Acyclic edge coloring through the Lovász Local Lemma

On the Algorithmic Lovász Local Lemma and Acyclic Edge Coloring

The algorithm for Lovász Local Lemma by Moser and Tardos gives a constructive way to prove the existence of combinatorial objects that satisfy a system of constraints. We present an alternative probabilistic analysis of the algorithm that does not involve counting witness-trees. We then apply our approach to Acyclic Edge Coloring to obtain a direct probabilistic proof that a graph with maximum ...

Lovász Local Lemma

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...

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...