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(n2d φd(d+1)/2). 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((nφ)(d - log d)·(1 - Θ(1/φ))) Pareto optimal solutions in expectation. For the bi-criteria case we present a higher lower bound of Ω(n2 φ1 - Θ(1/φ)), which almost matches the known upper bound of O(n2φ). © 2011 Springer-Verlag.