On the Separation of the Rank-1 Chvatal-Gomory Inequalities for the Fixed-Charge 0-1 Knapsack Problem

고정비용 0-1 배낭문제에 대한 크바탈-고모리 부등식의 분리문제에 관한 연구

  • 박경철 (명지대학교 경영학과) ;
  • 이경식 (한국외국어대학교 산업경영공학과)
  • Received : 2011.03.23
  • Accepted : 2011.05.09
  • Published : 2011.06.30

Abstract

We consider the separation problem of the rank-1 Chvatal-Gomory (C-G) inequalities for the 0-1 knapsack problem with the knapsack capacity defined by an additional binary variable, which we call the fixed-charge 0-1 knapsack problem. We analyze the structural properties of the optimal solutions to the separation problem and show that the separation problem can be solved in pseudo-polynomial time. By using the result, we also show that the existence of a pseudo-polynomial time algorithm for the separation problem of the rank-1 C-G inequalities of the ordinary 0-1 knapsack problem.

Keywords

Acknowledgement

Supported by : Hankuk University of Foreign Studies Research Fund

References

  1. Capara, A. and M. Fishetti, "{0, 1/2}-Chvatal-Gomory Cuts," Mathematical Programming, Vol.74, No.3(1996), pp.221-235. https://doi.org/10.1007/BF02592196
  2. Capara, A., M. Fishetti, and A.N. Letchford, "On the Separation of Maximally Violated mod-k Cuts," Mathematical Programming, Vol.87, No.1 (2000), pp.37-56. https://doi.org/10.1007/s101079900107
  3. Dash, S., O. Gunluk, and A. Lodi, "MIR Closures of Polyhedral Sets," Mathematical Programming, Vol.121, No.1(2010), pp.33-60. https://doi.org/10.1007/s10107-008-0225-x
  4. Daskin, M.S., Network and Discrete Location : Models, Algorithms, and Applications, Wiley, 1995.
  5. Eisenbrand, F., "On the Membership Problem for the Elementary Closure of a Polyhedron," Combinatorica, Vol.19, No.2(1999), pp.297-300. https://doi.org/10.1007/s004930050057
  6. Fischetti, M. and A. Lodi, "Optimizing over the first Chvatal Closure," Mathematical Programming, Vol.110, No.1(1999), pp.3-20.
  7. Glover, F., H.D. Sherali, and Y. Lee, "Generating Cuts from Surrogate Contraint Analysis for Zero-One and Multiple Choice Programming," Computational Optimization and Application, Vol.8(1997), pp.151-172. https://doi.org/10.1023/A:1008621204567
  8. Lee, K. and S. Park, "A Cut Generation Method for the (0, 1)-Knapsack Problem with a Variable Capacity," Journal of the Korean OR/MS Society, Vol.25, No.3(2000), pp.1-15.
  9. Nemhauser, G.L. and L.A. Wolsey, Integer and Combinatorial Optimization, Wiley, 1988.