Knapsack problem with objective value gaps

Abstract

International audienceWe study a 0-1 knapsack problem, in which the objective value is forbidden to take some values. We call gaps related forbidden intervals. The problem is NP-hard and pseudo-polynomially solvable independently on the measure of gaps. If the gaps are large, then the problem is polynomially non-approximable. A non-trivial special case with respect to the approximate solution appears when the gaps are small and polynomially close to zero. For this case, two fully polynomial time approximation schemes are proposed. The results can be extended for the constrained longest path problem and other combinatorial problems

Simple paths with exact and forbidden lengths

“…To the best of our knowledge, no literature exists on problems with the forbidden objective values gaps. The only exception is our recent paper (Dolgui, Kovalyov, & Quilliot, 2017) on a 0-1 knapsack problem with the forbidden objective function values. On the other hand, several exact value (or exact cost) combinatorial problems have been studied in the literature, which concern the existence of a solution with a given objective function value.…”