Smoothed Analysis for Graph Isomorphism

2024-10-08

Symposium on the Theory of Computing

AI Summary

Graph isomorphism testing lacks a known polynomial-time algorithm, yet combinatorial heuristics—such as color refinement—perform efficiently on most practical instances. This paper initiates the application of smoothed analysis to this problem, systematically studying isomorphism testing and canonical labeling under random perturbations and on Erdős–Rényi random graphs (G(n,p)). Leveraging probabilistic analysis, combinatorial graph theory, and the one-dimensional Weisfeiler–Leman (WL) algorithm, we rigorously prove that, under the (G(n,1/2)) model, naive color refinement produces a unique canonical labeling with probability (1 - o(1)), thereby solving graph isomorphism correctly with high probability in polynomial time. Our key contribution is the first smoothed analysis framework justifying the empirical efficiency of classical heuristic algorithms; moreover, we elevate the success probability of the WL algorithm from constant to asymptotically almost sure—establishing its robustness under realistic input distributions.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

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 for this phenomenon 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.

Problem

• Improving canonical labeling for randomly perturbed graphs via smoothed analysis
• Establishing polynomial-time canonical labeling for sparse random graphs G(n,p)
• Analyzing effectiveness of naive refinement algorithms after minimal random edge modifications

Innovation

• Enhanced naive refinement with O(n) random edge modifications
• Polynomial-time canonical labeling for all random graph densities
• Characterization of automorphism groups in typical sparse random graphs