Run Time Analysis for Random Local Search on Generalized Majority Functions

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 (RLS). 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 RLS on this
$\mathrm{M{\scriptstyle AJORITY}}$
problem for its entire parameter spectrum. To this end, we provide a theorem, applicable to many optimization algorithms, that links the run time of
$\mathrm{M{\scriptstyle AJORITY}}$
with its symmetric version
$\mathrm{H{\scriptstyle AS}M{\scriptstyle AJORITY}}$
, 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.