

Jun Fukuyama's P≠NP Page - vznvzn
http://junfukuyama.wordpress.com/
a new serious proof claim on P!=NP has been put forward by a Phd mathematician/computer scientist Jun Fukuyama last July 1 and has received very little public attention since then.<p>http://www.linkedin.com/pub/junichiro-fukuyama/36/2b9/88b<p>rumor is that its been submitted to a journal. it uses a known plausible approach based on monotone circuit theory for which there are some long established existing proofs of circuit lower bounds (dating to a celebrated 1985 proof by Razborov). Fukuyama has published several papers in computer science. it would be great if the online community could give this some attention as with the Deolalikar proof from 2.5yrs ago.
======
ColinWright
Very briefly:

Context: Some problems are easy, and known to be easy. Some problems appear to
be hard, and yet easy to verify alleged solutions. That seems odd - let's have
a closer look.

Example:

Consider being given a network and asked if there's a path that visits every
node exactly once and returns to its starting point. (This is the task of
asking if a Hamilton Cycle exists). If the network is big enough, deciding if
this can be done can be really hard. However, checking if someone has
successfully done it is easy.

Motivation for a definition of NP:

A problem for which it is efficient (for which read "polynomial") to check a
purported solution is said to be in "NP". Knowing whether a Hamilton cycle
exists in a network is one of many, many such problems.

Definition: A problem is in "NP" if there is a polynomial time algorithm to
test an alleged solution.

Definition: A problem is in "P" if there is a polynomial time algorithm to
solve it.

Lemma: All problems in "P" are also in "NP". Proof left to the interested
reader.

(Note: technically we are dealing with Decision Problems - problems where the
answer is always Yes or No. For example, I didn't ask you to find a Hamilton
Cycle, I only asked if there was one. Sometimes the Decision version can be
ramped up into actually finding the solution.)

Note that easy problems (for example, finding a cycle that visits every edge
exactly once - an Euler Cycle) is also "NP", but it's not a hard NP problem -
it's actually in "P". This point confuses a lot of people. NP does not
automatically mean hard - that comes in a minute.

Another example:

Similarly, factoring a large integer can be really hard, but if someone claims
to have a factor, checking it is as easy as doing the division.

Comparing problems:

Cleverly, it can be shown that if you can solve the Hamilton Circuit problem
efficiently, that can be converted into a way of solving the factoring
problem. In some very real sense, the Hamilton Circuit problem is harder than
(or equal to) the factoring problem.

Finding:

There are many, many NP problems that have been shown all to be equivalent to
each other, all to be easy to check solutions, and yet no algorithm for
solving in polynomial time known. Such problems are call NP-Complete (NPC)
because an efficient algorithm for any of them will completely solve all the
NP problems.

So:

* A problem is said to be in "P" if there is an algorithm that takes instances and produces answers in polynomial time.

* A problem is said to be in "NP" if there is an algorithm that takes only polynomial time to check a purported solution for correctness.

* It is unknown if there are any problems that are in "NP", but not in "P".

That is the P vs NP question, and the Clay Mathematics Institute has offered
$10^6 for its resolution. You could do this by showing that there is a
polynomial time algorithm for any of the NPC problems, or by showing that no
such algorithm exists.

Summary:

* P problems are "easy"

* NPC problems are thought to be hard

* There are lots of NPC problems, and solving any one of them would solve all NP problems.

* Integer factoring (IF) is thought to be outside P, but not as hard as NPC.

* There are problems that are harder than all NPC problems, these are called NP-Hard.

So we think that

    
    
      P < IF < NPC={ Clique, 3SAT, G3C, Knapsack, Ham, ...} <= NP-Hard
    

But we don't know if these are, in fact, all different. The submitted link is
to a paper that says yes, they are all different.

I hope that helps.

~~~
Aardwolf
Why do they call it decision problems, if it's really about finding a solution
that is much more than a boolean (like a path, ...)?

The whole idea of "verify" versus "find" makes no sense for problems with a
yes/no answer, because if you can verify a boolean answer in polynomial time,
you can always find the solution in polynomial time too, after all there are
only two possible solutions so you need to verify only one to find the answer.

If you need extra data to prove this "yes" in polynomial time, then the answer
isn't a boolean anymore, but a boolean plus the extra data, so how can you
still call that a yes/no problem?

~~~
btilly
_...if you can verify a boolean answer in polynomial time, you can always find
the solution in polynomial time too, after all there are only two possible
solutions so you need to verify only one to find the answer._

It doesn't work that way. The common characteristic of NP-complete problems is
that there are an exponential number of possible solutions, but verifying that
a purported solution is a solution takes only polynomial time.

For example consider SAT (see
<http://en.wikipedia.org/wiki/Boolean_satisfiability_problem> for more). This
is the problem of deciding whether there is a set of possible variable values
that make a given Boolean equation evaluate to _True_. For an equation of
length n there are in general O(n) variables, and therefore 2^O(n) possible
sets of true/false values that the variables can have. So a brute force search
takes exponential time.

But if you give me an equation, AND a set of values for the variables that you
claim will result in the final expression evaluating to _True_ , I can verify
it in time O(n).

Does that answer your question?

------
anandkulkarni
Before anyone gets too excited, I would caution that there there is a long
history of failed attempts at this problem, and an equally long history of
premature announcements. On the surface, here is what is encouraging about
this one:

\- The author is a theory professional and not an amateur or a crank

\- The author frames this as a possible proof where he's seeking feedback on
the method, rather than declaring a solution up front

\- The author builds on existing theory in a formerly mainstream approach that
has been shown to have problems (circuit lower bounds)

\- Looks like a serious and well-considered proof effort

Here is why HN should hold off on any excitement:

\- This style of proof was found to be unworkable in the past. In particular,
approaches to proving circuit lower bounds in this manner were shown by
Razborov and Rudich to be "natural proofs" in the 1990s, which is a category
of proof strategy that cannot demonstrate P/NP. It's not explained why this
strategy does not succumb to the same problem -- though the author surely has
some reasoning around it. (see point 6 of Scott Aaronson's "8 signs a proposed
P/NP proof is wrong": <http://www.scottaaronson.com/blog/?p=458>. You might
also see this comment from /r/math:
[http://www.reddit.com/r/compsci/comments/14mqqt/anyone_have_...](http://www.reddit.com/r/compsci/comments/14mqqt/anyone_have_the_math_to_check_out_this_p_np_proof/c7elwbz))
).

\- There is a very good community framework for evaluating proposed solutions
when the author's ready. Possibly the author has already submitted the work to
a journal, but it's not clear that he's confident enough yet to invite more
active discussion on his effort from the theory community. I can't find any
discussion of it online. It's likely he's pursuing these on his own, and
conceivably flaws have been found since the summer.

\- The author does not declare this as a completed solution in the same way
that Deolikar did two years ago. In that case, his announcement led to an
analysis by the theory community online that led to discovery of core errors.
We can wait here 'til the author gives a bit more information.

\- This proof joins dozens of other announced solutions to P/NP in both
directions (see, for example, <http://www.win.tue.nl/~gwoegi/P-versus-NP.htm>)
which emerge quite regularly, and close attention paid to each will leave no
time for original research :)

~~~
vznvzn
hi AK. maybe consider blogging about this? this is a well written but
superficial analysis. scott aaronson insists on his blog that a proof should
explain why it succeeds against "known barriers" eg razborov/rudich Natural
Proofs. but this is really an optional requirement of a proof. moreover the
actual barrier to "natural proofs" is very subtle and basically insists that a
proof, if it exists, should have a certain "intrinsic complexity" in its
constructions, and that many such constructions in the literature for class
separations do not have this "intrinsic complexity". but new researchers are
just suggesting that this only requires some new "intrinsic complexity"
function that hasnt been seen before; but that such functions do exist, they
just dont seem to be used in proofs that we know of. eminent authorities in
the field such as Lipton have argued that the near 20yr old Natural Proofs may
be overdramatized as a real barrier.

here is liptons blog on the subj:
[http://rjlipton.wordpress.com/2009/03/25/whos-afraid-of-
natu...](http://rjlipton.wordpress.com/2009/03/25/whos-afraid-of-natural-
proofs/)

here is chow in AMS:

"Nevertheless, it is my personal opinion that the optimistic approach is the
right one; that is, the Razborov–Rudich result should be regarded as a hint,
and not a barrier, to separating complex- ity classes. The only real barrier
is our lack of imagination."

<http://www.ams.org/notices/201111/rtx111101586p.pdf>

~~~
anandkulkarni
I may have come across as insistent on clearly stating ways around the natural
proofs barrier because this is Hacker News, where folks are liable to jump to
conclusions, and not, say, <http://cstheory.stackexchange.com>. Given the
large number of attempts against P vs. NP, frequently by folks who are not
active in complexity theory, it's often better to try to understand these
ideas on theory sites.

You make an reasonable point, vis-a-vis Chow; rather than being regarded as an
absolute barrier for lines of reasoning involving circuit complexity, the
requirement of being a sufficiently complex proof may provide a jumping-off
point for uncovering a working method.

------
thomasbk
A great site with 95 documented attempts at proving P ?= NP is over at
<http://www.win.tue.nl/~gwoegi/P-versus-NP.htm>, for those interested in the
subject.

------
B-Con
From other people's research (
[http://michaelnielsen.org/polymath1/index.php?title=Jun_Fuku...](http://michaelnielsen.org/polymath1/index.php?title=Jun_Fukuyama%27s_P%E2%89%A0NP_Paper)
) he seems to have a credible background.

Although I find it interesting that he works for Toyota:

> My name is Jun Fukuyama. I’m currently a researcher at Toyota InfoTechnology
> Center. I’m visiting WINLAB and Civil Engineering department at Rutgers
> University, working on algorithms and mathematical analysis related to
> vehicular communications and mobility modeling.

Unless he's been doing this research in his spare time, it seems like an odd
fit for Toyota to dedicate research to. Obviously large companies employ
theory-oriented researchers, but when I think Toyota I don't really think
about very abstract computer science. I'm curious if there are any particular
applications for the company one way or the other.

~~~
tomjakubowski
Toyota actually has opened a school in Chicago specifically for computer
science, in conjunction with the University of Chicago:

[http://en.wikipedia.org/wiki/Toyota_Technological_Institute_...](http://en.wikipedia.org/wiki/Toyota_Technological_Institute_at_Chicago)

~~~
jamesjporter
I go to UChicago; that place is sort of like an eerie mystery box that nobody
ever goes to or talks about. I didn't even find out about it's existence until
my 3rd year here. My one friend who's been in there described the experience
as really bizarre; it left him half convinced that its actually a front for
some sort of Israeli money laundering operation.

~~~
iskander
They have some very prominent machine learning folks there and I went to the
Machine Learning Summer School they hosted a few years back. Seemed legitimate
enough to me!

------
vznvzn
hi all. meant to post a comment but didnt understand this hackernews interface
so far, am brand new to this site. fukuyama states on his web page he's worked
on P vs NP for over 10 yrs, both inside and outside of his professional jobs
which include research and teaching. the web page is a proof [claimed/attempt]
that P!=NP posted on Jul 1. unfortunately its gotten very little to no online
attention since then, at this point so far. he doesnt seem to have announced
it anywhere in cyberspace, only created the blog.

~~~
mdxn
If you don't mind me asking, how and when did you find his blog post?

------
gburt
I wish I had the mathematical background to comprehend this. It's very
involved. My understanding is that he has shown that CLIQUE (a known NP-
complete problem) takes exponential time and thus is not solvable in polytime,
this would mean there is a problem outside of P but inside of NP, and this P
!= NP.

~~~
GoGoGalois
If a problem is NP complete then there is an algorithm to solve it in
nondeterminstic polynomial time. Any nondet algorithm can be transformed to an
exp poly time algorithm (just run through each of your poly time algorithms
one by one) so showing CLIQUE has an exp algorithm doesn't help at all. We
want a lower bound.

~~~
morsch
Yes, that's what he is saying, _every_ CLIQUE algorithm takes at least
exponential time.

~~~
GoGoGalois
Oh ok, I thought the first post was referring to a single algorithm, not every
possible algorithm. My mistake.

~~~
gburt
Sorry, if I could I would edit it to be more clear. We definitely need to show
that no algorithm can exist with poly runtime for all inputs.

------
btilly
I googled to see if Terry Tao had weighed in yet. He has not that I've seen.

However [http://vzn1.wordpress.com/2012/12/08/outline-for-a-np-
vsppol...](http://vzn1.wordpress.com/2012/12/08/outline-for-a-np-vsppoly-
proof-based-on-monotone-circuits-hypergraphs-and-factoring/) suggests that
this looks like a serious attempt. (Of course most serious attempts will
fail.)

------
breakyerself
Can we get a laymans explanation of what this is about? I understand what p np
is, but don't really understand what this person is claiming. Is this some
kind of claim about the overall concept of p np? Or proving a specific case?

~~~
notimetorelax
If I understand it right he showed that Clique cannot be solved in polynomial
time, since any NP complete problem is at least as hard as Clique none of
those problems can be solved in polynomial time. So he used Clique to draw
general conclusion that P!=NP.

~~~
sjg007
This only works if all NPC problems can be converted to CLIQUE?

~~~
qdog
All NP problems can be converted into the other NP problems, therefore if you
can prove it for one NP problem, you prove it for all NP problems.

~~~
cgray4
You missed a few words. Any NP-complete problem can be converted into any
other NP-complete problem in polynomial time.

~~~
qdog
Right, oops.

------
mdxn
Note: For the moment, I will assume that the NP != NC and above results are
correct enough to consider the validity of the latter portion of the paper.

As far as I can tell (from my initial reading), the author (and many many
commenters here) seem(s) to be confusing polynomial time (P) with polynomial
circuit size (P/poly). These are completely different complexity classes in
style and probably in power, too. Last time I checked, there was not a good
characterization of P in terms of circuit complexity (the attempt at doing so
is P/poly). So, logically, comparing P and NP is not even possible without
additional knowledge about P. The exception to this being showing that X != NP
where P <= X <= NP.

For reference: P <= BPP <= NP

We know that BPP (a randomized version of P) is (non-strictly) contained in
P/poly. We also know that proving P = BPP (which is conjectured to be true by
most) requires these classes require superpolynomial lower bounds for
Boolean|Arithmetic circuit size. As far as I know, proving an exponential
lower bound for an NP-complete problem doesn't immediately rule that P != NP
since P !=> only polynomial size circuit size (the circuit size only relates
to the size of the advice function). P might not contain problems that require
exponential circuit size, but this fact is not stated, proven, nor reference
by the author here.

Simply put: The author assumes that P != NP follows immediately from Theorem
6.1. This is not as obvious as they might think it is; many additional details
are needed.

Admittedly, I have not rigorously studied or concluded anything from the
flattening process the author proposes. I honestly don't believe I have to.
Skimming over it, I don't see any way that P is characterized in terms of
polynomial circuit sizes (which probably is an assumption as powerful as
P!=NP) here. I also do not see any mention of P/poly, which I believe the
author is attempting to use.

For a paper approaching the P vs NP issue using circuit complexity, I would
expect this much. Because of the lack of even a mention of BPP, P/poly, and
others, I would be highly surprised if many of these results in the paper hold
at all. Though, hopefully, I might be wrong.

EDIT: Accidentally used circuit depth when I was really thinking of size.

~~~
vznvzn
"as far as I know, proving an exponential lower bound for an NP complete
problem doesnt immediately rule that P!=NP". that statement is _FALSE_. P!=NP
is exactly a consequence of proving exponential lower time bounds. actually
P!=NP would be a consequence of even proving somewhat weaker _superpolynomial_
time bounds on any NP complete problem...

~~~
mdxn
I was clearly referring to exponential circuit sizes (note the sentence before
it), not exponential time.

By the way, you can edit your original reply to contain more information
rather than just making a bunch of new ones.

~~~
vznvzn
oops. thx for info. didnt notice that. new to this. sorry for multiple replies
that should just be edits.

------
jere
For a popular treatment of the subject and the consequences, see this film (I
assume the first film ever made about P=NP and it was released just this
year): <http://en.wikipedia.org/wiki/Travelling_Salesman_(2012_film)>

I just finished a section on NPC problems in my algorithms class (working on a
masters). For years, I've been confused about the true meaning of P, NP, and
NPC. Feels weird saying it, but I am so glad I've finally got a good grasp on
at least the definitions. One of the most gratifying moments in the class was
doing homework problems that required proving various problems were in NPC.

------
albertzeyer
Btw., what about Vinay Deolalikar's proof now?

The last update I could find:
[http://rjlipton.wordpress.com/2011/08/11/deolalikars-
claim-o...](http://rjlipton.wordpress.com/2011/08/11/deolalikars-claim-one-
year-later/) <http://news.ycombinator.com/item?id=2891710>

~~~
iso-8859-1
It is incorrect. I guess that after he realized it himself, he got discouraged
and didn't bother to tell people about it.

------
gburt
Has this been submitted to a journal or the Clay Institute or any peer review
yet?

------
Tipzntrix
I'm interested in knowing how many of you on here are going to read this in
one sitting. Not that it would be sufficient to analyze it for bugs, but
because you're that interested.

~~~
akavi
If I had the mathematical background necessary to understand it, I certainly
would.

... and I really wish I did have the mathematical background. Anyone have any
recommendations on how to go from a standard CS undergrad math background to
being able to understand proofs like these?

~~~
Tloewald
Read it and google as you go. Math is one of the best covered topics online.
For a potential millenium prize proof, it seems very approachable. That said,
that's not saying much.

~~~
m0nastic
I find math in some cases to be really hard to Google. Particularly with
expressions that I don't know the names of; I never know what to use as search
terms.

------
alwaysright
There is a great udacity course with all the background to the P/NP problem:
<http://www.udacity.com/overview/Course/cs313/> I found it very informative.
It also has a section on "should you try solving the P vs. NP problem" and the
answer is "not unless you have a lot of time, money, and though of something
no one tried before, or is a math / CS professor"

------
Zircom
YES, YOUR MAJESTY!

~~~
KaoruAoiShiho
All Hail Britannia

------
kiskis
damn, there is a typo on page 32 (in paper version 1.11). Stopped reading. I'm
wondering if that voids the proof. Let's the downvote begin.

~~~
bowyakka
This is why we cant have nice things

~~~
kiskis
now the funny thing is that there is really a typo on that page, so all of the
these hacker news people downvote just because I pointed out some small, but
truth parth of reality.

this is called witch-hunt

~~~
Mouq
Well, sure. It's more that you posted a comment not that there's a typo but
that this typo makes it so unbearable to read that you were simply forced to
stop. But anyway...

~~~
kiskis
it will be oil on fire, and you guys will already downvoted me into oblivion,
but the truth is the same, the paper has several typos, which makes in
unquestionable that it did not see a spell-checker.

~~~
ColinWright
Spell checking math, and in particular LaTeX math, is a complete nightmare. I,
for one, find that there are just as many substantive typographical errors
after spell checking my papers as before, and all I've done is wasted at a
minimum an hour trawling through false positives.

Besides, small typos in a paper that's substantially readable and well
crafted, written by someone in a language other than their first, are scarcely
a reason to stop reading.

