Connecting Efficient Knapsacks – Experiments with the Equally-Weighted Bi-Criteria Knapsack Problem

There are many applications of the classical knapsack problem in which the weight of the items being considered for the knapsack are identical, e.g., selecting successful applicants for grants, awarding scholarships to students, etc. Often there are multiple criteria for selecting items to be placed in the knapsack. This paper presents two new algorithms for finding efficient bi-criteria knapsacks when selecting from equally-weighted items. The success of these two algorithms depend on unproven characteristics of the set of efficient knapsacks. If there is a path of adjacent swaps between any two efficients knapsack that remains within the set of efficient knapsacks, then the first algorithm will find the complete set of efficient knapsacks. The second algorithm requires an even stronger condition to guarantee finding the complete set of efficient knapsacks, namely that the path of adjacent swaps not only remains within the set of efficient knapsacks, but that the swaps always increase one objective while decreasing the other. We compare the two algorithms with a bi-criteria shortest-path approach (that is known to find the set of efficient knapsacks) and show that the two algorithms produce the same solutions for a large test set of randomised problems. Our results suggest an interesting structure for the set of efficient knapsacks that may be exploited to find faster solution methods.