Smoothed analysis for graph isomorphism

TL;DR

The paper advances graph isomorphism testing by combining color refinement with smoothed analysis, showing that tiny random perturbations render naïve refinement effective whp. It extends these ideas to all densities by analyzing $G(n,p)$ and the 2-WL framework, delivering polynomial-time canonical labelling for most perturbations and many sparse regimes, and providing a refined automorphism description of typical random graphs. The approach hinges on views/universal covers, core/ kernel structure, disparity graphs, and multi-round sprinkling, plus a robust use of the 2-dimensional WL algorithm to propagate distinguishing information. The results bridge combinatorial refinement techniques with probabilistic graph models, yielding practical canonical labelling guarantees and contributing to the logical understanding of random graph structure via Weisfeiler–Leman dimensions. Overall, the work significantly strengthens the practical and theoretical understanding of graph isomorphism under perturbations and across densities, with potential implications for network alignment and related combinatorial algorithms.

Abstract

There is no known polynomial-time algorithm for graph isomorphism testing, but elementary combinatorial "refinement" algorithms seem to be very efficient in practice. Some philosophical justification is provided by a classical theorem of Babai, Erdős and Selkow: an extremely simple polynomial-time combinatorial algorithm (variously known as "naïve refinement", "naïve vertex classification", "colour refinement" or the "1-dimensional Weisfeiler-Leman algorithm") yields a so-called canonical labelling scheme for "almost all graphs". More precisely, for a typical outcome of a random graph $G(n,1/2)$, this simple combinatorial algorithm assigns labels to vertices in a way that easily permits isomorphism-testing against any other graph. We improve the Babai-Erdős-Selkow theorem in two directions. First, we consider randomly perturbed graphs, in accordance with the smoothed analysis philosophy of Spielman and Teng: for any graph $G$, naïve refinement becomes effective after a tiny random perturbation to $G$ (specifically, the addition and removal of $O(n\log n)$ random edges). Actually, with a twist on naïve refinement, we show that $O(n)$ random additions and removals suffice. These results significantly improve on previous work of Gaudio-Rácz-Sridhar, and are in certain senses best-possible. Second, we complete a long line of research on canonical labelling of random graphs: for any $p$ (possibly depending on $n$), we prove that a random graph $G(n,p)$ can typically be canonically labelled in polynomial time. This is most interesting in the extremely sparse regime where $p$ has order of magnitude $c/n$; denser regimes were previously handled by Bollobás, Czajka-Pandurangan, and Linial-Mosheiff. Our proof also provides a description of the automorphism group of a typical outcome of $G(n,p_n)$ (slightly correcting a prediction of Linial-Mosheiff).

Figures (2)

• Figure 1: An illustration of the colour refinement algorithm on the 4-edge path.
• Figure 2: An example graph $G$, and the depth-$3$ views for $u_{5}$ and $u_{2}$. As these trees are non-isomorphic, one can deduce that $u_{2}$ and $u_{5}$ are assigned different colours after three steps of the colour refinement algorithm (in fact, they already receive different colours after the first step). In the figure we show the multisets $\mathcal{L}^i(u_2,u_5)$, $\mathcal{L}^i(u_5,u_2)$ and the sets $\mathcal{S}^{i}(\{u_{2},u_{5}\})$ for $i\in \{0,1,2,3\}$.

Theorems & Definitions (125)

• Theorem 1.1
• Remark
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Definition 1.7
• Theorem 1.8
• Remark 1.9