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Random Knapsack in Expected Polynomial Time


Beier,  Rene
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

Vöcking,  Berthold
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Beier, R., & Vöcking, B. (2003). Random Knapsack in Expected Polynomial Time. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC-03) (pp. 232-241). New York, USA: ACM.

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 {\em dominating solutions\/} (i.e., Pareto-optimal 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 the effects of correlations between profits and weights. Our analysis confirms and explains practical studies showing that so-called {\em strongly correlated\/} instances are harder to solve than {\em weakly correlated\/} ones.