What Can Be Computed Locally Revisited: First-Order Logic on Sparse Graphs in Distributed Computing

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 $\Pi$ of problems, and for which classes $\mathcal{G}$ of graphs, all problems in $\Pi$ 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 Ne\v{s}et\v{r}il 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.