An experimental study of random knapsack problems

The size of the Pareto curve for the bicriteria version of the knapsack problem is polynomial on average. This has been shown for various random input distributions. We experimentally investigate the number of Pareto points for knapsack instances over n elements whose profits and weights are chosen at random according to various classes of input distributions. The numbers observed in our experiments are significantly smaller than the known upper bounds. For example, the upper bound for so-called uniform instances is O(n³). Based on our experiments, we conjecture that the number of Pareto points for these instances is only T(n²). We also study other structural properties for random knapsack instances that have been used in theoretical studies to bound the average-case complexity of the knapsack problem. Furthermore, we study advanced algorithmic techniques for the knapsack problem. In particular, we review several ideas that originate from theory as well as from practice. Most of the concepts that we use are simple and have been known for at least 20 years, but apparently have not been used in this combination. Surprisingly, the result of our study is a very competitive code that outperforms the best previous implementation Combo by orders of magnitude for various classes of random instances, including harder random knapsack instances in which profits and weights are chosen in a correlated fashion.