Distributed Model Checking in Graphs Classes of Bounded Expansion

We show that for every first-order logic (FO) formula $\varphi$, and every graph class $\mathcal{G}$ of bounded expansion, there exists a distributed (deterministic) algorithm that, for every $n$-node graph $G\in\mathcal{G}$ of diameter $D$, decides whether $G\models \varphi$ in $O(D+\log n)$ rounds under the standard CONGEST model. Graphs of bounded expansion encompass many classes of sparse graphs such as planar graphs, bounded-treedepth graphs, bounded-treewidth graphs, bounded-degree graphs, and graphs excluding a fixed graph $H$ as a minor or topological minor. Note that our algorithm is optimal up to a logarithmic additional term, as even a simple FO formula such as "there are two vertices of degree 3" already on trees requires $\Omega(D)$ rounds in CONGEST. Our result extends to solving optimization problems expressed in FO (e.g., $k$-vertex cover of minimum weight), as well as to counting the number of solutions of a problem expressible in a fragment of FO (e.g., counting triangles), still running in $O(D+\log n)$ rounds under the CONGEST model. This exemplifies the contrast between sparse graphs and general graphs as far as CONGEST algorithms are concerned. For instance, Drucker, Kuhn, and Oshman [PODC 2014] showed that the problem of deciding whether a general graph contains a 4-cycle requires $\Theta(\sqrt{n}/\log n)$ rounds in CONGEST. For counting triangles, the best known algorithm of Chang, Pettie, and Zhang [SODA 2019] takes $\tilde{O}(\sqrt{n})$ rounds. Finally, our result extends to distributed certification. We show that, for every FO formula~$\varphi$, and every graph class of bounded expansion, there exists a certification scheme for $\varphi$ using certificates on $O(\log n)$ bits. This significantly generalizes the recent result of Feuilloley, Bousquet, and Pierron [PODC 2022], which held solely for graphs of bounded treedepth.