A $(1-e^{-1}-\varepsilon)$-Approximation for the Monotone Submodular Multiple Knapsack Problem

We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint (SMKP) . The input is a set $I$ of items, each associated with a non-negative weight, and a set of bins, each having a capacity. Also, we are given a submodular, monotone and non-negative function $f$ over subsets of the items. The objective is to find a subset of items $A \subseteq I$ and a packing of the items in the bins, such that $f(A)$ is maximized. SMKP is a natural extension of both Multiple Knapsack and the problem of monotone submodular maximization subject to a knapsack constraint. Our main result is a nearly optimal polynomial time $(1-e^{-1}-\varepsilon)$-approximation algorithm for the problem, for any $\varepsilon>0$. Our algorithm relies on a refined analysis of techniques for constrained submodular optimization combined with sophisticated application of tools used in the development of approximation schemes for packing problems.