Approximations for Monotone and Nonmonotone Submodular Maximization with Knapsack Constraints

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 non-negative 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.