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If there was an algorithm that could solve any instance of the problem in a time that was a polynomial function of the number of vertices, then there would be an algorithm to solve any NP-C problem in polynomial time.

Since no one knows how to do that, no one knows how to write a deterministic algorithm that can solve any instance of clique in polynomial time.

This does not prevent there from being an algorithm that can quickly solve a particular instance of clique.

But it doesn't really make sense to say that a particular problem can be solved in exponential time,

Exponential time and polynomial time describes how the time to solve varies as function of the size of the problem.