A Local Isotopy Lemma

Constructive Lovasz Local Lemma

Let V be a finite set of independent random variables, and let A denote a finite set of events that are determined by V . That is, each event A ∈ A maps the set of assignments of V to {0, 1}. Definition 1. Given the set of independent random variables V and set of events A determined by the variables of V , define the relevant variables for an event A ∈ A, denoted vbl(A) ⊂ V to be the smallest ...

Lovász Local Lemma

A Sharper Local Lemma with Improved Applications

We give a new family of Lovász Local Lemmas (LLL), with applications. Shearer has given the most general condition under which the LLL holds, but the original condition of Lovász is simpler and more practical. Do we have to make a choice between practical and optimal? In this article we present a continuum of LLLs between the original and Shearer’s conditions. One of these, which we call Clique...

A Constructive Lovász Local Lemma for Permutations

While there has been significant progress on algorithmic aspects of the Lovász Local Lemma (LLL) in recent years, a noteworthy exception is when the LLL is used in the context of random permutations. The breakthrough algorithm of Moser & Tardos only works in the setting of independent variables, and does not apply in this context. We resolve this by developing a randomized polynomial-time algor...

Focused Stochastic Local Search and the Lovász Local Lemma

We develop tools for analyzing focused stochastic local search algorithms. These are algorithms which search a state space probabilistically by repeatedly selecting a constraint that is violated in the current state and moving to a random nearby state which, hopefully, addresses the violation without introducing many new ones. A large class of such algorithms arise from the algorithmization of ...