Alternating control tree search for knapsack/covering problems

Authors:
Lars Magnus Hvattum;Halvard Arntzen;Arne Løkketangen;Fred Glover
Affiliations:
Department of Industrial Economics and Technology Management, Norwegian University of Science and Technology, Trondheim, Norway;Molde University College, Molde, Norway;Molde University College, Molde, Norway;Leeds School of Business, University of Colorado, Boulder, USA
Venue:
Journal of Heuristics
Year:
2010

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Abstract

The Multidimensional Knapsack/Covering Problem (KCP) is a 0---1 Integer Programming Problem containing both knapsack and weighted covering constraints, subsuming the well-known Multidimensional Knapsack Problem (MKP) and the Generalized (weighted) Covering Problem. We propose an Alternating Control Tree Search (ACT) method for these problems that iteratively transfers control between the following three components: (1) ACT-1, a process that solves an LP relaxation of the current form of the KCP. (2) ACT-2, a method that partitions the variables according to 0, 1, and fractional values to create sub-problems that can be solved with relatively high efficiency. (3) ACT-3, an updating procedure that adjoins inequalities to produce successively more constrained versions of KCP, and in conjunction with the solution processes of ACT-1 and ACT-2, ensures finite convergence to optimality. The ACT method can also be used as a heuristic approach using early termination rules. Computational results show that the ACT-framework successfully enhances the performance of three widely different heuristics for the KCP. Our ACT-method involving scatter search performs better than any other known method on a large set of KCP-instances from the literature. The ACT-based methods are also found to be highly effective on the MKP.