Approximation speed-up by quadratization on leadingones

Abstract

We investigate the quadratization of LeadingOnes in the context of the landscape for local search. We prove that a standard quadratization (i.e., its expression as a degree-2 multilinear polynomial) of LeadingOnes transforms the search space for local search in such a way that faster progress can be made. In particular, we prove there is a $$\varOmega (n/\log n)$$ speed-up for constant-factor approximations by RLS when using the quadratized version of the function. This suggests that well-known transformations for classical pseudo-Boolean optimization might have an interesting impact on search heuristics. We derive and present numerical results that investigate the difference in correlation structure between the untransformed landscape and its quadratization. Finally, we report experiments that provide a detailed glimpse into the convergence properties on the quadratized function.