The smoothed number of Pareto-optimal solutions in bicriteria integer optimization

• Published in: Mathematical programming Vol. 200; no. 1; pp. 319 - 355
• Main Authors: Beier, René, Röglin, Heiko, Rösner, Clemens, Vöcking, Berthold
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2023; Springer; Springer Nature B.V
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Full Length Paper; Heuristic methods; Integers; Knapsack problem; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Optimization; Pareto efficiency; Pareto optimum; Polynomials; Random noise; Shortest-path problems; Theoretical; Smoothed analysis; Bicriteria optimization; Pareto-optimal solutions; Integer optimization
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-022-01885-6

A well-established heuristic approach for solving bicriteria optimization problems is to enumerate the set of Pareto-optimal solutions. The heuristics following this principle are often successful in practice. Their running time, however, depends on the number of enumerated solutions, which is exponential in the worst case. We study bicriteria integer optimization problems in the model of smoothed analysis, in which inputs are subject to a small amount of random noise, and we prove an almost tight polynomial bound on the expected number of Pareto-optimal solutions. Our results give rise to tight polynomial bounds for the expected running time of the Nemhauser-Ullmann algorithm for the knapsack problem and they improve known results on the running times of heuristics for the bounded knapsack problem and the bicriteria shortest path problem.