The Graph Isomorphism Problem: Local Certificates for Giant Action

TL;DR

This thesis explains Babai's quasi-polynomial algorithm for graph isomorphism, focusing on local certificates and extending previous explanations, ultimately clarifying the algorithm's complexity and group-theoretic techniques involved.

Contribution

It provides a detailed explanation of Babai's algorithm, especially the local certificates approach, and refines understanding of its complexity and group-theoretic methods.

Findings

Babai's algorithm has complexity exp(C(log n)^3).

Local certificates are crucial for handling cases beyond Luks' method.

Group and combinatorial arguments clarify Babai's approach.

Abstract

This thesis provides an explanation of L\'aszl\'o Babai's quasi-polynomial algorithm for the Graph Isomorphism Problem published in 2015 with a particular focus on the case of local certificates, i.e. the case that cannot be dealt with by Luks' method. The thesis extends the explanations provided by Harald Andr\'es Helfgott in 2017. It is concluded that the complexity of Babai's algorithm is $\exp\left(C \left(\log n\right)^3\right)$ for $n$ the number of vertices, $C$ a constant. Group theoretical and combinatorial arguments are used to give more details on Babai's method of local certificates. They treat Luks' barrier case in which the imprimitve permutation group $G$ can be mapped onto an alternating group with large domain.