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Approximating Multiobjective Knapsack Problems

Author

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  • Thomas Erlebach

    (Computer Engineering and Networks Laboratory, ETH Zürich, CH-8092 Zürich, Switzerland)

  • Hans Kellerer

    (University of Graz, Department of Statistics and Operations Research, Universitätsstr. 15, A-8010 Graz, Austria)

  • Ulrich Pferschy

    (University of Graz, Department of Statistics and Operations Research, Universitätsstr. 15, A-8010 Graz, Austria)

Abstract

For multiobjective optimization problems, it is meaningful to compute a set of solutions covering all possible trade-offs between the different objectives. The multiobjective knapsack problem is a generalization of the classical knapsack problem in which each item has several profit values. For this problem, efficient algorithms for computing a provably good approximation to the set of all nondominated feasible solutions, the Pareto frontier, are studied. For the multiobjective one-dimensional knapsack problem, a practical fully polynomial-time approximation scheme (FPTAS) is derived. It is based on a new approach to the single-objective knapsack problem using a partition of the profit space into intervals of exponentially increasing length. For the multiobjective m-dimensional knapsack problem, the first known polynomial-time approximation scheme (PTAS), based on linear programming, is presented.

Suggested Citation

  • Thomas Erlebach & Hans Kellerer & Ulrich Pferschy, 2002. "Approximating Multiobjective Knapsack Problems," Management Science, INFORMS, vol. 48(12), pages 1603-1612, December.
    • Handle: RePEc:inm:ormnsc:v:48:y:2002:i:12:p:1603-1612
    DOI: 10.1287/mnsc.48.12.1603.445
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    References listed on IDEAS

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    1. Frieze, A. M. & Clarke, M. R. B., 1984. "Approximation algorithms for the m-dimensional 0-1 knapsack problem: Worst-case and probabilistic analyses," European Journal of Operational Research, Elsevier, vol. 15(1), pages 100-109, January.
    2. Magazine, M. J. & Oguz, Osman, 1981. "A fully polynomial approximation algorithm for the 0-1 knapsack problem," European Journal of Operational Research, Elsevier, vol. 8(3), pages 270-273, November.
    3. Hans Kellerer & Ulrich Pferschy, 1999. "A New Fully Polynomial Time Approximation Scheme for the Knapsack Problem," Journal of Combinatorial Optimization, Springer, vol. 3(1), pages 59-71, July.
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    3. Fritz Bökler & Markus Chimani & Mirko H. Wagner, 2022. "On the rectangular knapsack problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(1), pages 149-160, August.
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    6. Caserta, Marco & Voß, Stefan, 2019. "The robust multiple-choice multidimensional knapsack problem," Omega, Elsevier, vol. 86(C), pages 16-27.
    7. Mavrotas, George & Florios, Kostas, 2013. "An improved version of the augmented epsilon-constraint method (AUGMECON2) for finding the exact Pareto set in Multi-Objective Integer Programming problems," MPRA Paper 105034, University Library of Munich, Germany.
    8. Florios, Kostas & Mavrotas, George & Diakoulaki, Danae, 2010. "Solving multiobjective, multiconstraint knapsack problems using mathematical programming and evolutionary algorithms," European Journal of Operational Research, Elsevier, vol. 203(1), pages 14-21, May.
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    10. Laumanns, Marco & Zenklusen, Rico, 2011. "Stochastic convergence of random search methods to fixed size Pareto front approximations," European Journal of Operational Research, Elsevier, vol. 213(2), pages 414-421, September.
    11. Aissi, Hassene & Bazgan, Cristina & Vanderpooten, Daniel, 2009. "Min-max and min-max regret versions of combinatorial optimization problems: A survey," European Journal of Operational Research, Elsevier, vol. 197(2), pages 427-438, September.
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    13. Liesio, Juuso & Mild, Pekka & Salo, Ahti, 2007. "Preference programming for robust portfolio modeling and project selection," European Journal of Operational Research, Elsevier, vol. 181(3), pages 1488-1505, September.
    14. Fündeling, C.-U. & Trautmann, N., 2010. "A priority-rule method for project scheduling with work-content constraints," European Journal of Operational Research, Elsevier, vol. 203(3), pages 568-574, June.
    15. Bazgan, Cristina & Hugot, Hadrien & Vanderpooten, Daniel, 2009. "Implementing an efficient fptas for the 0-1 multi-objective knapsack problem," European Journal of Operational Research, Elsevier, vol. 198(1), pages 47-56, October.
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