A new polynomial time algorithm for 0-1 multiple knapsack problem based on dominant principles

Abstract This paper presents a heuristic to solve the 0–1 multi-constrained knapsack problem (01MKP) which is NP-hard. In this heuristic the dominance property of the constraints is exploited to reduce the search space to find near optimal solutions of 01MKP. This heuristic is tested for 10 benchmark problems of sizes up to 105 and for seven classical problems of sizes up to 500, taken from the literature and the results are compared with optimum solutions. Space and computational complexity of solving 01MKP using this approach are also presented. The encouraging results especially for relatively large size test problems indicate that this heuristic can successfully be used for finding good solutions for highly constrained NP-hard problems.

[1]  John E. Beasley,et al.  OR-Library: Distributing Test Problems by Electronic Mail , 1990 .

[2]  Sartaj Sahni,et al.  Approximate Algorithms for the 0/1 Knapsack Problem , 1975, JACM.

[3]  A. Victor Cabot,et al.  An Enumeration Algorithm for Knapsack Problems , 1970, Oper. Res..

[4]  Wei Shih,et al.  A Branch and Bound Method for the Multiconstraint Zero-One Knapsack Problem , 1979 .

[5]  Hasan Pirkul,et al.  A heuristic solution procedure for the multiconstraint zero‐one knapsack problem , 1987 .

[6]  Thomas Bäck,et al.  The zero/one multiple knapsack problem and genetic algorithms , 1994, SAC '94.

[7]  Andreas Drexl,et al.  A simulated annealing approach to the multiconstraint zero-one knapsack problem , 1988, Computing.

[8]  John E. Beasley,et al.  A Genetic Algorithm for the Multidimensional Knapsack Problem , 1998, J. Heuristics.

[9]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[10]  A Volgenant,et al.  An Improved Heuristic for Multidimensional 0-1 Knapsack Problems , 1990 .

[11]  Hasan Pirkul,et al.  Efficient algorithms for solving multiconstraint zero-one knapsack problems to optimality , 1985, Math. Program..

[12]  Saïd Hanafi,et al.  An efficient tabu search approach for the 0-1 multidimensional knapsack problem , 1998, Eur. J. Oper. Res..

[13]  Fred W. Glover,et al.  Cutting and Surrogate Constraint Analysis for Improved Multidimensional Knapsack Solutions , 2002, Ann. Oper. Res..

[14]  E. Balas An Additive Algorithm for Solving Linear Programs with Zero-One Variables , 1965 .

[15]  E. Balas,et al.  Pivot and Complement–A Heuristic for 0-1 Programming , 1980 .

[16]  Fred Glover,et al.  Critical Event Tabu Search for Multidimensional Knapsack Problems , 1996 .

[17]  Osman Oguz,et al.  A heuristic algorithm for the multidimensional zero-one knapsack problem , 1984 .

[18]  Paolo Toth,et al.  Knapsack Problems: Algorithms and Computer Implementations , 1990 .

[19]  Ralph E. Gomory,et al.  The Theory and Computation of Knapsack Functions , 1966, Oper. Res..