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.

中文翻译：

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通过自动机理论镜头的分布式图问题

图问题的局部性是最小距离 $T$，这样每个节点都可以根据其半径-$T$ 邻域选择自己的解决方案部分。在许多情况下，当且仅当局部性较小时，可以使用分布式或并行算法有效地解决图问题。在这项工作中，我们寻求自动化可解性和局部性的研究：给定图问题 $\Pi$ 的描述，我们想确定 $\Pi$ 是否可解以及 $\Pi$ 的渐近局部性是什么图形大小的函数。换句话说，我们寻求自动合成高效的分布式和并行算法来解决 $\Pi$。我们专注于局部可检查图问题；在这些问题中，如果一个解决方案在所有恒定半径邻域中看起来都可行，那么它就是全局可行的。之前关于此类问题的工作带来的主要是坏消息：与局部性相关的问题通常是不可判定的，即使我们专注于标记路径和循环的情况，确定局部性也是 $\mathsf{PSPACE}$-hard（Balliu 等人., PODC 2019)。对于未标记的路径和循环，以及作为扩展，对于有根树，我们用有效的算法补充了先前的负面结果。我们引入了一种新的自动机理论视角来研究局部可检查图问题。我们将局部可检查问题 $\Pi$ 表示为一元字母表上的非确定性有限自动机 $\mathcal{M}$。我们确定了自动机 $\mathcal{M}$ 的多项式时间可计算属性，它几乎完全捕获了 $\Pi$ 在循环和路径中的可解性和局部性，