The Smoothed Number of Pareto Optimal Solutions in Bicriteria Integer Optimization

Abstract

A well established heuristic approach for solving various bicriteria optimization problems is to enumerate the set of Pareto optimal solutions, typically using some kind of dynamic programming approach. The heuristics following this principle are often successful in practice. Their running time, however, depends on the number of enumerated solutions, which can be exponential in the worst case. In this paper, we prove an almost tight bound on the expected number of Pareto optimal solutions for general bicriteria integer optimization problems in the framework of smoothed analysis. Our analysis is based on a semi-random input model in which an adversary can specify an input which is subsequently slightly perturbed at random, e. g., using a Gaussian or uniform distribution. Our results directly imply tight polynomial bounds on the expected running time of the Nemhauser/Ullmann heuristic for the 0/1 knapsack problem. Furthermore, we can significantly improve the known results on the running time of heuristics for the bounded knapsack problem and for the bicriteria shortest path problem. Finally, our results also enable us to improve and simplify the previously known analysis of the smoothed complexity of integer programming.