Reduced costs propagation in an efficient implicit enumeration for the 01 multidimensional knapsack problem

Yannick Vimont (1) , Sylvain Boussier (1) , Michel Vasquez (1)

1 LGI2P - Laboratoire de Génie Informatique et Ingénierie de Production

In a previous work we proposed a variable fixing heuristics for the 0-1 Multidimensional knapsack problem (01MDK). This approach uses fractional optima calculated in hyperplanes which contain the binary optimum. This algorithm obtained best lower bounds on the OR-Library benchmarks. Although it is very attractive in terms of results, this method does not prove the optimality of the solutions found and may fix variables to a non-optimal value. In this paper, we propose an implicit enumeration based on a reduced costs analysis which tends to fix non-basic variables to their exact values. The combination of two specific constraint propagations based on reduced costs and an efficient enumeration framework enable us to fix variables on the one hand and to prune significantly the search tree on the other hand. Experimentally, our work provides two main contributions: (1) we obtain several new optimal solutions on hard instances of the OR-Library and (2) we reduce the bounds of the number of items at the optimum on several harder instances.

Mots clés: Multidimensional knapsack problem; Implicit enumeration; Variable fixing; Reduced costs; Constraint propagation

Domaines: Recherche opérationnelle [math.OC]

Yannick Vimont, Sylvain Boussier, Michel Vasquez. Reduced costs propagation in an efficient implicit enumeration for the 01 multidimensional knapsack problem. Journal of Combinatorial Optimization, 2008, 15 (2), pp.165-178. ⟨10.1007/s10878-007-9074-4⟩. ⟨hal-00353906⟩