A multi-level search strategy for the 0-1 Multidimensional Knapsack Problem

  • Authors:

  • Sylvain Boussier;Michel Vasquez;Yannick Vimont;Saïd Hanafi;Philippe Michelon

  • Affiliations:

  • LIA, Université d'Avignon et des Pays de Vaucluse, 339 chemin des Meinajaries, BP 1228, 84911 Avignon Cedex 9, France;LGI2P, ícole des Mines d'Alès, Parc scientifique Geroges Besse, 30035 Nímes, France;LGI2P, ícole des Mines d'Alès, Parc scientifique Geroges Besse, 30035 Nímes, France;LAMIH, Université de Valenciennes et du Hainaut-Cambrésis Le Mont Houy - BP 311, 59304 Valenciennes, France;LIA, Université d'Avignon et des Pays de Vaucluse, 339 chemin des Meinajaries, BP 1228, 84911 Avignon Cedex 9, France

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

We propose an exact method based on a multi-level search strategy for solving the 0-1 Multidimensional Knapsack Problem. Our search strategy is primarily based on the reduced costs of the non-basic variables of the LP-relaxation solution. Considering that the variables are sorted in decreasing order of their absolute reduced cost value, the top level branches of the search tree are enumerated following Resolution Search strategy, the middle level branches are enumerated following Branch & Bound strategy and the lower level branches are enumerated according to a simple Depth First Search enumeration strategy. Experimentally, this cooperative scheme is able to solve optimally large-scale strongly correlated 0-1 Multidimensional Knapsack Problem instances. The optimal values of all the 10 constraint, 500 variable instances and some of the 30 constraint, 250 variable instances of the OR-Library were found. These values were previously unknown.