Random Walks that Find Perfect Objects and the Lov\'{a}sz Local Lemma

TL;DR

This paper introduces an algorithmic approach based on random walks to efficiently find perfect objects under the Lovász Local Lemma, bypassing traditional probabilistic methods and working in unstructured spaces.

Contribution

It provides a new sufficient condition for rapid convergence of random walks to sinks, inspired by Moser's entropic proof of the Lovász Local Lemma, applicable to unstructured state spaces.

Findings

Established a sufficient condition for quick sink reachability.

Demonstrated the method's applicability to unstructured spaces.

Connected entropy bounds with the inevitability of reaching a sink.

Abstract

We give an algorithmic local lemma by establishing a sufficient condition for the uniform random walk on a directed graph to reach a sink quickly. Our work is inspired by Moser's entropic method proof of the Lov\'{a}sz Local Lemma (LLL) for satisfiability and completely bypasses the Probabilistic Method formulation of the LLL. In particular, our method works when the underlying state space is entirely unstructured. Similarly to Moser's argument, the key point is that the inevitability of reaching a sink is established by bounding the entropy of the walk as a function of time.