Lower Bounds for the Smoothed Number of Pareto Optimal Solutions

Abstract

AbstractIn 2009, Röglin and Teng showed that the smoothed number of Pareto optimal solutions of linear multi-criteria optimization problems is polynomially bounded in the number n of variables and the maximum density φ of the semi-random input model for any fixed number of objective functions. Their bound is, however, not very practical because the exponents grow exponentially in the number d + 1 of objective functions. In a recent breakthrough, Moitra and O’Donnell improved this bound significantly to \(O \big ( n^{2d} \phi^{d(d+1)/2}\big )\).An “intriguing problem”, which Moitra and O’Donnell formulate in their paper, is how much further this bound can be improved. The previous lower bounds do not exclude the possibility of a polynomial upper bound whose degree does not depend on d. In this paper we resolve this question by constructing a class of instances with Ω( (n φ)(d − logd) ·(1 − Θ1/φ) ) Pareto optimal solutions in expectation. For the bi-criteria case we present a higher lower bound of Ω( n 2 φ 1 − Θ1/φ ), which almost matches the known upper bound of O( n 2 φ).KeywordsMaximum DensityKnapsack ProblemPareto Optimal SolutionSplit StepSmooth NumberThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.