An EPTAS for Cardinality Constrained Multiple Knapsack via Iterative Randomized Rounding

We study the Uniform Cardinality Constrained Multiple Knapsack problem (CMK), a natural generalization of Multiple Knapsack with applications ranging from cloud computing to radio networks. The input is a set of items, each has a value and a weight, and a set of uniform capacity bins. The goal is to assign a subset of the items of maximum total value to the bins such that $(i)$ the capacity of any bin is not exceeded, and $(ii)$ the number of items assigned to each bin satisfies a given cardinality constraint. The best known approximation ratio for CMK is $1-\frac{\ln (2)}{2} -\epsilon \approx 0.653$, which follows from a result for a generalization of the problem. Our main contribution is an efficient polynomial time approximation scheme (EPTAS) for CMK. This essentially resolves the complexity status of the problem, since the existence of a fully polynomial time approximation scheme (FPTAS) is ruled out. Our technique is based on the following simple algorithm: in each iteration, solve a configuration linear program (LP) of the problem; then, sample configurations (i.e., feasible subsets of items for a single bin) according to a distribution specified by the LP solution. The algorithm terminates once each bin is assigned a configuration. We believe that our generic technique may lead to efficient approximations for other assignment problems.