A Fast Approximation Scheme for the Multiple Knapsack Problem

  • Klaus Jansen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7147)


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 \(2^{O(1/\epsilon \log^2(1/\epsilon))} + poly(n)\).


Linear Program Relaxation Full Paper Polynomial Time Approximation Scheme Medium Item Fully Polynomial Time Approximation Scheme 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Klaus Jansen
    • 1
  1. 1.Institut für InformatikChristian-Albrechts-Universität zu KielKielGermany

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