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I was going through the lecture of prof. Erik Demaine and he said that a problem X is NP-Hard if it is at-least as hard as every problem Y that belongs to NP class.

He further said that if we can prove that there exists a problem that belongs to NP but not to P, then we can absolutely be sure that NP hard problems don't belong to P class.

Here is my question... What happens if P=NP. Thus it mean that all NP-hard problems become polynomially solvable? Will all NP hard problems reduce to polynomial order if P=NP?

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No. A problem is Np-Hard if all NP problems are reducible to an instance of that problem in polynomial time. Some NP-Hard problems cannot be solved in nondeterministic polynomial time, and are not in NP. Then these problems will not be polynomial time solvable regardless of whether or not P=NP.

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The halting problem is NP-hard: given any language $L$ in P accepted by some nondeterministic machine $M$, we can come up with a polytime reduction that on input $x$ constructs a Turing machine which runs $M$ on $x$ and all possible witnesses, halting if $M$ accepts any of them, and going into an infinite loop otherwise.

We know unconditionally that the halting problem is not computable, and so in particular, isn’t in P.

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So just as your Prof. wikipedia says (emphasis mine):

NP-hardness [...]is the defining property of a class of problems that are, informally, "at least as hard as the hardest problems in NP"

(Still Informally) you can think of the ‚at least‘ as:

NP $\leq$ NP-Hard

So you have to distinguish those NP-hard problems for which the equality holds from those which are ,strictly‘ harder. I.e problems that are NP-hard and are also in NP (a.k.a NP-complete) vs problems which are NP-hard but not in NP.

Looking at it this way, it should be realtively easy to reason about your question. Namely, if P=NP:

  • For problems that are NP-hard and are also in NP it will obviously mean they are also in P (since P=NP by assumption)
  • For problems that are NP-hard but not in NP it will mean nothing.

This euler diagram from wikipedia might also help to clear things up

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