Approximations for Monotone and Non-monotone 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 {\em extension by expectation} of the submodular function. Formally, we show that, for any non-negative submodular function, an $\alpha$-approximation algorithm for the continuous relaxation implies a randomized $(\alpha -\text{eps})$-approximation algorithm for the discrete problem. We use this relation to improve the best known approximation ratio for the problem to $1\mathrm{/}4-\text{eps}$, for any $\text{eps}>0$, and to obtain a nearly optimal $(1-e^{-1}-\text{eps})-$approximation ratio for the monotone case, for any $\text{eps}>0$. 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.