New bounds for the Moser-Tardos distribution release_i2ljgsqctfck5gngmj2jwsatme

by David G. Harris

Released as a article .

2016  

Abstract

The Lovász Local Lemma (LLL) is a probabilistic principle which has been used to show the existence of structures that have good "local" properties. Nearly all applications in combinatorics have been turned into efficient algorithms. The simplest, variable-based setting of the LLL was covered by the seminal algorithm of Moser & Tardos (2010). This was extended by Harris & Srinivasan (2014) to random permutations, and more recently by Achlioptas & Ilioupoulos (2014) and Harvey & Vondrák (2015) to general probability spaces. One can similarly define for these algorithms an "MT-distribution," which is the distribution at the termination of the Moser-Tardos algorithm. Hauepler et al. (2011) showed bounds on the MT distribution which essentially match the LLL-distribution for the variable-assignment setting; Harris & Srinivasan showed similar results for the permutation setting. In this work, we show new bounds on the MT-distribution which are significantly stronger than the LLL-distribution. In the variable-assignment setting, we show a tighter bound on the probability of a disjunctive event or singleton event. As a consequence, in k-SAT instances with bounded variable occurrence, the MT-distribution satisfies an ϵ-approximate j-wise independence condition asymptotically stronger than the LLL-distribution. We use this to show a nearly tight bound on the minimum implicate size of a CNF boolean formula. Another noteworthy application is constructing independent transversals which avoid a given subset of vertices; this provides a constructive analogue to a result of Rabern (2014). In the permutation LLL setting, we show a new bound which is similar to the cluster-expansion LLL criterion of Bissacot et al. (2011). We illustrate by showing new, stronger bounds on low-weight Latin transversals and partial Latin transversals.
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Date   2016-11-01
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arXiv  1610.09653v2
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