Barriers and local minima in energy landscapes of stochastic local search

Abstract

A local search algorithm operating on an instance of a Boolean constraint satisfaction problem (in particular, k-SAT) can be viewed as a stochastic process traversing successive adjacent states in an ``energy landscape'' defined by the problem instance on the n-dimensional Boolean hypercube. We investigate analytically the worst-case topography of such landscapes in the context of satisfiable k-SAT via a random ensemble of satisfiable ``k-regular'' linear equations modulo 2. We show that for each fixed k=3,4,..., the typical k-SAT energy landscape induced by an instance drawn from the ensemble has a set of 2^{Ω(n)} local energy minima, each separated by an unconditional Ω(n) energy barrier from each of the O(1) ground states, that is, solution states with zero energy. The main technical aspect of the analysis is that a random k-regular 0/1 matrix constitutes a strong boundary expander with almost full GF(2)-linear rank, a property which also enables us to prove a 2^{Ω(n)} lower bound for the expected number of steps required by the focused random walk heuristic to solve typical instances drawn from the ensemble. These results paint a grim picture of the worst-case topography of k-SAT for local search, and constitute apparently the first rigorous analysis of the growth of energy barriers in a random ensemble of k-SAT landscapes as the number of variables n is increased.