Correction to: Directed Lovász Local Lemma and Shearer’s Lemma

Lovász Local Lemma

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...

Domatic partitions and the Lovász local lemma

We resolve the problem posed as the main open question in [4]: letting δ(G), ∆(G) and D(G) respectively denote the minimum degree, maximum degree, and domatic number (defined below) of an undirected graph G = (V,E), we show that D(G) ≥ (1−o(1))δ(G)/ ln(∆(G)), where the “o(1)” term goes to zero as ∆(G) → ∞. A dominating set of G is any set S ⊆ V such that for all v ∈ V , either v ∈ S or some nei...

Commutativity in the Algorithmic Lovász Local Lemma

Recently, Achlioptas and Iliopoulos [1] proposed a new “Random Walk” framework for finding objects that avoid “bad features”, or “flaws”. It extends the Moser-Tardos resampling algorithm [12] to more general discrete spaces. At each step the method picks a flaw present in the current state and “resamples” it. However, it is less flexible than the Moser-Tardos method since [1] requires a specifi...