np-complete


Is it possible to find the probability to a solution of NP-complete problems?


The title covers the entirety of the question. Is it possible to derive a function to say with certainty that a proposed solution to a NP-complete problem has a m percent chance of being correct?
I doubt it. For example a random function's probability to be optimal is numberOfOptimalSolutions / searchSpace. So if you know the answer to that, you can deduce the numberOfOptimalSolutions (which is IIRC always harder to know than finding 1 optimal solution to an NP complete problem).
The book "In pursuit of the traveling salesman" lists for each construction heuristic in TSP how close (or far) it is in the worst case scenario from the optimal solution. It also mentions the average percentage of being correct for (some of) those algo's, but IIRC it's probably based on sampling and it gets worse as the problem scales out.

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