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A multiple knapsack constraint over a set of items is defined by a set ofbins of arbitrary capacities, and a weight for each of the items. An assignmentfor the constraint is an allocation of subsets of items to the bins whichadheres to bin capacities. In this paper we present a unified algorithm thatyields efficient approximations for a wide class of submodular and modularoptimization problems involving multiple knapsack constraints. One notableexample is a polynomial time approximation scheme for Multiple-Choice MultipleKnapsack, improving upon the best known ratio of $2$. Another example isNon-monotone Submodular Multiple Knapsack, for which we obtain a$(0.385-\varepsilon)$-approximation, matching the best known ratio for a singleknapsack constraint. The robustness of our algorithm is achieved by applying anovel fractional variant of the classical linear grouping technique, which isof independent interest.

Modular and Submodular Optimization with Multiple Knapsack Constraints via Fractional Grouping