The Multidimensional Knapsack Problem: Structure and Algorithms

TLDR

The authors investigate the multidimensional knapsack problem, presenting theoretical insights and evaluating ILP‑based, metaheuristic, and collaborative solution approaches. They analyze LP relaxation gaps, introduce a novel core concept, and develop ILP‑based and memetic algorithms, benchmarking them against leading methods on standard MKP instances. Extensive experiments demonstrate that the proposed methods achieve highly competitive results in markedly shorter run times than previously described approaches.

Abstract

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.