  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 which
adheres to bin capacities. In this paper we present a unified algorithm that
yields efficient approximations for a wide class of submodular and modular
optimization problems involving multiple knapsack constraints. One notable
example is a polynomial time approximation scheme for Multiple-Choice Multiple
Knapsack, improving upon the best known ratio of $2$. Another example is
Non-monotone Submodular Multiple Knapsack, for which we obtain a
$(0.385-\varepsilon)$-approximation, matching the best known ratio for a single
knapsack constraint. The robustness of our algorithm is achieved by applying a
novel fractional variant of the classical linear grouping technique, which is
of independent interest.
