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Abstract

The Multiple Knapsack Problem (MKP) (with equal capacities) can be defined as follows: Given a set of n items with positive integer weights and profits, a subset has to be selected such that the items in this subset can be packed into m knapsacks of equal capacities and such that the total profit of all items in the knapsacks is maximized. For m = 1 (MKP) reduces to the classical 0-1 single knapsack problem. It is known that (MKP) admits no fully polynomial-time approximation scheme even for m = 2 unless  \(\mathcal{P} = \mathcal{NP}\). In this paper we present a polynomial time approximation scheme for (MKP) even if m is part of the input. This solves an important open problem in the field of knapsack problems.

Keywords

Knapsack Problem Critical Item Polynomial Time Approximation Scheme Small Item Fully Polynomial Time Approximation Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Caprara, A., Kellerer, H., Pferschy, U.: The multiple subset sum problem. Technical Report, Faculty of Economics, University of Graz (1998) (submitted)Google Scholar
  2. 2.
    Kellerer, H., Pferschy, U.: A new fully polynomial approximation scheme for the knapsack problem. In: Jansen, K., Rolim, J.D.P. (eds.) APPROX 1998. LNCS, vol. 1444, pp. 123–134. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  3. 3.
    Lenstra, H.W.: Integer programming with a fixed number of variables. Mathematics of Operations Research 8, 538–548 (1983)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Martello, S., Toth, P.: Knapsack problems: Algorithms and computer implementations. J. Wiley & Sons, Chichester (1990)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Hans Kellerer
    • 1
  1. 1.Institut für Statistik und Operations ResearchUniversität GrazGrazAustria

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