A Fast Approximation Scheme for the Multiple Knapsack Problem

Abstract

In this paper we propose an improved efficient approximation scheme for the multiple knapsack problem (MKP). Given a set ${\mathcal A}$ of n items and set ${\mathcal B}$ of m bins with possibly different capacities, the goal is to find a subset $S \subseteq{\mathcal A}$ of maximum total profit that can be packed into ${\mathcal B}$ without exceeding the capacities of the bins. Chekuri and Khanna presented a PTAS for MKP with arbitrary capacities with running time $n^{O(1/\epsilon^8 \log(1/\epsilon))}$ . Recently we found an efficient polynomial time approximation scheme (EPTAS) for MKP with running time $2^{O(1/\epsilon^5 \log(1/\epsilon))} poly(n)$ . Here we present an improved EPTAS with running time $2^{O(1/\epsilon \log^4(1/\epsilon))} + poly(n)$ . If the integrality gap between the ILP and LP objective values for bin packing with different sizes is bounded by a constant, the running time can be further improved to $2^{O(1/\epsilon \log^2(1/\epsilon))} + poly(n)$ .