Bounds and Algorithms for the Knapsack Problem with Conflict Graph

We study the Knapsack Problem with Conflict Graph (KPCG), an extension of the 0-1 Knapsack Problem, in which a conflict graph describing incompatibilities between items is given. The goal of the KPCG is to select the maximum profit set of compatible items while satisfying the knapsack capacity constraint. We present a Branch-and-Bound approach to derive optimal solutions to the KPCG in short computing times. Alternative dual bound procedures are compared, as well as different branching strategies, and new effective ones are proposed. Extensive computational experiments are reported, showing that the proposed method outperforms an existing approach from the literature and a Mixed Integer Programming formulation tackled through a general purpose solver.