New bounds for the Moser-Tardos distribution
release_i2ljgsqctfck5gngmj2jwsatme
by
David G. Harris
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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