A Simple Algorithm for a Computationally Hard Problem

TL;DR

This paper introduces a simple, fast randomized algorithm for the graph isomorphism problem, achieving near-quadratic time for most graph pairs, with a focus on non-strongly co-det adjacency matrices.

Contribution

It presents a novel randomized algorithm that improves efficiency for a broad class of graph pairs in solving the graph isomorphism problem.

Findings

Achieves O(n^2.373) runtime for most graph pairs.

Effectively distinguishes non-strongly co-det adjacency matrices.

Provides a practical approach for a historically hard problem.

Abstract

Graph isomorphism problem is a known hard problem. In this paper, a novel randomized algorithm is proposed for this problem which is very simple and fast. It solves the graph isomorphism problem with running time O(n^2.373) for any pair of n-vertex graphs whose adjacency matrices are not strongly co-det. Strongly co-det pair of matrices have very special symmetric structure which can be disarranged to be not strongly co-det by manipulating one element of the matrices.