Pareto Optimal Solutions for Smoothed Analysts

TL;DR

This paper improves the expected bound on the number of Pareto optimal solutions in smoothed analysis for multi-objective binary optimization, reducing the dependence on the dimension from exponential to polynomial.

Contribution

It introduces a new analysis technique with two algorithms that significantly tightens the bound on the expected number of Pareto optima in smoothed analysis.

Findings

Expected number of Pareto optima is bounded by n^{2d} in smoothed analysis.

New proof method analyzes algorithms to reconstruct solutions from testimonies.

Improves previous bounds with worse dependence on dimension d.

Abstract

Consider an optimization problem with $n$ binary variables and $d+1$ linear objective functions. Each valid solution $x \in \{0,1\}^n$ gives rise to an objective vector in $\R^{d+1}$, and one often wants to enumerate the Pareto optima among them. In the worst case there may be exponentially many Pareto optima; however, it was recently shown that in (a generalization of) the smoothed analysis framework, the expected number is polynomial in $n$. Unfortunately, the bound obtained had a rather bad dependence on $d$; roughly $n^{d^d}$. In this paper we show a significantly improved bound of $n^{2d}$. Our proof is based on analyzing two algorithms. The first algorithm, on input a Pareto optimal $x$, outputs a "testimony" containing clues about $x$'s objective vector, $x$'s coordinates, and the region of space $B$ in which $x$'s objective vector lies. The second algorithm can be regarded as a {\em speculative} execution of the first -- it can uniquely reconstruct $x$ from the testimony's clues and just \emph{some} of the probability space's outcomes. The remainder of the probability space's outcomes are just enough to bound the probability that $x$'s objective vector falls into the region $B$.