On the Locality of the Lov\'asz Local Lemma

TL;DR

This paper introduces a novel local analysis of Lovász Local Lemma algorithms, enabling more efficient parallel and distributed algorithms by determining variable assignments from small neighborhoods.

Contribution

The paper provides a new analysis technique for resampling algorithms, improving complexity bounds in parallel and distributed models for the constructive Lovász Local Lemma.

Findings

Achieves $O( ext{log log}_{1/p} n)$ local complexity in the LOCAL model.

Develops algorithms with $d^{O( ext{log log}_{1/p} n)}$ probes in LCA and VOLUME models.

Introduces an $O( ext{log log log}_{1/p} n)$ round algorithm for CONGESTED CLIQUE and MPC models.

Abstract

The Lov\'asz Local Lemma is a versatile result in probability theory, characterizing circumstances in which a collection of $n$ `bad events', each occurring with probability at most $p$ and dependent on a set of underlying random variables, can be avoided. It is a central tool of the probabilistic method, since it can be used to show that combinatorial objects satisfying some desirable properties must exist. While the original proof was existential, subsequent work has shown algorithms for the Lov\'asz Local Lemma: that is, in circumstances in which the lemma proves the existence of some object, these algorithms can constructively find such an object. One main strand of these algorithms, which began with Moser and Tardos's well-known result (JACM 2010), involves iteratively resampling the dependent variables of satisfied bad events until none remain satisfied. In this paper, we present a novel analysis that can be applied to resampling-style Lov\'asz Local Lemma algorithms. This analysis shows that an output assignment for the dependent variables of most events can be determined only from $O(\log \log_{1/p} n)$-radius local neighborhoods, and that the events whose variables may still require resampling can be identified from these neighborhoods. This allows us to improve randomized complexities for the constructive Lov\'asz Local Lemma (with polynomial criterion) in several parallel and distributed models. In particular, we obtain: 1) A LOCAL algorithm with $O(\log\log_{1/p} n)$ node-averaged complexity (while matching the $O(\log_{1/p} n)$ worst-case complexity of Chung, Pettie, and Su). 2) An algorithm for the LCA and VOLUME models requiring $d^{O(\log\log_{1/p} n)}$ probes per query. 3) An $O(\log\log\log_{1/p} n)$-round algorithm for CONGESTED CLIQUE, linear space MPC, and Heterogenous MPC.