Witness Trees in the Moser–Tardos Algorithmic Lovász Local Lemma and Penrose Trees in the Hard-Core Lattice Gas

Abstract

We point out a close connection between the Moser-Tardos algorithmic version of the Lovász Local Lemma, a central tool in probabilistic combinatorics, and the cluster expansion of the hard core lattice gas in statistical mechanics. We show that the notion of witness trees given by Moser and Tardos is essentially coincident with that of Penrose trees in the Cluster expansion scheme of the hard core gas. Such an identification implies that the Moser Tardos algorithm is successful in a polynomial time if the Cluster expansion converges.

Moser-Tardos resampling algorithm, entropy compression method and the subset gas

“…End while. It is also worth to mention that Kolipaka and Szegedy showed in [35] that algorithm Resampling is successful in polynomial time also if Shearer conditions hold (see also [6] for a similar result).…”

Moser–Tardos resampling algorithm, entropy compression method and the subset gas

“…ƒ in an expected total number of steps less than or equal to P e2F e . It is also worth to mention that Kolipaka and Szegedy showed in [33] that the algorithm RESAMPLING is successful in polynomial time also if Shearer conditions hold (see also [6] for a similar result).…”

A local lemma via entropy compression

“…Within this "variable setting" Moser and Tardos proved [17] an algorithmic version of the LLL identical to Theorem 1.2, They also proved that their MT-algorithm finds an evaluation of the variable in V such that none of the bad events of the family A occur in an expected time proportional to the number of events in the family A. This algorithmic version of the LLL proposed by Moser and Tardos has been very recently improved by Pegden [19] by replacing condition (1.1) with condition (1.8) using once again the connection with cluster expansion (see also [2] and [13]). Pegden's result is as follows.…”