Algorithmic Improvements of the Lovász Local Lemma via Cluster Expansion

The Lovász Local Lemma (LLL) is a powerful tool that can be used to prove that an object having none of a set of bad properties exists, using the probabilistic method. In many applications of the LLL it is also desirable to explicitly construct the combinatorial object. Recently it was shown that this is possible using a randomized algorithm in the full asymmetric LLL setting [R. Moser and G. Tardos, 2010]. A strengthening of the LLL for the case of dense local neighborhoods proved in [R. Bissacot et al., 2010] was recently also made constructive in [W. Pegden, 2011]. In another recent work [B. Haupler, B. Saha, A. Srinivasan, 2010], it was proved that the algorithm of Moser and Tardos is still efficient even when the number of events is exponential. Here we prove that these last two contributions can be combined to yield a new version of the LLL. 1998 ACM Subject Classification G.2.1 Combinatorics; G.3 Probabilistic Algorithms