The Smoothed Number of Pareto-Optimal Solutions in Non-integer Bicriteria Optimization

Pareto-optimal solutions are one of the most important and well-studied solution concepts in multi-objective optimization. Often the enumeration of all Pareto-optimal solutions is used to filter out unreasonable trade-offs between different criteria. While in practice, often only few Pareto-optimal solutions are observed, for almost every problem with at least two objectives there exist instances with an exponential number of Pareto-optimal solutions. To reconcile theory and practice, the number of Pareto-optimal solutions has been analyzed in the framework of smoothed analysis, and it has been shown that the expected value of this number is polynomially bounded for linear integer optimization problems. In this paper we make the first step towards extending the existing results to non-integer optimization problems. Furthermore, we improve the previously known analysis of the smoothed number of Pareto-optimal solutions in bicriteria integer optimization slightly to match its known lower bound.