A Multi-Level Search Strategy for the 0-1 Multidimensional Knapsack Problem.

Sylvain Boussier (1) , Michel Vasquez (2) , Yannick Vimont (2) , Saïd Hanafi (3) , Philippe Michelon (1)

1 LIA - Laboratoire Informatique d'Avignon
2 LGI2P - Laboratoire de Génie Informatique et Ingénierie de Production
3 LAMIH - Laboratoire d'Automatique, de Mécanique et d'Informatique industrielles et Humaines - UMR 8201

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.

Mots clés: Resolution Search; Branch & Bound; Reduced costs; 0-1 Multidimensional Knapsack Problem

Domaines: Recherche opérationnelle [math.OC]

Sylvain Boussier, Michel Vasquez, Yannick Vimont, Saïd Hanafi, Philippe Michelon. A Multi-Level Search Strategy for the 0-1 Multidimensional Knapsack Problem.. Discrete Applied Mathematics, 2010, 158, pp.97-109. ⟨10.1016/j.dam.2009.08.007⟩. ⟨hal-00814762⟩