Revisiting the Graph Isomorphism Problem with Semidefinite Programming

We present a new algorithm for the graph isomorphism problem which solves an equivalent maximum clique formulation via semidefinite programming. Following previous results, we show that graph isomorphism is equivalent to determining whether an auxiliary graph structure has a clique of specific order. We show that the problem of determining whether that graph contains such a clique can be formulated as a semidefinite programming problem, and thus can be (almost exactly) solved in polynomial time. Our algorithm provides a significant complexity result in graph isomorphism testing, and also represents the first use of semidefinite programming for solving this problem.