Fixed Set Search Matheuristic Applied to the Knapsack Problem with Forfeits

This paper focuses on solving the knapsack problem with forfeits (KPF). This variation of the knapsack problem includes soft conflicts or forfeits, where forfeit pairs consist of two items and an associated penalty. The proposed method combines the fixed set search (FSS) metaheuristic's learning mechanism with integer programming to solve subproblems. A new ground set of elements for the KPF is introduced to maximize the information provided by the fixed set, and the method for creating fixed sets is adjusted to increase the diversity of solutions. The conducted computational experiments show that the proposed method significantly outperforms current state-of-the-art methods and finds a large number of best-known solutions for the standard test instances. Moreover, the method performs well across a wide range of parameter values. The proposed approach does not significantly exploit any specific properties of the KPF and could potentially be applied to other 0-1 problems, such as the minimum vertex cover problem or the facility location problem, without significant modifications. Additionally, the method's simplicity makes it suitable for hybridization with other metaheuristic approaches or the incorporation of specific KPF properties to improve its performance