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I understand the complexity of the issue in the event that P = NP: in this case, we couldn't prove it "by example" - it's simply impossible to sort through all possible tasks. The proof that, for all NP-complete problems, checking a solution is not easier than finding it, apparently, should be very non-trivial. However, if P =/= NP, is it not possible to prove this by simply providing an example of a problem for which it's always "easier" to check the correctness of the solution than to find the solution itself?

Do we really not have a single example of a problem for which finding a solution would be more difficult than checking it? Are there at least heuristic arguments in favor of the existence of such problems?

Thanks in advance.

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  • $\begingroup$ No we don't have a single example. That would be an extraordinary event in the CS community. $\endgroup$
    – user16034
    Commented Feb 11, 2023 at 14:42
  • $\begingroup$ We know a lot of problems with an easy verification but nothing else than a difficult resolution. The heurisitics are more in favor of non-existence of easy solutions. For instance, nobody has been able to find a better bound than $\Omega(1.3^n)$ for 3SAT, despite deep research. $\endgroup$
    – user16034
    Commented Feb 11, 2023 at 14:51
  • $\begingroup$ You might like this question and this question too. $\endgroup$
    – Juho
    Commented Feb 12, 2023 at 8:32

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If we had a single example where we can prove that finding a solution is more difficult then checking it, then this would immediately yield a proof of P != NP (well, with some assumptions). So, when people say it is hard to prove that P != NP, they are also saying it is hard to come up with even a single example where we can prove that.

Now, we have plenty of examples that are good candidates: where it is easy to check a solution, but it appears to be hard to find a solution. The thing is, we have no proof that it truly is hard to find a solution. Maybe there is some incredibly clever algorithm we just haven't thought up yet. That doesn't seem too likely, but we have no proof there is no such algorithm.

For instance, 3SAT is a good candidate. Or, any problem that is NP-complete is a good candidate.

There are certainly heuristic arguments. For instance, people have tried for decades to find an efficient, polynomial-time algorithm for 3SAT, but have not succeeded yet. This could be viewed as a heuristic that suggests that perhaps no such algorithm exists. But it is not a proof.

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