The Complexity Landscape of Distributed Locally Checkable Problems on Trees

TL;DR

This paper investigates the complexity gaps of locally checkable problems on bounded-degree trees, establishing new bounds, a constructive decision algorithm, and hardness results that deepen understanding of distributed problem complexities.

Contribution

It proves the existence of complexity gaps for LCL problems on trees, provides a sequential algorithm to identify problem complexity, and establishes EXPTIME-hardness for distinguishing certain complexities.

Findings

Confirmed gaps between $\Theta(n^{1/(k-1)})$ and $\Theta(n^{1/k})$ for all $k \geq 2$

Developed a sequential algorithm to classify LCL problems into complexity classes

Proved EXPTIME-hardness of distinguishing $\Theta(1)$ and $\Theta(n)$ round complexities

Abstract

Recent research revealed the existence of gaps in the complexity landscape of locally checkable labeling (LCL) problems in the LOCAL model of distributed computing. For example, the deterministic round complexity of any LCL problem on bounded-degree graphs is either $O(\log^\ast n)$ or $\Omega(\log n)$ [Chang, Kopelowitz, and Pettie, FOCS 2016]. The complexity landscape of LCL problems is now quite well-understood, but a few questions remain open. For bounded-degree trees, there is an LCL problem with round complexity $\Theta(n^{1/k})$ for each positive integer $k$ [Chang and Pettie, FOCS 2017]. It is conjectured that no LCL problem has round complexity $o(n^{1/(k-1)})$ and $\omega(n^{1/k})$ on bounded-degree trees. As of now, only the case of $k = 2$ has been proved [Balliu et al., DISC 2018]. In this paper, we show that for LCL problems on bounded-degree trees, there is indeed a gap between $\Theta(n^{1/(k-1)})$ and $\Theta(n^{1/k})$ for each $k \geq 2$. Our proof is constructive in the sense that it offers a sequential algorithm that decides which side of the gap a given LCL problem belongs to. We also show that it is EXPTIME-hard to distinguish between $\Theta(1)$-round and $\Theta(n)$-round LCL problems on bounded-degree trees. This improves upon a previous PSPACE-hardness result [Balliu et al., PODC 2019].