# Revisiting the Graph Isomorphism Problem with Semidefinite Programming

## TL;DR

This paper introduces a novel polynomial-time algorithm for the graph isomorphism problem using semidefinite programming, offering an efficient and nearly exact solution to a longstanding computational challenge.

## Contribution

It is the first to formulate the graph isomorphism problem as a semidefinite programming problem and solve it with a rounding technique, advancing the computational approach to this problem.

## Key findings

- Formulates graph isomorphism as a semidefinite programming problem
- Provides a polynomial-time algorithm for clique detection in auxiliary graphs
- Demonstrates near-exact solutions through rounding of SDP results

## Abstract

It is well-known that the graph isomorphism problem can be posed as an equivalent problem of determining whether an auxiliary graph structure contains a clique of specific order. However, the algorithms that have been developed so far for this problem are either not efficient or not exact. In this paper, we present a new algorithm which solves this equivalent formulation via semidefinite programming. Specifically, we show that the problem of determining whether the auxiliary graph contains a clique of specific order can be formulated as a semidefinite programming problem, and can thus be (almost exactly) solved in polynomial time. Furthermore, we show that we can determine if the graph contains such a clique by rounding the optimal solution to the nearest integer. Our algorithm provides a significant complexity result in graph isomorphism testing, and also represents the first use of semidefinite programming for solving this problem.