maximum facility location, and marketing over social networks. In this paper we consider the problem of maximizing any submodular function subject to d knapsack constraints, where d is a fixed constant. We establish a strong relation between the discrete problem and its continuous relaxation, obtained through extension by expectation of the submodular function. Formally, we show that, for any nonnegative submodular function, an α-approximation algorithm for the continuous relaxation implies a randomized (α-ε)-approximation algorithm for the discrete problem. We use this relation to obtain an (e-1-ε)-approximation for the problem, and a nearly optimal (1-e-1-ε)-approximation ratio for the monotone case, for any ε>0. We further show that the probabilistic domain defined by a continuous solution can be reduced to yield a polynomial-size domain, given an oracle for the extension by expectation. This leads to a deterministic version of our technique.