Lopsidependency in the Moser-Tardos framework: Beyond the Lopsided Lovasz Local Lemma

The Lopsided Lov\'{a}sz Local Lemma (LLLL) is a powerful probabilistic principle which has been used in a variety of combinatorial constructions. While originally a general statement about probability spaces, it has recently been transformed into a variety of polynomial-time algorithms. The resampling algorithm of Moser & Tardos (2010) is the most well-known example of this. A variety of criteria have been shown for the LLLL; the strongest possible criterion was shown by Shearer, and other criteria which are easier to use computationally have been shown by Bissacot et al (2011), Pegden (2014), Kolipaka & Szegedy (2011), and Kolipaka, Szegedy, Xu (2012). We show a new criterion for the Moser-Tardos algorithm to converge. This criterion is stronger than the LLLL criterion; this is possible because it does not apply in the same generality as the original LLLL; yet, it is strong enough to cover many applications of the LLLL in combinatorics. We show a variety of new bounds and algorithms. A noteworthy application is for $k$-SAT, with bounded occurrences of variables. As shown in Gebauer, Sz\'{a}bo, and Tardos (2011), a $k$-SAT instance in which every variable appears $L \leq \frac{2^{k+1}}{e (k+1)}$ times, is satisfiable. Although this bound is asymptotically tight (in $k$), we improve it to $L \leq \frac{2^{k+1} (1 - 1/k)^k}{k-1} - \frac{2}{k}$ which can be significantly stronger when $k$ is small. We introduce a new parallel algorithm for the LLLL. While Moser & Tardos described a simple parallel algorithm for the Lov\'{a}sz Local Lemma, and described a simple sequential algorithm for a form of the Lopsided Lemma, they were not able to combine the two. Our new algorithm applies in nearly all settings in which the sequential algorithm works this includes settings covered by our new stronger LLLL criterion.