Run Time Analysis for Random Local Search on Generalized Majority Functions

TL;DR

This paper analyzes how neutrality affects the run time of random local search on generalized majority functions within the W-model framework, providing theoretical bounds and new analytical tools.

Contribution

It introduces a theoretical analysis of neutrality's impact on run time, offering upper bounds and generalized drift and Wald's theorems for the W-model.

Findings

Upper bounds for run time on MAJORITY functions.

A theorem linking MAJORITY and HASMAJORITY run times.

Generalized drift theorems and Wald's equation.

Abstract

Run time analysis of evolutionary algorithms recently makes significant progress in linking algorithm performance to algorithm parameters. However, settings that study the impact of problem parameters are rare. The recently proposed W-model provides a good framework for such analyses, generating pseudo-Boolean optimization problems with tunable properties. We initiate theoretical research of the W-model by studying how one of its properties -- neutrality -- influences the run time of random local search. Neutrality creates plateaus in the search space by first performing a majority vote for subsets of the solution candidate and then evaluating the smaller-dimensional string via a low-level fitness function. We prove upper bounds for the expected run time of random local search on this MAJORITY problem for its entire parameter spectrum. To this end, we provide a theorem, applicable to many optimization algorithms, that links the run time of MAJORITY with its symmetric version HASMAJORITY, where a sufficient majority is needed to optimize the subset. We also introduce a generalized version of classic drift theorems as well as a generalized version of Wald's equation, both of which we believe to be of independent interest.