Distributed graph problems through an automata-theoretic lens

SIROCCO 2021 · 28th International Colloquium on Structural Information and Communication Complexity, online, June–July 2021 ·

Abstract

The locality of a graph problem is the smallest distance $T$ such that each node can choose its own part of the solution based on its radius-$T$ neighborhood. In many settings, a graph problem can be solved efficiently with a distributed or parallel algorithm if and only if it has a small locality.

In this work we seek to automate the study of solvability and locality: given the description of a graph problem $\Pi$, we would like to determine if $\Pi$ is solvable and what is the asymptotic locality of $\Pi$ as a function of the size of the graph. Put otherwise, we seek to automatically synthesize efficient distributed and parallel algorithms for solving $\Pi$.

We focus on locally checkable graph problems; these are problems in which a solution is globally feasible if it looks feasible in all constant-radius neighborhoods. Prior work on such problems has brought primarily bad news: questions related to locality are undecidable in general, and even if we focus on the case of labeled paths and cycles, determining locality is PSPACE-hard (Balliu et al., PODC 2019).

We complement prior negative results with efficient algorithms for the cases of unlabeled paths and cycles and, as an extension, for rooted trees. We study locally checkable graph problems from an automata-theoretic perspective by representing a locally checkable problem $\Pi$ as a nondeterministic finite automaton $\mathcal{M}$ over a unary alphabet. We identify polynomial-time-computable properties of the automaton $\mathcal{M}$ that near-completely capture the solvability and locality of $\Pi$ in cycles and paths, with the exception of one specific case that is co-NP-complete.

Publication

Tomasz Jurdziński and Stefan Schmid (Eds.): Structural Information and Communication Complexity, 28th International Colloquium, SIROCCO 2021, Wrocław, Poland, June 28 – July 1, 2021, Proceedings, volume 12810 of Lecture Notes in Computer Science, pages 31–49, Springer, Berlin, 2021

ISBN 978-3-030-79527-6

Journal Version

This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author’s copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.