Algorithms for the bounded set-up knapsack problem

The Bounded Set-up Knapsack Problem (BSKP) is a generalization of the Bounded Knapsack Problem (BKP), where each item type has a set-up weight and a set-up value that are included in the knapsack and the objective function value, respectively, if any copies of that item type are in the knapsack. This paper provides three dynamic programming algorithms that solve BSKP in pseudo-polynomial time and a fully polynomial-time approximation scheme (FPTAS). A key implication from these results is that the dynamic programming algorithms and the FPTAS can also be applied to BKP. One of the dynamic programming algorithms presented solves BKP with the same time and space bounds of the best known dynamic programming algorithm for BKP. Moreover, the FPTAS improves the worst-case time bound for obtaining approximate solutions to BKP as compared to using FPTASs designed for BKP or the 0–1 Knapsack Problem.