Lower Bounds for the Smoothed Number of Pareto optimal Solutions

TL;DR

This paper establishes new lower bounds on the number of Pareto optimal solutions in smoothed multi-criteria optimization, showing that the bounds grow polynomially with problem size and density, and nearly match known upper bounds.

Contribution

It constructs instances demonstrating near-matching lower bounds for the smoothed number of Pareto solutions, addressing an open question about the bounds' dependence on the number of objectives.

Findings

Lower bounds grow polynomially with problem size and density.

Bi-criteria case lower bound nearly matches known upper bound.

Constructs instances with a large number of Pareto optimal solutions.

Abstract

In 2009, Roeglin 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 $\phi$ 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 $O(n^{2d} \phi^{d(d+1)/2})$. 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 $\Omega ((n \phi)^{(d-\log{d}) \cdot (1-\Theta{1/\phi})})$ Pareto optimal solutions in expectation. For the bi-criteria case we present a higher lower bound of $\Omega (n^2 \phi^{1 - \Theta{1/\phi}})$, which almost matches the known upper bound of $O(n^2 \phi)$.