A new combinatorial branch-and-bound algorithm for the Knapsack Problem with Conflicts

Abstract

We study the Knapsack Problem with Conflicts, a generalization of the Knapsack Problem in which a set of conflicts specifies pairs of items which cannot be simultaneously selected. In this work, we propose a novel combinatorial branch-and-bound algorithm for this problem based on an n-ary branching scheme. Our algorithm effectively combines different procedures for pruning the branch-and-bound nodes based on different relaxations of the Knapsack Problem with Conflicts. Its main elements of novelty are: (i) the adoption of the branching-and-pruned set branching scheme which, while extensively used in the maximum-clique literature, was never successfully employed for solving the Knapsack Problem with Conflicts; (ii) the adoption of the Multiple-Choice Knapsack Problem for the derivation of upper bounds used for pruning the branch-and-bound tree nodes; and (iii) the design of a new upper bound for the latter problem which can be computed very efficiently. Key to our algorithm is its high pruning potential and the low computational effort that it requires to process each branch-and-bound node. An extensive set of experiments carried out on the benchmark instances typically used in the literature shows that, for edge densities ranging from 0.1 to 0.9, our algorithm is faster by up to two orders of magnitude than the state-of-the-art method and by up to several orders of magnitude than a state-of-the-art mixed-integer linear programming solver.