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A Faster Isomorphism Test for Graphs of Small Degree

Martin Grohe, Daniel Neuen, Pascal Schweitzer

TL;DR

This work gives an improved isomorphism test for graphs of small degree: their algorithms runs in time n^O((log d)^c), where n is the number of vertices of the input graphs, d is the maximum degree of theinput graphs, and c is an absolute constant.

Abstract

In a recent breakthrough, Babai (STOC 2016) gave a quasipolynomial time graph isomorphism test. In this work, we give an improved isomorphism test for graphs of small degree: our algorithms runs in time $n^{O((\log d)^{c})}$, where $n$ is the number of vertices of the input graphs, $d$ is the maximum degree of the input graphs, and $c$ is an absolute constant. The best previous isomorphism test for graphs of maximum degree $d$ due to Babai, Kantor and Luks (FOCS 1983) runs in time $n^{O(d/ \log d)}$.

A Faster Isomorphism Test for Graphs of Small Degree

TL;DR

This work gives an improved isomorphism test for graphs of small degree: their algorithms runs in time n^O((log d)^c), where n is the number of vertices of the input graphs, d is the maximum degree of theinput graphs, and c is an absolute constant.

Abstract

In a recent breakthrough, Babai (STOC 2016) gave a quasipolynomial time graph isomorphism test. In this work, we give an improved isomorphism test for graphs of small degree: our algorithms runs in time , where is the number of vertices of the input graphs, is the maximum degree of the input graphs, and is an absolute constant. The best previous isomorphism test for graphs of maximum degree due to Babai, Kantor and Luks (FOCS 1983) runs in time .

Paper Structure

This paper contains 35 sections, 50 theorems, 51 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Outline
  3. Preliminaries
  4. Graphs and other structures
  5. Group Theory
  6. Permutation groups
  7. Algorithms for permutation groups
  8. Groups with restricted composition factors
  9. String Isomorphism and Luks's algorithm
  10. Recursion
  11. The structure of primitive groups in $\mathop{\mathrm{\widehat{\Gamma}}}\nolimits_d$
  12. The O'Nan-Scott Theorem
  13. I. Affine Groups.
  14. II. Almost Simple Groups.
  15. III. Simple Diagonal Action.
  16. ...and 20 more sections

Key Result

Theorem 1.1

The Graph Isomorphism Problem for graphs of maximum degree $d$ can be solved in time $n^{\mathcal{O}((\log d)^c)}$, for an absolute constant $c$.

Figures (1)

  • Figure 4.1: A visualization of $\Gamma$. Here $\Omega=[9]$, $k=3$, and $\mathfrak{B}_0 = \{\Omega\}$, $\mathfrak{B}_1=\{\{1,2,3\}, \{4,5,6\},\{7,8,9\}\}$, $\mathfrak{B}_3=\{\{1\},\ldots,\{9\}\}$. Furthermore, $I = \{1,2\}$, $m_i=3$ and $t_i=2$; we ignore the condition $t_i\le m_i/2$ for illustration purposes.

Theorems & Definitions (91)

  • Theorem 1.1
  • Theorem 2.1: cf. Seress03
  • Lemma 2.2: BCP82, Lemma 2.2
  • Definition 2.3
  • Lemma 2.4: Luks luks82
  • Theorem 2.5: BCP82
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 81 more