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Probabilistic ANalysis of Discrete . . .

By René Beier

Abstract

We investigate the performance of exact algorithms for hard optimization problems under random inputs. In particular, we prove various structural properties that lead to two general average-case analyses applicable to a large class of optimization problems. In the first part we study the size of the Pareto curve for binary optimization problems with two objective functions. Pareto optimal solutions can be seen as trade-offs between multiple objectives. While in the worst case, the cardinality of the Pareto curve is exponential in the number of variables, we prove polynomial upper bounds for the expected number of Pareto points when at least one objective function is linear and exhibits sufficient randomness. Our analysis covers general probability distributions with finite mean and, in its most general form, can even handle different probability distributions for the coefficients of the objective function. We apply this result to the constrained shortest path problem and to the knapsack problem. There are algorithms for both problems that can enumerate all Pareto optimal solutions very efficiently, so that our polynomial upper bound on the size of the Pareto curve implies that the expected running time of these algorithms is polynomial as well. For example, we obtain a bound of O(n 4) for uniformly random knapsack instances, where n denotes the number of available items. In th

Year: 2004
OAI identifier: oai:CiteSeerX.psu:10.1.1.134.9380
Provided by: CiteSeerX
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