The question of 'what can be computed locally?' lies at the heart of distributed computing in networks. As established in Naor and Stockmeyer's seminal paper (STOC 1993), this question is undecidable, even for graph problems whose solutions can be checked locally. In this paper, we adopt a novel perspective on the question, by asking for which classes $Π$ of problems, and for which classes $\mathcal{G}$ of graphs, all problems in $Π$ can be solved efficiently in a distributed manner in all graphs of $\mathcal{G}$. This paper focuses on two natural candidates for such an approach, namely the class of problems expressible in first-order logic (FO), because of their intrinsic form of locality thanks to Gaifman's theorem, and the class of graphs with bounded expansion, because they form a large class of graphs encompassing, e.g., planar, bounded-treewidth, and bounded-degree graphs. The starting point of our work is the decade-old open question of Nešetřil and Ossona de Mendez (Distributed Computing 2016) on the distributed complexity of local FO formula on graphs of bounded expansion, in the standard CONGEST model of distributed computing. A formula $\varphi(x)$ is local if the satisfaction of $\varphi(x)$ depends only on the $r$-neighborhood of its free variable $x$, for some fixed $r$. For instance, the formula '$x$ belongs to a triangle' is local. We resolve the open problem positively by showing that, for every local FO formula $\varphi(x)$, and for every graph class $\mathcal{G}$ of bounded expansion, there exists a deterministic algorithm that identifies, for every $n$-vertex graph $G\in \mathcal{G}$, all vertices $v$ of $G$ such that $G\models \varphi(v)$, in $O(\log n)$ rounds. When dropping the locality condition, we show that $O(D+\log n)$ rounds are sufficient for deciding any FO formula $\varphi$ on graphs of bounded expansion.

翻译：“何谓局部可计算？”这一问题是网络分布式计算的核心。正如Naor与Stockmeyer的开创性论文（STOC 1993）所确立的，即使对于解可在局部验证的图问题，该问题也是不可判定的。本文采用全新视角探讨该问题：针对哪些问题类$Π$以及哪些图类$\mathcal{G}$，$Π$中的所有问题均可在$\mathcal{G}$的所有图中以分布式方式高效求解。本文聚焦于该研究路径的两个自然候选对象：一是可用一阶逻辑（FO）表达的问题类（借助Gaifman定理，这类问题具有固有的局部性形式）；二是具有有界扩张度的图类（因其构成了包含平面图、有界树宽图、有界度图等在内的广泛图类）。我们工作的起点是Nešetřil与Ossona de Mendez十年前提出的开放性问题（Distributed Computing 2016），该问题关注标准CONGEST分布式计算模型下，有界扩张度图上局部FO公式的分布式计算复杂度。若公式$\varphi(x)$的满足性仅取决于其自由变量$x$的$r$-邻域（$r$为固定常数），则该公式是局部的。例如，“$x$属于三角形”这一公式即为局部公式。我们通过证明以下结论正面解决了该开放问题：对于任意局部FO公式$\varphi(x)$及任意有界扩张度图类$\mathcal{G}$，存在确定性算法，可在$O(\log n)$轮内，对任意$n$顶点图$G\in \mathcal{G}$，识别所有满足$G\models \varphi(v)$的顶点$v$。若取消局部性条件，我们证明对于有界扩张度图上的任意FO公式$\varphi$，$O(D+\log n)$轮计算即可完成判定。