Decision Making Based on Approximate and Smoothed Pareto Curves

We consider bicriteria optimization problems and investigate the relationship between two standard approaches to solving them: (i) computing the Pareto curve and (ii) the so-called decision maker’s approach in which both criteria are combined into a single (usually non-linear) objective function. Previous work by Papadimitriou and Yannakakis showed how to efficiently approximate the Pareto curve for problems like Shortest Path, Spanning Tree, and Perfect Matching. We wish to determine for which classes of combined objective functions the approximate Pareto curve also yields an approximate solution to the decision maker’s problem. We show that an FPTAS for the Pareto curve also gives an FPTAS for the decision maker’s problem if the combined objective function is growth bounded like a quasi-polynomial function. If the objective function, however, shows exponential growth then the decision maker’s problem is NP-hard to approximate within any polynomial factor. In order to bypass these limitations of approximate decision making, we turn our attention to Pareto curves in the probabilistic framework of smoothed analysis. We show that in a smoothed model, we can efficiently generate the (complete and exact) Pareto curve with a small failure probability if there exists an algorithm for generating the Pareto curve whose worst case running time is pseudopolynomial. This way, we can solve the decision maker’s problem w. r. t. any non-decreasing objective function for randomly perturbed instances of, e. g., Shortest Path, Spanning Tree, and Perfect Matching.