An Almost Optimal Approximation Algorithm for Monotone Submodular Multiple Knapsack

We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint. The input is a set $I$ of items, each has a non-negative weight, and a set of bins of arbitrary capacities. Also, we are given a submodular, monotone and non-negative function $f$ over subsets of the items. The objective is to find a packing of a subset of items $A \subseteq I$ in the bins such that $f(A)$ is maximized. Our main result is an almost optimal polynomial time $(1-e^{-1}-\varepsilon)$-approximation algorithm for the problem, for any $\varepsilon>0$. The algorithm relies on a structuring technique which converts a general multiple knapsack constraint to a constraint in which the bins are partitioned into groups of exponentially increasing cardinalities, each consisting of bins of uniform capacity. We derive the result by combining structuring with a refined analysis of techniques for submodular optimization subject to knapsack constraints.