Distributed Treewidth Computation and Courcelle's Theorem in the CONGEST Model

TL;DR

This paper extends Courcelle's theorem to the distributed CONGEST model, enabling efficient decision and optimization of MSO-expressible properties in graphs of bounded treewidth, with algorithms for treewidth approximation and separator computation.

Contribution

It introduces the first distributed algorithms for MSO property decision, optimization, and treewidth approximation in the CONGEST model, leveraging low-congestion shortcuts and separator computations.

Findings

Decides MSO properties in or graphs of bounded treewidth in rom diameter D.

Provides a distributed algorithm for treewidth approximation with or width and depth.

Develops an or computing small vertex separators in graphs of bounded treewidth.

Abstract

Algorithmic meta-theorems, stating that graph properties expressible in some particular logic can be decided efficiently in graph classes having some specific structural properties, are now standard in sequential graph algorithms. One of the most classic examples is Courcelle's theorem: all properties expressible in Monadic Second-Order logic (MSO) are decidable in linear time in graphs of bounded treewidth. We provide here a distributed version of Courcelle's theorem, in the standard CONGEST model for distributed computing: For any MSO formula $\varphi$ and any constant $k$, there is a CONGEST algorithm that, given an input communication network $G$ of treewidth at most $k$ and of diameter $D$, decides if $G$ satisfies property $\varphi$ in $\tilde O(D)$ rounds. Simple examples show that the dependency on $D$ is unavoidable. Also, if we drop the assumption of bounded treewidth, deciding MSO properties such as 3-colorability are known to require $\tilde{\Omega}(n^2)$ rounds in the CONGEST model. Our results extend to optimization problems (e.g., computing a maximum size independent set, or a minimum dominating set) and counting (e.g. triangle counting). As usual, the $\tilde{O}$ notation hides polylogarithmic factors in $n$; here it also hides a constant factor depending on $k$ and on the MSO formula $\varphi$. We also give a distributed algorithm producing a linear approximation for treewidth: For any $k$, it decides that the treewidth of the input network $G$ is larger than $k$ or computes a tree decomposition of width $O(k)$ and depth $O(\log n)$, in $\tilde O(k^{O(k)} D)$ rounds in CONGEST. Our algorithms make use of the low-congestion shortcuts framework introduced by Ghaffari and Haeupler [SODA 2016], and our main technical tool is an $\tilde O(k^4 D)$ algorithm for computing $(s,t)$-vertex separators of size at most $k+1$ in graphs of treewidth at most $k$.