alt Hacker NewsWhile this has no immediate bearing on P-vs-NP, it is interesting to note:
- Early CS researchers like Donald Knuth entered CS in the days when regularly someone would come up with a surprising faster algorithm for some problem, and their intuitions were that most problems had faster algorithms than you'd expect.
- Later generations of CS researchers, as all the easy algorithms were discovered, and post Karp's definition of NP-completeness (and the Garey & Johnson book, say), began to specialize in proving hardness, and complexity theory with a “zoo” of complexity classes became more about separation and lower bounds, e.g. I've seen algorithms like Arora's PTAS for Euclidean TSP being described as a “failed attempt to prove a lower bound”. So an entire generation of researchers' intuition was mostly in the direction of an elegant universe with neat separation results (even if we couldn't get them).
- So e.g. surveys of computer scientists showed most of them believing that P≠NP, and someone like Knuth has always seemed a bit of an outlier for saying that maybe P=NP: his argument was simply that there are so many possible polynomial-time algorithms that we haven't explored, and maybe one of them with a large exponent could solve NP-complete problems. (In response to arguments that many researchers have failed to find any such algorithms, he would say that many researchers have failed to prove that P≠NP.) E.g. in 2017 when Scott Aaronson wrote his survey (https://www.scottaaronson.com/papers/pnp.pdf = https://www.scottaaronson.com/papers/pnp-kindle.pdf) he wrote:
> Admittedly, it can be hard to tell whether declarations that P = NP are meant seriously, or are merely attempts to be contrarian. However, we can surely agree with Knuth and Lipton that we’re far from understanding the limits of efficient computation, and that there are further surprises in store.
- What we've seen in this month is surprising unexpected algorithms, like o(n log n) algorithms for FFT and integer multiplication, and now this sub–exponential-time algorithm for 3-SAT. (Yes the ETH has had less support than P≠NP but I still gather it's unexpected.)
So (though I don't expect intuitions on P≠NP to change, and probably that's still the right intuition… but I'm just an interested amateur), maybe there will be a bit of a swing of the pendulum back to the idea that maybe clever algorithms exist that we just haven't thought of.
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Part of the problem with the current centralization of the world's math capability within AI labs is that if they do find that P=NP via an efficient algorithm, it could have implications on all modern public key based cryptography. So it's not completely clear that they will (be allowed to) release these results.
We had O(n lg n) since 2019 already; now we (supposedly) know the limit is less than that (but of course at least linear).
> In 2019, Harvey and van der Hoeven came up with a galactic algorithm with complexity O ( n log n ) {\displaystyle O(n\log n)}. This matches a guess by Schönhage and Strassen that this would be the optimal bound.
> In September 2026, OpenAI released a preprint claiming an algorithm with complexity O ( n ( log n ) α ) {\displaystyle O(n(\log n)^{\alpha })} with α < 1 {\displaystyle \alpha <1}
https://en.wikipedia.org/wiki/Multiplication_algorithm
That said, it's interesting to contemplate that there is "space between" polynomial and exponential time complexities.
I believe this stems from an incomplete understanding of the problem space on my part (excuse the multiple puns); but, I am skeptical Knuth really said P=NP. There are so many mathemtical examples of equations that clearly show P≠NP. Computing in itself was originally formed from using iterative methods to solve hard problems. Least-squares methods, sin() cos(), traveling salesman, so many to list? To say P=NP for all problems seems like a fools errand to me.
There are hard problems where the worst case may be exponentially hard, but the average case is much faster. Linear programming and the traveling salesman problem are like that. The faster approaches are considered "probabilistic solutions".
In cryptography, most of the problems believed to be exponentially hard, such as factoring, don't have proofs of an exponential lower bound. There's some risk of a breakthrough that makes those problems easier. A probabilistic solution would be good enough to break many cryptosystems.
I suspect that people at the cryptanalysis agencies worldwide are staying up late tonight, trying to figure out the impacts and risks of this new result.