DOI QR코드

DOI QR Code

A Lagrangian Heuristic for the Multidimensional 0-1 Knapsack Problem

다중 배낭 문제를 위한 라그랑지안 휴리스틱

  • 윤유림 (서울대학교 컴퓨터공학부) ;
  • 김용혁 (광운대학교 컴퓨터소프트웨어학과)
  • Received : 2010.07.12
  • Accepted : 2010.11.20
  • Published : 2010.12.25

Abstract

In general, Lagrangian method for discrete optimization is a kind of technique to easily manage constraints. It is traditionally used for finding upper bounds in the branch-and-bound method. In this paper, we propose a new Lagrangian search method for the 0-1 knapsack problem with multiple constraints. A novel feature of the proposed method different from existing Lagrangian approaches is that it can find high-quality lower bounds, i.e., feasible solutions, efficiently based on a new property of Lagrangian vector. We show the performance improvement of the proposed Lagrangian method over existing ones through experiments on well-known large scale benchmark data.

일반적으로 이산 최적화에서의 라그랑지안 방법은 제약조건을 쉽게 다루기 위한 기법이다. 이 방법은 전형적으로 분지한계법에서 상한을 찾을 때 사용한다. 본 논문은 여러 개의 제약조건이 있는 다중 배낭 문제를 위한 새로운 라그랑지안 방법을 제안한다. 기존 라그랑지안 접근법과는 달리 제안한 방법은 라그랑지안 벡터의 새로운 특징에 기초하여 품질 좋은 하한(즉, 가능 해)을 효율적으로 찾을 수 있다. 잘 알려진 큰 규모의 벤치마크 데이터에서 실험을 하였고 제안한 라그랑지안 방법은 기존 방법의 성능을 개선하였다.

Keywords

References

  1. M. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, 1979.
  2. A. Caprara, H. Kellerer, U. Pferschy, and D. Pisinger, "Approximation algorithms for knapsack problems with cardinality constraints," European Journal of Operational Research, Vol.123, pp. 333-345, 2000. https://doi.org/10.1016/S0377-2217(99)00261-1
  3. H. Kellerer and U. Pferschy, "A new fully polynomial approximation scheme for the knapsack problem," In Proceedings of the 1st International Workshop on Approximation Algorithms for Combinatorial Optimization, pp. 123-134, 1998.
  4. E. L. Lawler, "Fast approximation algorithms for knapsack problems," In Proceedings of the 17th Annual Symposium on Foundations of Computer Science, pp. 206-213, 1997.
  5. S. Sahni, "Approximate algorithms for the 0/1 knapsack problem," Journal of the ACM, Vol. 22, pp. 115-124, 1975. https://doi.org/10.1145/321864.321873
  6. C. Chekuri and S. Khanna, "A PTAS for the multiple knapsack problem," In Proceedings of the Symposium on Discrete Algorithms, pp.213-222, 2000.
  7. P. C. Chu, A genetic algorithm approach for combinatorial optimization problems. PhD thesis, The management school, imperial college of science, London, 1997.
  8. A. Freville and G. Plateau, "An efficient preprocessing procedure for the multidimensional 0-1 knapsack problem," Discrete Applied Mathematics, Vol. 49, pp. 189-212, 1994. https://doi.org/10.1016/0166-218X(94)90209-7
  9. A. M. Frieze and M. R. B. Clarke, "Approximation algorithms for the m-dimensional 0-1 knapsack problem: Worst-case and probabilistic analyses," European Journal of Operational Research, Vol. 15, pp. 100-109, 1984. https://doi.org/10.1016/0377-2217(84)90053-5
  10. B. Gavish and H. Pirkul, "Efficient algorithms for solving multiconstraint zero-one knapsack problems to optimality," Mathematical Programming, Vol. 31, pp. 78-105, 1985. https://doi.org/10.1007/BF02591863
  11. M. J. Magazine and O. Oguz, "A heuristic algorithm for the multidimensional zero-one knapsack problem," European Journal of Operational Research, Vol. 16, pp. 319-326, 1984. https://doi.org/10.1016/0377-2217(84)90286-8
  12. G. R. Raidl, "An improved genetic algorithm for the multiconstrained 0-1 knapsack problem," In Proceedings of the IEEE Conference on Evolutionary Computation, pp. 207-211, 1998.
  13. M. Vasquez and J.-K. Hao, "A hybrid approach for the 0-1 multidimensional knapsack problem," In Proceedings of the 17th International Joint Conference on Articial Intelligence, pp. 328-333, 2001.
  14. D. G. Luenberger, Optimization by Vector Space Methods. John Wiley & Sons, Inc., 1969.
  15. G. L. Nemhauser and L. A. Wolsey, Inter and Combinatorial Optimization. John Wiley & Sons, Inc., 1988.
  16. L. A. Wolsey, Integer Programming, John Wiley & Sons, Inc., 1998.
  17. Y.-J. Chang and B. W. Wah, "Lagrangian techniques for solving a class of zero-one integer linear programs," In Proceedings of the Computer Software and Applications Conference, pp. 156-161, 1995.
  18. B. Gavish, "On obtaining the 'best' multipliers for a Lagrangean relaxation for integer programming," Computers & Operations Research, Vol. 5, pp. 55-71, 1978. https://doi.org/10.1016/0305-0548(78)90018-7
  19. A. M. Geoffrion, "Lagrangian relaxation for integer programming," Mathematical Programming Study," Vol. 2, pp. 82-114, 1974. https://doi.org/10.1007/BFb0120690
  20. D. Schuurmans, F. Southey, and R. C. Holte, "The exponentiated subgradient algorithm for heuristic boolean programming," In Proceedings of the International Joint Conferences on Artificial Intelligence, pp. 334-341, 2001.
  21. B. W. Wah and Y. Shang, "A discrete Lagrangian-based global-search method for solving satisability problems," In D.-Z. Du, J. Gu, and P. Pardalos, editors, Satisability Problem: Theory and Applications, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pp. 365-392. 1997.
  22. B. W. Wah and Z. Wu, "The theory of discrete Lagrange multipliers for nonlinear discrete optimization," In Principles and Practice of Constraint Programming, pp. 28-42, 1999.
  23. S. Martello and P. Toth, Knapsack Problems: Algorithms and Computer Implementations. John Wiley & Sons, Inc., 1990.
  24. G. R. Raidl, "Weight-codings in a genetic algorithm for the multiconstraint knapsack problem," In Proceedings of the Congress on Evolutionary Computation, Vol. 1, pp. 596-603, 1999.
  25. R. K. Martin, Large Scale Linear and Integer Optimization: A Unified Approach. Kluwer Academic Publishers, 1998.
  26. M. L. Fisher, "The Lagrangian relaxation method for solving integer programming problems," Management Science, Vol. 27, No. 1, pp. 1-18, 1981. https://doi.org/10.1287/mnsc.27.1.1
  27. J. F. Shapiro, "A survey of Lagrangian techniques for discrete optimization," Annals of Discrete Mathematics, Vol. 5, pp. 113-138, 1979. https://doi.org/10.1016/S0167-5060(08)70346-7
  28. P. C. Chu and J. E. Beasley, "A genetic algorithm for the multidimensional knapsack problem," Journal of Heuristics, Vol. 4, pp. 63-86, 1998. https://doi.org/10.1023/A:1009642405419