Canonizing Graphs of Bounded Rank-Width in Parallel via Weisfeiler--Leman
Michael Levet, Puck Rombach, Nicholas Sieger
TL;DR
The paper addresses the problem of graph canonization for graphs of bounded rank-width by developing a parallel algorithm that runs in TC^2 time. It combines the Weisfeiler--Leman framework with a rank-decomposition-based recursion, aided by split-pairs and flip extensions, to identify and canonically label graphs in parallel while controlling recursion depth via the $O( ext{log} n)$ height. A key technical achievement is showing that $(6k+3)$-WL identifies rank-width-$k$ graphs in $O( ext{log} n)$ rounds, which yields an NC/isomorphism-test route and enables FO+C definability with $6k+4$ variables. The canonical labeling Can$(G)$ is computed by a parallel, divide-and-conquer procedure along the rank decomposition, achieving TC depth $O(( ext{log} n)^2)$ and size $n^{O(16^k)}$, hence TC^2 for fixed $k$. The work also improves the descriptive complexity for graphs of bounded treewidth and raises open questions about the remaining gaps between WL dimension and the depth required for polylogarithmic rounds, as well as potential L- or FPT-strengthening of these results.
Abstract
In this paper, we show that computing canonical labelings of graphs of bounded rank-width is in $\textsf{TC}^{2}$. Our approach builds on the framework of Köbler & Verbitsky (CSR 2008), who established the analogous result for graphs of bounded treewidth. Here, we use the framework of Grohe & Neuen (ACM Trans. Comput. Log., 2023) to enumerate separators via split-pairs and flip functions. In order to control the depth of our circuit, we leverage the fact that any graph of rank-width $k$ admits a rank decomposition of width $\leq 2k$ and height $O(\log n)$ (Courcelle & Kanté, WG 2007). This allows us to utilize an idea from Wagner (CSR 2011) of tracking the depth of the recursion in our computation. Furthermore, after splitting the graph into connected components, it is necessary to decide isomorphism of said components in $\textsf{TC}^{1}$. To this end, we extend the work of Grohe & Neuen (ibid.) to show that the $(6k+3)$-dimensional Weisfeiler--Leman (WL) algorithm can identify graphs of rank-width $k$ using only $O(\log n)$ rounds. As a consequence, we obtain that graphs of bounded rank-width are identified by $\textsf{FO} + \textsf{C}$ formulas with $6k+4$ variables and quantifier depth $O(\log n)$. Prior to this paper, isomorphism testing for graphs of bounded rank-width was not known to be in $\textsf{NC}$.