The unbounded knapsack problem

Abstract

This paper presents a survey of the unbounded knapsack problem. We focus on the techniques for obtaining the optimal solutions, particularly those using the periodic structure of the optimal solutions when the knapsack weight-carrying capacity b is sufficiently large. In addition to reviewing existing algorithms on the subject, the paper also includes two new algorithms, one for finding the onset of the optimal periodic solutions in time O(nw 1), where w1 is the weight of the best item, i.e. the item with the highest value-to-weight ratio, and a second one for finding the optimal solutions when the capacity b is below the critical value where the optimal periodic solution begins. The second algorithm has a worst-case time complexity of O(nw1 v1), where v1 is the value of the best item. © 2009 Springer Berlin Heidelberg.