tight approximations for modular and submodular optimization with $d$-resource multiple knapsack constraints

A multiple knapsack constraint over a set of items is defined by a set of bins of arbitrary capacities, and a weight for each of the items. An assignment for the constraint is an allocation of subsets of items to the bins, such that the total weight of items assigned to each bin is bounded by the bin capacity. We study modular (linear) and submodular maximization problems subject to a constant number of (i.e., $d$-resource) multiple knapsack constraints, in which a solution is a subset of items, along with an assignment of the selected items for each of the $d$ multiple knapsack constraints. Our results include a {\em polynomial time approximation scheme} for modular maximization with a constant number of multiple knapsack constraints and a matroid constraint, thus generalizing the best known results for the classic {\em multiple knapsack problem} as well as {\em d-dimensional knapsack}, for any $d \geq 2$. We further obtain a tight $(1-e^{-1}-\epsilon)$-approximation for monotone submodular optimization subject to a constant number of multiple knapsack constraints and a matroid constraint, and a $(0.385-\epsilon)$-approximation for non-monotone submodular optimization subject to a constant number of multiple knapsack constraints. At the heart of our algorithms lies a novel representation of a multiple knapsack constraint as a polytope. We consider this key tool as a main technical contribution of this paper.