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This argument is useless since polynomial time is just as intractable as exponential time if the exponent is large. For example, lets say that we found a way to

by Klockan 10y ago

This argument is useless since polynomial time is just as intractable as exponential time if the exponent is large. For example, lets say that we found a way to solve all np problems in n^1000 time, would that change anything? No, there aren't even 2^1000 atoms in the universe so we will only be able to solve it for the case n=1.

JadeNB 10y ago
> This argument is useless since polynomial time is just as intractable as exponential time if the exponent is large. For example, lets say that we found a way to solve all np problems in n^1000 time, would that change anything? No, there aren't even 2^1000 atoms in the universe so we will only be able to solve it for the case n=1. That seems to be an argument for the uselessness of the distinction between P-with-large-exponent and not-P, not for the uselessness of having a reduction of NP to P. Also, it's worth noting that, if one is a theoretical computer scientist, then there is more of interest in the study of computing than what can be done in the universe. (I am a mathematician, and am, as I suspect most of my colleagues, serenely unconcerned with the applicability or non- of my work.)
paulddraper 10y ago
It might be useless for solving NP problems, but that's not the "point" of P=NP per say. P=NP is an intrinsically interesting math problem. And the parent is pointing out that if P=NP, we already have a construction. So (if his opinion was presented accurately), Knuth is wrong about a positive proof being non-constructive. --- I believe there are already known problems whose best solution is O(n^k) for arbitrarily high k. So yeah, naturally not everything will be practically tractable.
qmalzp 10y ago
If the minimum exponent of a problem as natural as 3-SAT ended up being ~1000, that might be even more fascinating than the fact that it's polynomial.
btilly 10y ago
Your argument supports the view that P is a useless concept, not that that argument is pointless.
- VLM 10y ago
  Often a "proof" that P does not equal NP is that it would be terribly inconvenient if solvable in n to the second or whatever, and therefore in a sort of cosmic censorship saves us by claiming it clearly can't be the case because it would be so annoying. A good argument can be provided by an analysis of statistical thermodynamics gas laws. It highly disturbs some people to think that atoms being in motion implies that all the oxygen atoms in the room air might coincidentally cluster into a small corner of the room leaving the victim to suffocate. There is no reason to think that impossible under most thermodynamics theories. However, it is nearly infinitely unlikely and you'd merely have to wait a ten to the hundred times the lifespan of the universe for it to coincidentally happen. I suppose it scales quite strongly with room volume and linearly with air pressure, you have one atom in the ISS airlock and being in one half the air lock at any given moment is a very predictable 50:50 odds... Likewise if the smallest polynomial solution to a NP problem takes a google's worth of universes to calculate, then even if it is poly you need not worry too much about mere human inconvenience. I'm not saying its a wise way to go thru life, but there exists a large set of people who get hung up on whats convenient or beautiful must be right, even if whats right turns out to be very ugly or inconvenient, so for them "P=NP but your bank account is still secure" is a very big deal.

by Klockan 10y ago

JadeNB 10y ago

> This argument is useless since polynomial time is just as intractable as exponential time if the exponent is large. For example, lets say that we found a way to solve all np problems in n^1000 time, would that change anything? No, there aren't even 2^1000 atoms in the universe so we will only be able to solve it for the case n=1. That seems to be an argument for the uselessness of the distinction between P-with-large-exponent and not-P, not for the uselessness of having a reduction of NP to P. Also, it's worth noting that, if one is a theoretical computer scientist, then there is more of interest in the study of computing than what can be done in the universe. (I am a mathematician, and am, as I suspect most of my colleagues, serenely unconcerned with the applicability or non- of my work.)

paulddraper 10y ago

It might be useless for solving NP problems, but that's not the "point" of P=NP per say. P=NP is an intrinsically interesting math problem. And the parent is pointing out that if P=NP, we already have a construction. So (if his opinion was presented accurately), Knuth is wrong about a positive proof being non-constructive. --- I believe there are already known problems whose best solution is O(n^k) for arbitrarily high k. So yeah, naturally not everything will be practically tractable.

qmalzp 10y ago

If the minimum exponent of a problem as natural as 3-SAT ended up being ~1000, that might be even more fascinating than the fact that it's polynomial.

btilly 10y ago

Your argument supports the view that P is a useless concept, not that that argument is pointless.

VLM 10y ago
Often a "proof" that P does not equal NP is that it would be terribly inconvenient if solvable in n to the second or whatever, and therefore in a sort of cosmic censorship saves us by claiming it clearly can't be the case because it would be so annoying. A good argument can be provided by an analysis of statistical thermodynamics gas laws. It highly disturbs some people to think that atoms being in motion implies that all the oxygen atoms in the room air might coincidentally cluster into a small corner of the room leaving the victim to suffocate. There is no reason to think that impossible under most thermodynamics theories. However, it is nearly infinitely unlikely and you'd merely have to wait a ten to the hundred times the lifespan of the universe for it to coincidentally happen. I suppose it scales quite strongly with room volume and linearly with air pressure, you have one atom in the ISS airlock and being in one half the air lock at any given moment is a very predictable 50:50 odds... Likewise if the smallest polynomial solution to a NP problem takes a google's worth of universes to calculate, then even if it is poly you need not worry too much about mere human inconvenience. I'm not saying its a wise way to go thru life, but there exists a large set of people who get hung up on whats convenient or beautiful must be right, even if whats right turns out to be very ugly or inconvenient, so for them "P=NP but your bank account is still secure" is a very big deal.