
Predicting When P=NP Is Resolved - furcyd
https://rjlipton.wordpress.com/2019/12/22/predicting-when-pnp-is-resolved/
======
joe_the_user
My pure amateur, possibly muddled understanding is that P=!NP is rather
different from most open conjectures.

A) It's extremely general or even "foundational" in the sense that it's asking
whether any algorithm at all exists to solve an extremely general sort of
problem in polynomial time.

B) All of the standard methods used to solve problems of this sort have at
least been pronounced exhausted at this point.

C) Very few theorems that inherently limit the speed of a class of problems
actually have ever been proven. Very few methods for proving these constraints
are known.

D) A lot of famous theorems have yielded results through being embedded in a
larger, different field where they are just one result of many proved with a
new machinery (Fermet most prominently). But given P=!NP is so general it
can't really embedded in a larger, tractible space - lots of things are
equivalent to it but all these things are kind of the same.

~~~
someguy12342
Great post. Here are a few things pointers if you want more information about
your points:

B) The situation is actually worse than exhausted. We know (have proven) that
the regular methods simply won't work (check up "the reletaivzation barrier"
and "natural proofs" and "algebrization")

C) We actually can construct problems which take at least a certain amount of
time to solve (though admittedly many of them are kind of unnatural). Search
up "time hierarchy theorem". This is probably one of the theorems you mention.
However, you're right, as far as natural problems go, it appears to be extreme
difficult to prove most natural problems takes at least linear time
([https://mathoverflow.net/questions/4953/super-linear-time-
co...](https://mathoverflow.net/questions/4953/super-linear-time-complexity-
lower-bounds-for-any-natural-problem-in-np)). Note the log*... factor grows
extremely slowly.

In fact, it is not known whether SAT requires more than linear time. Since SAT
is NP-complete, we expect it to have no polynomial time algorithm... and yet
here we are wondering whether it requires even more than linear time. If you
are interested in this sort of thing, maybe look up fine-grained complexity.
Also note that to solve P != NP, we will have to atleast shown P != PSPACE,
since PSPACE contains NP. However, even this seemingly easier problem has had
no progress and seems like it would require a huge breakthrough.

~~~
raverbashing
From this it seems that the question itself might have been hopelessly naive

Is there any evidence that P might be equal to NP? This seems different from
other famous conjectures like Fermat's Last Theorem or Riemann's Theorem where
what's hard is finding a counterexample that disproves the theory.

~~~
someguy12342
I think RJ Lipton (blog author) and Knuth believe it could be true. However,
most complexity theorists don't (I think Fortnow does a survey of theorists on
this every few years). Knuth says something like "well all it would take is a
single algorithm for any of these hundreds of problems". A fine take for the
godfather of algorithms. At the same time, there are many people who have in
some sense tried to find such an algorithm, even non-explicitly (e.g.
factoring would be in P if P = NP). Moreover, there is a lot of complexity
theoretic work which makes it hard to believe it to be true (I think Scott
Aaronson's blog has some presentable information about this)

~~~
lonelappde
[http://www.informit.com/articles/article.aspx?p=2213858&WT.m...](http://www.informit.com/articles/article.aspx?p=2213858&WT.mc_id=Author_Knuth_20Questions)

Note that Knuth does not believe that the constant factor on the P algorithm
be will be less than the size of the entire Universe.

~~~
saagarjha
On the first P algorithm found, or _any_ P algorithm?

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AwesomeLemon
Should we even care for a mathematical proof that P=NP? We know that in
practice we haven't been able to come up with algorithms that solve NP
problems in polynomial time. Suppose we'll be told that this is possible (i.e.
P=NP): will this help up us to invent such algorithms? I don't see how, unless
the proof will be by construction.

~~~
joe_the_user
Working programmers and even people concerned with producing useful algorithm
probably shouldn't care about P=NP?. It seems very likely to be true and if it
isn't true, it seems like a polynomial algorithm would be unwieldy indeed.
Further, P!=NP is a theory entirely about worst case scenarios. Many NP
complete problems are actually quite solvable in the average case and the
worst case can have little relation to the average case (SAT solvers that are
quite fast on average exist now, for example).

For logicians and mathematicians, a proof of N=NP? would be an incredible
accomplishment simply because it's a problem that at this point no one know
where to start on and so by definition, the proof would be a piece of
remarkable and surprising mathematics giving people much to think on.

~~~
AnimalMuppet
P=NP is very likely to be _true_? I would say it's very likely to be _false_.

Why do I say that? Because it's possible to construct, say, SAT3 problems such
that it seems impossible to solve them in polynomial time. If _some_ problems
can't be solved in polynomial time, then P!=NP.

Why do you think that it's "very likely" that P=NP?

~~~
norrius
Given the next sentence (...if it isn't true, it seems like a polynomial
algorithm would be unwieldy indeed), I think the GP actually meant P!=NP to be
true.

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jacobkg
In my mind I always regard P/NP in the same category as the Riemann
Hypothesis^. That has been open since 1859 (160 years) and shows no signs of
cracking. So in other words we may be in for a long wait.

^Perhaps because one of the hypothesized examples of a hard problem is the
factoring of multiples of large primes (though not proved NP Complete). Or
perhaps because they are both extremely famous open problems.

~~~
hinkley
One of the things that slows scientific progress is the lack of cross-
discipline learning. Every so often you hear about someone taking an old
concept from another field and applying it. Quite a few breakout companies
have combined interdisciplinary knowledge into one product and made a mint.

So many people are affected by NP complete problems that you'd think that
someone would have a problem that appeared simpler on the surface and gave
insights into a P solution, but either nobody has, or the right phrasing about
it has caught nobody's attention.

There's an XKCD lamenting something pretty similar. You solved some unsolved
problem but nobody will ever know because it's buried in a bug fix for your
obscure little product.

~~~
fovc
[https://m.xkcd.com/664/](https://m.xkcd.com/664/)

~~~
raverbashing
I think Academia is more likely to criticize your technique because it doesn't
generalize to any ring or you didn't prove a certain assumption you had.

Of course the real bug is in some IEEE 754 quirk you forgot to consider.

------
kibwen
Is it possible to prove that a proof exists for either P=NP or P!=NP?
Conversely, is it possible to prove that the conjecture is unprovable?

~~~
marcosdumay
Since an algorithm for running NP-complete problems in P-time would be enough
to prove whether P=NP, wouldn't the impossibility to prove it just prove that
P!=NP?

~~~
ookdatnog
Surprisingly, a theorem being unprovable does not mean that the theorem is
untrue, even if you can prove its unprovability! This was shown by Goedel
([https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...](https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems)).
He showed that there exists no system powerful enough to express theorems
about the natural numbers which can prove all facts about natural numbers
_and_ be sound (ie not prove things that are untrue).

~~~
marcosdumay
That's not exactly what I'm saying. What I am saying is that it's impossible
for it to be true and unprovable. If it is true, it means a proof exists.

Also, I am talking about this specific problem.

~~~
lonelappde
Can you prove that?

"impossibility to prove" is different from "proven impossible" because hiding
behind the loose English are different models of logics. That's what Godel
theorems at e about.

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skissane
I have no idea what will be proven nor when. (My uneducated guess is not any
time soon.)

But I hope it isn't "here's a proof that P!=NP". That'd be just too
_boring_... Much cooler possibilities would be "here's a non-constructive
proof that P=NP" or "here's a proof that P=NP is independent of ZFC".

~~~
enriquto
for me the most exciting would be "here's a constructive proof that P=NP. It
uses polynomials of degree one million".

~~~
chx
While Knuth has a different take on the P=?NP problem than most others, namely
he believes P=NP, nonetheless, he is also saying that the degree won't ever be
known and not even an upper bound will be known for it.
[http://www.informit.com/articles/article.aspx?p=2213858&WT.m...](http://www.informit.com/articles/article.aspx?p=2213858&WT.mc_id=Author_Knuth_20Questions)
Certainly "if P=NP it'll be a pure non-constructive proof with no practical
consequence whatsoever" feels right.

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copperx
What is the difference between solving and resolving an open problem?

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einpoklum
Next top story on HackerNews:

"Predicting when a new article will be published about how P=NP is not solved
yet."

My prediction: Pretty soon :-(

