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Classification of Graph Isomorphism relative to NP-complete

Determine whether the Graph Isomorphism problem is NP-complete or belongs to another complexity class, and identify efficient algorithms accordingly, including potential quantum approaches if applicable.

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Background

The authors discuss prospects for efficient algorithms on problems whose classification within NP-complete is unknown and that lack known efficient classical solutions. They cite Graph Isomorphism as a canonical example of such a problem and reference current quasi-polynomial-time progress.

Clarifying the exact complexity status of Graph Isomorphism remains a central open question in computational complexity, with implications for both classical and quantum algorithm development.

References

Even though it is not expected that quantum computers, via quantum algorithms, will be able to solve NP-complete problems in a manner that is exact and efficient, there is a possibility of finding efficient algorithms for those problems for which we do not know whether they belong to a class of NP-complete problems and do not have known and efficient classical algorithms, like, for example, the problem of "checking whether two graphs are isomorphic, known as Graph Isomorphism\footnote{The current state of the art is the algorithm by László Babai, for which it is claimed to have a quasi-polynomial time. ".

— On the Theory of Quantum and Towards Practical Computation (2403.09682 - Kudelić, 7 Feb 2024) in Section Few Last Words