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Comment by not2b | Hacker News Reader

I believe that P != NP. But suppose someone finds a way to provably solve SAT in O(n**1000000), thus proving P = NP. It's polynomial, but still unusable for real problems.

Edit: fixing damage to my post by Hacker News, making people think I don't understand O-notation.

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If SAT is solvable in polynomial time it’s only a matter of time until we have drastically better solutions than Y=1000000. Not only that, every commercial server in the world is optimized for long-running P-time implementations.

Everything in the digital world relies on P =/= NP. If that falls apart, we go back to the drawing board and start over.

> it’s only a matter of time until we have drastically better solutions than Y=1000000

Why? I'm likely missing something (and I'd love to know what!) but it seems plenty possible that some group of problems have a lower bound that's polynomial and still big.

> it seems plenty possible that some group of problems have a lower bound that's polynomial and still big.

Yes, that is a definite fact: https://en.wikipedia.org/wiki/Time_hierarchy_theorem

10000000 isn't that big a number anyway. We may be up against horrendous numbers that are so large that there is no concise way to write them down, even with recursive formulas such as Ackermann's function, or that sort of thing.

O(n*1000000) is linear, polynomial would be O(N^Y), where Y is the degree of polynomial.

linear time complexity problems are a subset of polynomial time complexity problems, if we're being pedantic.

I think we can all agree that being pedantic is part of the foundation that makes HN be HN.

in the case where Y=1 you both agree.

O(n*1000000) is O(n) isn't it?

Yes

Right, but apparently Hacker News changes a * followed by another * into one *, even though several languages use two asterisks as the exponentiation operator.

Typo? Did you mean O(n^1000000) ?

No, I typed two '*' characters, and it appears that this site made one of them go away. I meant n to the millionth power.

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