Heuristic and exact fixation-based approaches for the discounted 0-1 knapsack problem

In this paper we consider the discounted 0-1 knapsack problem (DKP), which is an extension of the classical knapsack problem where a set of items is decomposed into groups of three items. At most one item can be chosen from each group and the aim is to maximize the total profit of the selected items while respecting the knapsack capacity constraint. The DKP is a relatively recent problem in the literature. We propose several efficient heuristic ways to solve the DKP using fixation techniques. One of the objectives of this work is to show that heuristic approaches dealing with the DKP can be strongly improved by using both heuristic and exact fixation rules. We propose several greedy algorithms and a new variable neighborhood search (VNS) to solve the DKP, and we evaluate the behavior of these methods on two sets of available instances when they are applied with or without fixation. Experiments show that VNS with fixation globally dominates approaches from the literature. Finally, we consider a dynamic programming-based approach where the fixation is incorporated, leading to a very efficient heuristic providing near optimal solutions and an exact approach to solve all the existing instances in less than 2 seconds.