The randomized local computation complexity of the Lov\'asz local lemma

TL;DR

This paper establishes the randomized local computation complexity of the Lovász Local Lemma as Θ(log n), provides a speed-up for locally checkable problems, and analyzes the volume complexity of tree coloring.

Contribution

It proves the Θ(log n) complexity for LLL in constant degree graphs, improves the speed-up bounds for locally checkable problems, and determines the volume complexity of tree coloring.

Findings

LCA complexity of LLL is Θ(log n) on constant degree graphs.

Any o(√log n) probe LCA algorithm can be converted to O(log* n) probes deterministically.

Deterministic volume complexity of c-coloring bounded degree trees is Θ(n).

Abstract

The Local Computation Algorithm (LCA) model is a popular model in the field of sublinear-time algorithms that measures the complexity of an algorithm by the number of probes the algorithm makes in the neighborhood of one node to determine that node's output. In this paper we show that the randomized LCA complexity of the Lov\'asz Local Lemma (LLL) on constant degree graphs is $\Theta(\log n)$. The lower bound follows by proving an $\Omega(\log n)$ lower bound for the Sinkless Orientation problem introduced in [Brandt et al. STOC 2016]. This answers a question of [Rosenbaum, Suomela PODC 2020]. Additionally, we show that every randomized LCA algorithm for a locally checkable problem with a probe complexity of $o(\sqrt{\log{n}})$ can be turned into a deterministic LCA algorithm with a probe complexity of $O(\log^* n)$. This improves exponentially upon the currently best known speed-up result from $o(\log \log n)$ to $O(\log^* n)$ implied by the result of [Chang, Pettie FOCS 2017] in the LOCAL model. Finally, we show that for every fixed constant $c \geq 2$, the deterministic VOLUME complexity of $c$-coloring a bounded degree tree is $\Theta(n)$, where the VOLUME model is a close relative of the LCA model that was recently introduced by [Rosenbaum, Suomela PODC 2020].