An EPTAS for Cardinality Constrained Multiple Knapsack via Iterative Randomized Rounding
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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-ln (2)/2 -ϵ≈ 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.
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