**From:** Matt Mahoney (*matmahoney@yahoo.com*)

**Date:** Wed Oct 21 2009 - 09:57:49 MDT

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Pavitra wrote:

*> Thus, "there exists at least one NP-complete problem solvable in
*

*> polynomial time" is equivalent to "P = NP", and "there exists at least
*

*> one NP-complete problem not solvable in polynomial time" is equivalent
*

*> to "P != NP".
*

No. "Traveling Salesman" is an NP-complete problem. But suppose I have n cities equally spaced at 1 mile intervals on a circle. Then I know (and can prove in polynomial time) that the shortest path for these instances is n miles.

Other instances may have more subtle shortcuts, but you don't know which ones. A proof of P != NP would only say that not all instances of a problem do.

-- Matt Mahoney, matmahoney@yahoo.com

----- Original Message ----

From: Pavitra <celestialcognition@gmail.com>

To: sl4@sl4.org

Sent: Wed, October 21, 2009 12:00:58 AM

Subject: Re: [sl4] to-do list for strong, nice AI

Matt Mahoney wrote:

*> Pavitra wrote:
*

*>> Just to check: I think you mean "...even if it turns out that P =
*

*>> NP"?
*

*>
*

*> No, I mean P != NP. Suppose it were proven. You would know that some
*

*> instances of, say, SAT or traveling salesman required exponential
*

*> time to solve, but you wouldn't know which ones. There are heuristics
*

*> that can solve lot of NP-complete problems quickly, just not all of
*

*> them. You don't know that any particular instance is hard because
*

*> there might be another heuristic that makes it easy.
*

I thought the definition of NP-complete was that if any single

NP-complete problem is solvable in polynomial time (i.e, is in P), then

any problem in NP is solvable in polynomial time.

Thus, "there exists at least one NP-complete problem solvable in

polynomial time" is equivalent to "P = NP", and "there exists at least

one NP-complete problem not solvable in polynomial time" is equivalent

to "P != NP".

That is, I believe that "there exists at least one NP-complete problem

solvable in polynomial time, and at least one other NP-complete problem

not solvable in polynomial time" has been mathematically disproven.

Am I completely missing the point?

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