This is an amazing and unexpected result. To be clear though, it does not imply that NP is subexponential: "Note that runtime bounds of the form 2^o(n) do not transfer under general polynomial-time reductions (2^n^o(1) would be needed for this), so our result does not imply that general NP problems
admit subexponential time algorithms.""
Wow. Complexity class separations are really really rare.
Can an expert in sat solving comment if there's any immediate applications of the method from the paper? Or is SAT solving in general pretty far away from worst-case times, and this is a theoretical algorithm which would be hard to apply in practice.
It's surprising that Exponential Time Hypothesis would be false. Quoting a paper from 2022[1]:
> We prove that the time complexity of self-attention is necessarily quadratic in the input length, unless the Strong Exponential Time Hypothesis (SETH) is false[...]
If OpenAI's proof is correct, does that mean self-attention can be implemented in sub-quadratic time? I.e. we could eventually have all the most advanced LLMs running on cheap hardware? To put it more playfully, is ETH being false actually bearish for Nvidia?
If they had instead found that it were true, it'd prove P!=NP (since 3SAT is in NP and proving ETH would imply it can't be solved in polytime)
Instead we now know that NP is a strict subset of EXP (there are problems in NP that can be solved in subexponential time). And in fact they provided a constructive proof that 3SAT (and thus any problem in NP) can be solved in subexponential (but still super-quasipolynomial, let alone polynomial) time.
Edit: Originally wrote the hypothesis being true would imply NP=EXPTIME, which is wrong, since there can be problems in EXPTIME which aren't in NP.
I mean the result that k-SAT cannot be solved in 2^{o(n)} clearly would imply P!=NP, but how does ETH imply NP = EXP? And how does \neg ETH imply NP != EXP?
It doesn't that was my mistake, since edited to remove the part about implying NP = EXP for ETH were true.
But with the hypothesis being false, NP being a strict subset of EXP does imply NP != EXP
This defeats a few hypotheses that were widely relied on.
strong exponential time hypothesis, and the titular exponential time hypothesis
It's not saying that these problems are easy, but it does demonstrate that they are much easier than previously believed. A decade ago if a human had published this kind of thing, and importantly if it held up, they would be almost certainly a strong candidate for some sort of award(s).
Assuming P != NP (and that this NP != EXP proof holds up), what's the next most "natural" candidate for how fast NP must grow? This paper appears to show Omega(2^(n/sqrt(log(n))) with a couple additional fudge factors which would be a very disappointing answer.
This proves that EXP != NP, which was a stronger claim than P != NP.
In a way it shows that NP is perhaps less distant from P than imagined. Whether it can be _reduced_ to P in some gigabrain way, or whether that can be shown to be impossible remains to be seen.
It is definitely progress towards an answer to P ?= NP.
If I am understanding this correctly (and I am still reading this!) and if this is true, this means NP is sub-exponential. Sub-exponential does not mean P=NP. But yes, this is huge if true.
This means hard problems aren't as hard as we thought, and it's still possible P=NP. If it turns out to be true P=NP, we're a little closer to that. (I hope P!=NP!)
Every paper which assumes ETH is true might now have their conclusions be disproven.
See this from the paper: "Note that runtime bounds of the form 2^o(n) do not transfer under general polynomial-time reductions (2^n^o(1) would be needed for this), so our result does not imply that general NP problems
admit subexponential time algorithms."
In computer science, the satisfiability problem is the task of determining whether a boolean problem can be satisfied, that is, made to be true, by some combination of inputs. For example, the expression: A AND NOT B is satisfied whenever A is true and B is false, while the expression: A AND NOT A cannot be satisfied.
A naive approach to satisfiablilty would be to check every possible input and see if the expression evaluates to true.
However, the number of combinations that must be checked is 2n, where n is the number of variables in the expression. This is what is meant by exponential time.
The exponential time hypothesis conjectures that you cannot do better, that is, the number of operations needed to solve satisfiability will always be exponential in the size of the problem, no matter what algorithm is chosen.
If this hypothesis is true, it would imply that P≠NP, because satisfiability has been shown to be in NP, and in order for it to be in P there would have to exist an algorithm that solves it in polynomial time.
The exponential time hypothesis is essentially that NP-complete problems' best solutions are all exponential.
NP-complete problems are essentially those where you need to make some kind of choice over a superlinearly-growing set of options where each option can be solved in linear time. Like, for example, finding a route between cities on a map under a certain total distance. You might need to go over all choices of routes.
The ETH essentially assumes that the best solution (in terms of complexity, i.e. how it grows) you can get for these kinds of problems is exponential. OpenAI just proved that this is false, you can have solutions that are below exponential complexity (but still above polynomial - if anyone finds a polynomial solution then we'd have proven that P=NP)
Is this really as interpetable math is fundamentally. Like wtf is this accessibility? who is this for? Do top mathematicians read this like a chapter book? Absolutely ridiculous they have all this compute and creative resources and cant make something more intuitive to help people understand it more natively
Can an expert in sat solving comment if there's any immediate applications of the method from the paper? Or is SAT solving in general pretty far away from worst-case times, and this is a theoretical algorithm which would be hard to apply in practice.
> We prove that the time complexity of self-attention is necessarily quadratic in the input length, unless the Strong Exponential Time Hypothesis (SETH) is false[...]
If OpenAI's proof is correct, does that mean self-attention can be implemented in sub-quadratic time? I.e. we could eventually have all the most advanced LLMs running on cheap hardware? To put it more playfully, is ETH being false actually bearish for Nvidia?
[1]: https://arxiv.org/abs/2209.04881
Instead we now know that NP is a strict subset of EXP (there are problems in NP that can be solved in subexponential time). And in fact they provided a constructive proof that 3SAT (and thus any problem in NP) can be solved in subexponential (but still super-quasipolynomial, let alone polynomial) time.
Edit: Originally wrote the hypothesis being true would imply NP=EXPTIME, which is wrong, since there can be problems in EXPTIME which aren't in NP.
It doesn't that was my mistake, since edited to remove the part about implying NP = EXP for ETH were true. But with the hypothesis being false, NP being a strict subset of EXP does imply NP != EXP
strong exponential time hypothesis, and the titular exponential time hypothesis
It's not saying that these problems are easy, but it does demonstrate that they are much easier than previously believed. A decade ago if a human had published this kind of thing, and importantly if it held up, they would be almost certainly a strong candidate for some sort of award(s).
In a way it shows that NP is perhaps less distant from P than imagined. Whether it can be _reduced_ to P in some gigabrain way, or whether that can be shown to be impossible remains to be seen.
It is definitely progress towards an answer to P ?= NP.
This means hard problems aren't as hard as we thought, and it's still possible P=NP. If it turns out to be true P=NP, we're a little closer to that. (I hope P!=NP!)
Every paper which assumes ETH is true might now have their conclusions be disproven.
Still, an impressive accomplishment.
NP-complete problems are essentially those where you need to make some kind of choice over a superlinearly-growing set of options where each option can be solved in linear time. Like, for example, finding a route between cities on a map under a certain total distance. You might need to go over all choices of routes.
The ETH essentially assumes that the best solution (in terms of complexity, i.e. how it grows) you can get for these kinds of problems is exponential. OpenAI just proved that this is false, you can have solutions that are below exponential complexity (but still above polynomial - if anyone finds a polynomial solution then we'd have proven that P=NP)
Fwiw never thought it was a well-named conjecture
This proves that EXP != NP