New bounds for the Moser-Tardos distribution: Beyond the Lovasz Local Lemma

The Lovász Local Lemma (LLL) is a probabilistic principle which has been used in a variety of combinatorial constructions to show the existence of structures that have good “local” properties. In many cases, one wants more information about these structures, other than that they exist. In such case, using the “LLL-distribution”, one can show that the resulting combinatorial structures have good global properties in expectation. While the LLL in its classical form is a general existential statement about probability spaces, 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 those known to hold for the LLL-distribution. In the variable-assignment setting, we show a tighter bound on the probability of a disjunctive event. As a consequence, in k-SAT instances with bounded variable occurrence, the MT-distribution satisfies an -approximate jwise 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 type of bound which is similar to the clusterexpansion LLL criterion of Bissacot et al. (2011), but is stronger and takes advantage of the extra structure in permutations. We illustrate by showing new, stronger bounds on low-weight Latin transversals and partial Latin transversals. ∗Department of Computer Science, University of Maryland, College Park, MD 20742. Research supported in part by NSF Awards CNS 1010789 and CCF 1422569. Email: [email protected] 0 ar X iv :1 61 0. 09 65 3v 5 [ m at h. C O ] 3 0 Ju n 20 17