On the Locality of the Lovász Local Lemma

TL;DR

The paper presents a locality-based analysis of resampling-style Lovász Local Lemma algorithms, showing that the final values of most dependent variables can be determined from $O(\log\log_{1/p} n)$-radius neighborhoods. This enables exponential improvements in randomized complexities across LOCAL, LCA/VOLUME, and parallel models such as CONGESTED CLIQUE and MPC, by identifying and exploiting insecure/risky nodes and shattering the dependency graph. The main technical tool is a refined study of witness trees, including the notions of $R$-possible and $\varepsilon$-narrow trees, which bound the propagation of resampling and bound the region of influence. As a result, the authors achieve a LOCAL node-averaged complexity of $O(\log\log_{1/p} n)$, $d^{O(\log\log_{1/p} n)}$ probes per query in LCA/VOLUME, and $O(\log\log\log_{1/p} n)$ rounds in CONGESTED CLIQUE, linear-space MPC, and heterogeneous MPC, marking substantial improvements over prior results and offering a path toward even tighter locality-based bounds in distributed LLL. The work thus advances practical constructive LLL methods in distributed settings and provides a rich framework for future exploration of asymmetric criteria, derandomization, and targeted LLL applications.

Abstract

The Lovász 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 Lovász 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 Lovász 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 Lovász 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.

Theorems & Definitions (34)

• Theorem 1: Lovász Local Lemma EL74Shearer85
• Lemma 2: Insecure events
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 6
• Definition 7
• Definition 8
• Lemma 9
• proof