Commutativity in the Algorithmic Lovász Local Lemma

Abstract

We consider the recent formulation of the algorithmic Lovász Local Lemma [N. Harvey and J. Vondrák, in Proceedings of FOCS, 2015, pp. 1327--1345; D. Achlioptas and F. Iliopoulos, in Proceedings of SODA, 2016, pp. 2024--2038; D. Achlioptas, F. Iliopoulos, and V. Kolmogorov, A Local Lemma for Focused Stochastic Algorithms, arXiv preprint, 2018] for finding objects that avoid “bad features,” or “flaws.” It extends the Moser--Tardos resampling algorithm [R. A. Moser and G. Tardos, J. ACM, 57 (2010), 11] to more general discrete spaces. At each step the method picks a flaw present in the current state and goes to a new state according to some prespecified probability distribution (which depends on the current state and the selected flaw). However, the recent formulation is less flexible than the Moser--Tardos method since it requires a specific flaw selection rule, whereas the algorithm of Moser and Tardos allows an arbitrary rule (and thus can potentially be implemented more efficiently). We formulate a new “commutativity” condition and prove that it is sufficient for an arbitrary rule to work. It also enables an efficient parallelization under an additional assumption. We then show that existing resampling oracles for perfect matchings and permutations do satisfy this condition.