Before revision:  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.
After revision:  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.