Random knapsack in expected polynomial time

In this paper, we present the first average-case analysis proving an expected polynomial running time for an exact algorithm for the 0/1 knapsack problem. In particular, we prove, for various input distributions, that the number of dominating solutions (i.e., Paretooptimal knapsack fillings) to this problem is polynomially bounded in the number of available items. An algorithm by Nemhauser and Ullmann can enumerate these solutions very efficiently so that a polynomial upper bound on the number of dominating solutions implies an algorithm with expected polynomial running time. The random input model underlying our analysis is very general and not restricted to a particular input distribution. We assume adversarial weights and randomly drawn profits (or vice versa). Our analysis covers general probability distributions with finite mean, and, in its most general form, can even handle different probability distributions for the profits of different items. This feature enables us to study ...

• Type: Conference
• Year: 2003
• Where: STOC
• Authors: René Beier, Berthold Vöcking
• Keywords: Algorithms, Paretooptimal Knapsack Fillings, Polynomial Running Time, Polynomial Upper Bound, STOC 2003