Where are the hard knapsack problems?

The knapsack problem is believed to be one of the “easier” N P -hard problems. Not only can it be solved in pseudo-polynomial time, but also decades of algorithmic improvements have made it possible to solve nearly all standard instances from the literature. The purpose of this paper is to give an overview of all recent exact solution approaches, and to show that the knapsack problem still is hard to solve for these algorithms for a variety of new test problems. These problems are constructed either by using standard benchmark instances with larger coefficients, or by introducing new classes of instances for which most upper bounds perform badly. The first group of problems challenge the dynamic programming algorithms while the other group of problems are focused towards branch-and-bound algorithms. Numerous computational experiments with all recent state-of-art codes are used to show that the (KP) is still difficult to solve for a wide number of problems. One could say that the previous benchmark tests were limited to a few highly structured instances, which do not show the full characteristics of knapsack problems.