Distributed graph problems through an automata-theoretic lens

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 $mathsf{PSPACE}$-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 introduce a new automata-theoretic perspective for studying locally checkable graph problems. We represent 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 $mbox{co-$mathsf{NP}$}$-complete.