The local loop 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

Derandomizing the Lovasz Local Lemma more effectively

The famous Lovasz Local Lemma [EL75] is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. Kratochvil et al. applied this technique to prove that a k-CNF in which each variable appears at most 2^k/(ek) times is always satisfiable [KST93]. In a breakthrough paper, Beck found that if we lower the occurrences to O(2^(k/48...

Hypergraph colouring and the Lovász Local Lemma

The Lov~sz Local Lemma yields sufficient conditions for a hypergraph to be 2-colourable, that is, to have a colouring of the points blue or red so that no edge is monochromatic. The method yields a general theorem, which shows for example that, if H is a hypergraph in which each edge contains at least 9 points and each point is contained in at most 11 edges, then H is 2-colourable. Here we see ...

The Lovász Local Lemma and Satisfiability

We consider boolean formulas in conjunctive normal form (CNF). If all clauses are large, it needs many clauses to obtain an unsatisfiable formula; moreover, these clauses have to interleave. We review quantitative results for the amount of interleaving required, many of which rely on the Lovász Local Lemma, a probabilistic lemma with many applications in combinatorics. In positive terms, we are...