Lower Bounds for the Average and Smoothed Number of Pareto Optima

TL;DR

This paper establishes new lower bounds on the expected number of Pareto optima in multiobjective 0-1 linear optimization, using geometric and probabilistic techniques, with implications for various combinatorial problems.

Contribution

It provides the first known lower bounds for natural multiobjective problems and introduces a flexible method for deriving such bounds under general distribution conditions.

Findings

Lower bound of (n^{d-1}) for Pareto optima in d-objective problems

First lower bound for natural multiobjective optimization problems like maximum spanning tree

Smoothed lower bound for 0-1 knapsack problem with d profits and (n^{d-1.5}^{(d- ext{log} d)(1- heta(1/\u03c6))})

Abstract

Smoothed analysis of multiobjective 0-1 linear optimization has drawn considerable attention recently. The number of Pareto-optimal solutions (i.e., solutions with the property that no other solution is at least as good in all the coordinates and better in at least one) for multiobjective optimization problems is the central object of study. In this paper, we prove several lower bounds for the expected number of Pareto optima. Our basic result is a lower bound of \Omega_d(n^(d-1)) for optimization problems with d objectives and n variables under fairly general conditions on the distributions of the linear objectives. Our proof relates the problem of lower bounding the number of Pareto optima to results in geometry connected to arrangements of hyperplanes. We use our basic result to derive (1) To our knowledge, the first lower bound for natural multiobjective optimization problems. We illustrate this for the maximum spanning tree problem with randomly chosen edge weights. Our technique is sufficiently flexible to yield such lower bounds for other standard objective functions studied in this setting (such as, multiobjective shortest path, TSP tour, matching). (2) Smoothed lower bound of min {\Omega_d(n^(d-1.5) \phi^{(d-log d) (1-\Theta(1/\phi))}), 2^{\Theta(n)}}$ for the 0-1 knapsack problem with d profits for phi-semirandom distributions for a version of the knapsack problem. This improves the recent lower bound of Brunsch and Roeglin.