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 A of n items and set B of m bins with possibly different capacities, the goal is to find a subset S ⊆ A of maximum total profit that can be packed into 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/∈8 log(1/∈)). Recently we found an efficient polynomial time approximation scheme (EPTAS) for MKP with running time 2 O(1/∈5 log(1/∈)) poly(n). Here we present an improved EPTAS with running time 2 O(1/∈ log4(1/∈)) + 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/∈ log2(1/∈)) + poly(n). © 2012 Springer-Verlag.