
P might be NP: A Polynomial Time Algorithm for the Hamilton Circuit Problem - Filligree
http://arxiv.org/abs/1305.5976
======
ColinWright
Just looking at the abstract:

    
    
        In this paper, we introduce a so-called Multistage graph Simple Path (MSP)
        problem and show that the Hamilton Circuit (HC) problem can be polynomially
        reducible to the MSP problem.
    

That would imply that the MSP is NP-hard. So far, so good.

    
    
        To solve the MSP problem, we propose a polynomial algorithm ...
    

That would imply that P=NP, and hence this would be a _major_ result, with
potentially wide-reaching consequences.

    
    
        ... and prove its NP-completeness.
    

Pause. This doesn't make sense. If you have a polynomial algorithm then it's
in P. If you've reduced HC to MSP then you've already shown MSP is NP-Hard.

They use the word "its" - to what are they referring? The algorithm? That
doesn't make sense, as an algorithm is not something that's NPC. The MSP
problem? Earlier claimed results show that it's NP-Hard, now they're showing
it's P, so to "prove its NP-completeness" doesn't fit.

However, English is not their first language (I assume) so perhaps I'm over-
thinking irrelevant detail.

    
    
        Our result implies NP=P.
    

Yes, yes it would.

Now I'm off to see if I, as a non-specialist, can make any sense of it.

~~~
pedrocr
From the Introduction:

    
    
      We will introduce a so-called 'Multistage graph Simple 
      Path' (MSP) problem and prove its NP-completeness.
    

So what they claim to be NP-complete is their new problem and not the
algorithm as it should be. I'll still say this will turn out to not be real. I
won't even try to follow their paper but it's not written in LaTeX so it can't
be real math... :)

~~~
ColinWright
They claim to have a new problem (MSP) which is NP-Hard because HC reduces to
it, and they claim to have a polynomial algorithm to solve MSP.

That's all very reasonable. What's less reasonable is that all of the
references bar one are to their own papers, and a paper from 2010 claims to
have been presented at a conference, and to prove this result.

It's not passing the sniff test, but I'll still see if the first few pages
make sense.

------
jacobparker
ArXiv has many "proofs" that P=NP and that P!=NP. ArXiv does not do any
vetting of correctness (it is beyond their scope.) We don't need a HN
submission for this and the title is inaccurate/clickbait.

~~~
mixedbit
I wonder why authors of P=NP 'proofs' won't attach a program that solves some
NP hard problem for a collection of large inputs in reasonable time.

This of course wouldn't prove the work is valid, but should be enough to draw
attention and have the paper reviewed by qualified people.

~~~
dagw
Mathematically you can prove that an algorithm with certain properties exists,
without being able to implement it. Also, as others have pointed out,
polynomial does not necessarily mean "fast".

~~~
mixedbit
Interesting, are you aware of any such algorithm (proven to exist but not
discovered yet)?

~~~
philix001
There are cases where the lower bound to solve some problems (e.g. sorting)
have been proved to be less than any known algorithm at the time.

If you prove that the lower bound for any NP-Complete problem is O(p) where p
is a polynomial, then P=NP and you do not necessarily have the algorithm.

<http://en.wikipedia.org/wiki/Upper_and_lower_bounds>

------
scythe
If you scroll to the last page, you'll find that this "Xinwen Jiang" character
cites: a textbook on computational complexity from the '70s, plus himself (in
the apparently nonexistent _Computer Technology and Automation_ ), more than
ten times. Furthermore, he's been at this since 1993. A dedicated crank if
I've ever seen one.

~~~
stiff
Spot the error or shut up. If Ramajuan would write a paper himself he could
only cite some rather obscure mathematical encyclopedia, he still uncovered
wide areas of mathematics unknown to his contemporaries. There might be one
Ramajuan for 100 000 cranks, but it still sucks to dismiss someones work based
on the fucking bibliography...

~~~
ColinWright

        > Spot the error or shut up.
    

That's not really how it works. I'm deciding whether it's worth my time trying
to understand this. If it's really the breakthrough it purports to be, I can
expect there to be some deep ideas and difficult tricks - I expect this to
take both time and effort.

To decide whether I'll bother I apply several heuristics, many of which are
well-known and informally documented. If the first page or two just seems like
obvious stuff, or is subtly nonsensical, then I won't bother.

But I am certainly concerned that he doesn't cite any other significant work
at all. More, the claims toward the end are rather, well, indistinct. It's
doing very well against the ten heuristics in this blog post:

<http://www.scottaaronson.com/blog/?p=304>

~~~
stiff
OK, but I would love for those heuristics to focus on assessing the content of
the paper, not the reputation of the author, his peers and quotations etc. You
can get some vague impression of what's going on in this paper in some 30
minutes, there are many parts that seem strange (he says for 4-stage graphs
one can manually verify that his "prooving algorithm" is correct and uses this
as an assumption later in his proof), so I would love to hear something about
that.

~~~
ColinWright
The heuristics I apply go well beyond the ones given in that post. In this
case I have taken on board the fact that so many references are just to his
own earlier work, and that work is old enough that it should have made a
bigger splash. That's a clear indication that this is not likely to be
correct, so I'm looking for mistakes, I'm not looking for deep ideas.

And I'm not finding deep ideas. I spent 10 minutes getting a feel for the
approach, and I'll come back when I have another 10 or 15 minutes to spare,
but it's really, really not looking good.

------
crntaylor
I'm very skeptical, for the following reasons:

1\. The author first released this paper (well, a version of it) in April
2009. It seems unlikely that it would have gone unnoticed for four years if
the proof was valid.

2\. Aside from a 1979 textbook and a 2010 paper, the author only cites himself
(10 times!)

3\. The author does not use TeX (see 1. at "Ten Signs A Claimed Mathematical
Breakthrough is False" <http://www.scottaaronson.com/blog/?p=304>)

~~~
glomph
>At some point, there might be nothing left to do except to roll up your
sleeves, brew some coffee, and tell your graduate student to read the paper
and report back to you.

Hahaha.

------
mixedbit
From the author home page:

'It seems our algorithm is a polynomial one. So we would like to discuss with
more people.'

It's weird if a proof doesn't even convince the author.

<https://sites.google.com/site/xinwenjiang/>

~~~
samolang
"Beware of bugs in the above code; I have only proved it correct, not tried
it." - Donald Knuth

~~~
devrelm
This appears to be a corollary: "Beware of bugs in the above code; I have only
tried it, not proved it correct."

------
mebassett
the arxiv doesn't peer-review papers. it has lots of proofs of famous
conjectures, et cetera.

this paper, in particular, smells fishy. the author doesn't cite anyone but
himself for significant result (he does cite someone else for referencing
material that's standard for specialists.)

related: <http://primes.utm.edu/notes/crackpot.html>

------
dburgoyne
Jiang found a polynomial-time algorithm for solving what he calls "labelled
multistage graphs"

Note that these graphs are partially ordered sets
(<http://en.wikipedia.org/wiki/Partially_ordered_set>)

and every partially ordered set has a unique corresponding comparability graph
(<http://en.wikipedia.org/wiki/Comparability_graph>)

and the problem of finding a Hamiltonian cycle in a comparability graph is
known not to be NP-complete.
(<http://link.springer.com/article/10.1007%2FBF00571188>)

Thus Jiang _may_ have found a polynomial-time algorithm, but it solves a
problem that is not NP-complete.

------
stiff
I spent some forty minutes on this, I think his "multistage graph simple path
problem" is a very obfuscated version of a simpler problem where there is no
need for this complicated labelling of vertices, with the information being
instead reflected in the connections between vertices. I think the problem he
discusses is simply the problem of determining whether two vertices in a graph
are connected via a path or not. Did anyone else understand his problem
formulation and had any similar impressions? I have a microscopic hope of this
being a valid P=NP proof, still would be nice to know wtf is this about.

~~~
ColinWright
His "Multistage Graph" seems poorly defined. I'm slowly coming to grasp what I
think it is, but he defines is as

    
    
        G = <V,E,S,D,L>
    

but pulls the _E(v)_ from nowhere.

It's not looking good.

~~~
stiff
Yes, I think the intention is for E(v) to be one of the inputs of the
algorithm (I contemplated E(v) for half an hour to guess this is what he
means). I thought maybe the following is possible: I create a new graph with
the same vertices as his graph, but instead of having an E(v) I connect each
vertex to the members of E(v) via an edge. Afterwards, if two vertices are not
connected via an edge in his original graph, I remove this edge from my new
graph. I cannot really get to the heart of it, I just had a vague impression
this might be equivalent, either way I think the problem could likely be
formalized in a more effective way.

------
raverbashing
Yes, not looking good

It will certainly be amazing if it is true, but it doesn't look right

The style, the constructs in the article, etc

But yes, translating one NP problem to another NP problem is a n allowed
approach and has been done between several NP problems (if you prove your
translation is correct, and valid all the time, of course)

------
numbfits
This video may help to understand the paper more vividly:
[http://www.youtube.com/watch?v=NYWgrjWQx60&feature=youtu...](http://www.youtube.com/watch?v=NYWgrjWQx60&feature=youtu.be)

