Graph isomorphisms in quasi-polynomial time

TL;DR

This paper discusses Babai's breakthrough in graph isomorphism, providing a quasi-polynomial time algorithm that advances the understanding of graph symmetries and automorphisms.

Contribution

It explains Babai's quasi-polynomial time algorithm for graph isomorphism, building on Luks' earlier work on bounded degree graphs, and addresses a long-standing open problem.

Findings

Graph isomorphism can be decided in quasi-polynomial time.

Babai's algorithm generalizes Luks' bounded degree approach.

The method advances the theoretical understanding of graph symmetries.

Abstract

Let us be given two graphs $\Gamma_1$, $\Gamma_2$ of $n$ vertices. Are they isomorphic? If they are, the set of isomorphisms from $\Gamma_1$ to $\Gamma_2$ can be identified with a coset $H\cdot\pi$ inside the symmetric group on $n$ elements. How do we find $\pi$ and a set of generators of $H$? The challenge of giving an always efficient algorithm answering these questions remained open for a long time. Babai has recently shown how to solve these problems -- and others linked to them -- in quasi-polynomial time, i.e. in time $\exp\left(O(\log n)^{O(1)}\right)$. His strategy is based in part on the algorithm by Luks (1980/82), who solved the case of graphs of bounded degree.