An effective dynamic programming algorithm for the minimum-cost maximal knapsack packing problem

Given a set of n items with profits and weights and a knapsack capacity C, we study the problem of finding a maximal knapsack packing that minimizes the profit of selected items. We propose for the first time an effective dynamic programming (DP) algorithm which has O(nC) time complexity and O(n+C) space complexity. We demonstrate the equivalence between this problem and the problem of finding a minimal knapsack cover that maximizes the profit of selected items. In an extensive computational study on a large and diverse set of benchmark instances, we demonstrate that the new DP algorithm outperforms the use of a state-of-the-art commercial mixed integer programming (MIP) solver applied to the two best-performing MIP models from the literature.