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I need to provide one simple evidence that graph isomorphism (GI) is not NP-complete.

I saw a number of papers on google scholar and answers on StackExchange. However, I have very limited knowledge of graph isomorphism, and I would like to just provide one simple evidence which I both understand and can explain clearly.

Below is what I can think of. Is this a valid argument?

Currently, we can solve GI in Quasipolynomial Time. If GI is NP-complete, then we should be able to solve other NP-complete problems in Quasipolynomial Time as well. However, right now, we are unable to solve any NP-complete problem in Quasipolynomial Time. Therefore, GI cannot be in np-complete.

Discrete lizard
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user60472
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1 Answers1

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The type of argument you are looking for is as follows:

If graph isomorphism were NP-complete, then some widely believed complexity assumption fails.

There are at least two such arguments:

  1. Schöning showed that if graph isomorphism is NP-complete then the polynomial hierarchy collapses to the second level (equivalently, $\Sigma_2^P = \Pi_2^P$).

  2. Babai's quasipolynomial time algorithm implies that if graph isomorphism is NP-complete then the exponential time hypothesis fails.

Whether such arguments are "valid" or not is a sociological question, not a mathematical one.

Yuval Filmus
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