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The Multidimensional Knapsack Problem: Structure and Algorithms

NICTA Victoria Research Laboratory, Department of Computer Science and Software Engineering, University of Melbourne, Melbourne, Victoria, 3010 Australia
Institute of Computer Graphics and Algorithms, Vienna University of Technology, A-1040 Vienna, Austria
Department of Statistics and Operations Research, University of Graz, A-8010 Graz, Austria
jakobp{at}csse.unimelb.edu.au
raidl{at}ads.tuwien.ac.at
pferschy{at}uni-graz.at

We study the multidimensional knapsack problem, present some theoretical and empirical results about its structure, and evaluate different integer linear programming (ILP)-based, metaheuristic, and collaborative approaches for it. We start by considering the distances between optimal solutions to the LP relaxation and the original problem and then introduce a new core concept for the multidimensional knapsack problem (MKP), which we study extensively. The empirical analysis is then used to develop new concepts for solving the MKP using ILP-based and memetic algorithms. Different collaborative combinations of the presented methods are discussed and evaluated. Further computational experiments with longer run times are also performed to compare the solutions of our approaches to the best-known solutions of another so-far leading approach for common MKP benchmark instances. The extensive computational experiments show the effectiveness of the proposed methods, which yield highly competitive results in significantly shorter run times than do previously described approaches.

Key words: multidimensional knapsack problem; integer linear programming; heuristics
History: received March 2007; revised July 2008; accepted April 2009.