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We study the problem of maximizing a monotone submodular function subject toa Multiple Knapsack constraint. The input is a set $I$ of items, each has anon-negative weight, and a set of bins of arbitrary capacities. Also, we aregiven a submodular, monotone and non-negative function $f$ over subsets of theitems. 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 convertsa general multiple knapsack constraint to a constraint in which the bins arepartitioned into groups of exponentially increasing cardinalities, eachconsisting of bins of uniform capacity. We derive the result by combiningstructuring with a refined analysis of techniques for submodular optimizationsubject to knapsack constraints.

An Almost Optimal Approximation Algorithm for Monotone Submodular Multiple Knapsack