Computational complexity of the graph isomorphism problem

Determine the precise computational complexity classification of the graph isomorphism problem, which asks whether two finite graphs are isomorphic. Specifically, ascertain whether graph isomorphism admits a polynomial-time algorithm, is NP-complete, or otherwise characterize its exact complexity class to resolve its open status in computational complexity.

Background

The paper discusses graph matching and notes that the quadratic assignment formulation is NP-hard. Within this context, the authors highlight that even the exact graph matching case—graph isomorphism—has unresolved computational complexity status, underscoring a foundational open question in the field.

Despite significant progress, such as Babai’s quasi-polynomial-time algorithm, the exact classification of the problem remains unsettled. This open problem provides broader context for the challenges faced by practical and theoretical graph matching approaches, including those considered in this work.