Parameterized approximation scheme for the multiple knapsack problem

Abstract

The multiple knapsack problem (MKP) is a well-known generalization of the classical knapsack problem. We are given a set A of n items and set B of m bins (knapsacks) such that each item a ∈ A has a size size(a) and a profit value profit(a), and each bin b ∈ B has a capacity c(b). The goal is to find a subset U ⊂ A of maximum total profit such that U can be packed into B without exceeding the capacities. The decision version of MKP is strongly NP-complete, since it is a generalization of the classical knapsack and bin packing problem. Furthermore, MKP does not admit an FPTAS even if the number m of bins is two. Kellerer gave a PTAS for MKP with identical capacities and Chekuri and Khanna presented a PTAS for MKP with general capacities with running time nO(log(1/e)/e8). In this paper we propose an EPTAS with parameterized running time 2O(log(1/e)/e5) · poly(n) + O(m) for MKP. This solves also an open question by Chekuri and Khanna.