Introduction to NP-Completeness of Knapsack Problems

Abstract

The reader may have noticed that for all the considered variants of the knapsack problem, no polynomial time algorithm have been presented which solves the problem to optimality. Indeed all the algorithms described are based on some kind of search and prune methods, which in the worst case may take exponential time. It would be a satisfying result if we somehow could prove it is not possible to find an algorithm which runs in polynomial time, somehow having evidence that the presented methods are “as good as we can do”. However, no proof has been found showing that the considered variants of the knapsack problem cannot be solved to optimality in polynomial time.