Lower Bounds for the Smoothed Number of Pareto Optimal Solutions

Abstract

In 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).

A part of this work was done at Maastricht University and was supported by a Veni grant from the Netherlands Organisation for Scientific Research.