A Branch-and-Price Algorithm for the Multiple Knapsack Problem

The multiple knapsack problem is a well-studied combinatorial optimization problem with several practical and theoretical applications. It consists of packing some subset of n items into m knapsacks such that the total profit of the chosen items is maximum. A new formulation of the problem is presented, where a Lagrangian relaxation is derived, and we prove that it dominates the commonly used relaxations for this problem. We also present a Dantzig-Wolfe decomposition of the new formulation that we solve to optimality using a branch-and-price algorithm, where its main advantage comes from the fact that it is possible to control whether an item is included in some knapsack or not. An improved algorithm for solving the resulting packing subproblems is also introduced. Computational experiments then show that the new approach achieves state-of-the-art results. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the Canadian Natural Sciences and Engineering Research Council (NSERC) [Grants 2017-06054 and 2021-04037]. This support is gratefully acknowledged.