
Polynomial Time Algorithm for Graph Isomorphism Testing - amichail
http://arxiv.org/abs/1004.1808
======
gjm11
This would be big news if true (though not P=NP-big; graph isomorphism is an
apparently-hard problem that isn't known to be NP-hard).

From a cursory glance, it doesn't look terribly promising, but I haven't made
a serious attempt to see what he's doing and how it might work. Examples of
not-terribly-promising things:

1\. He proves what he calls "Assertion 1", then says that it "suggests" some
other propositions that he calls "Conclusion 1", "Conclusion 2", "Conclusion
3". He doesn't offer any proofs of these; they don't seem to be obvious
consequences of Assertion 1; they don't even seem very likely to be true. In
particular, Assertion 1 has the form "if graphs G and G' are isomorphic, then
[linear algebra stuff]" and Conclusion 2 (which he uses later) has the form
"if [linear algebra stuff] then graphs G and G' are isomorphic".

2\. He then defines this thing he calls a W-matrix, which seems like it
basically encodes for a given vertex of a given graph which other vertices are
at any given distance from it. He then appears to claim -- I'm not quite sure,
because his notation is eccentric and it's late at night -- that equality of
W-matrices is basically the same thing as similarity of vertices (i.e.,
whether some graph automorphism carries one to the other), and that doesn't
look right to me.

I repeat that I haven't looked at this carefully, and I'm not an expert in the
field anyway. But it's not compelling enough to make me want to spend much
more time on it; I expect that if it turns out to be valid someone smarter
than me will check it over and let the world know :-).

~~~
pmb
Anyone can post to the ArXiv. So now wait a few weeks. One test: does his
algorithm work for bipartite graphs? Another test: his algorithm uses real
numbers. This is sketchy, as real numbers can't actually be represented in a
computer.

The paper is pretty poorly written for two reasons: English is clearly not
this guy's first language (no crime there, as it is said that the language of
science is "heavily accented english") but he is also using a little too much
handwaving for my comfort. This is a pretty math-heavy result, and this is not
an airtight proof in any sense.

Basically: I agree with everything you said about it not being compelling
enough for me to spend more time with it.

Fun fact so that I'll say more than "I agree": if GI is in P, the the
polynomial hierarchy collapses to a level that I forget, and if GI is NPC it
collapses to a PI_2! Neat!

~~~
mt2
> does his algorithm work for bipartite graphs?

Please, download the program and test any graph you like ;)

------
gjm11
The author appears to have done a fair bit of self-promotion in Wikipedia,
which seems like a bad sign (though of course it doesn't prove anything):
<http://en.wikipedia.org/wiki/User_talk:Tim32> and
[http://en.wikipedia.org/w/index.php?title=Graph_isomorphism&...](http://en.wikipedia.org/w/index.php?title=Graph_isomorphism&action=history)
(see, e.g., a reversion from David Eppstein, who most certainly knows his
stuff, saying "Persistent long-term self-promotion; blocking may be
indicated.").

~~~
gruseom
I can't help comparing this to the extreme _absence_ of self-promotion in the
most prominent example of somebody doing this who turned out to be the real
thing (Perelman).

~~~
gjm11
Yeah. But you've got to be careful; lots of people dismissed Louis de Branges'
claim to have proved the Bieberbach conjecture because he also had a long
history of self-promotion (and, indeed, of claiming to have proved things when
his proofs really had holes). But he really _did_ prove the Bieberbach
conjecture.

Then again, he also keeps claiming to have proved the Riemann hypothesis and
no one believes him :-).

~~~
gruseom
You obviously know more about this than I do. That's a very interesting case.

------
DarkShikari
One thing particularly interesting about this is that Graph Isomorphism is in
a hypothetical category of problems sometimes dubbed "NP-intermediate": they
are _believed_ to be hard, but nobody has proven they're NP-complete. Linear
programming used to be one of these, but was proven (rather brilliantly) to be
in P by Leonid Khachiyan in 1979.

Notably, as far as I know, it hasn't been proven that NP-intermediate _even
exists_. It's quite possible that all problems believed to be in that category
are actually either P or NP-complete. If this proof is on the right track,
this possibility may be all the more likely.

------
leif
Context:
[http://en.wikipedia.org/wiki/Graph_isomorphism#Algorithmic_a...](http://en.wikipedia.org/wiki/Graph_isomorphism#Algorithmic_approach)

tl;dr: Graph isomorphism recognition is an old and well-known problem that is
in NP. It was not known whether it was NP-complete or in P, but this paper
suggests that it is in P. This is kind of Big, assuming the paper's correct.

------
mt2
Hi, I’m Michael Trofimov, the author of this preprint.Thank you for your
interest, you are welcome to ask your questions. More info about me you can
find, for example, in Intel site:
<http://software.intel.com/sites/blackbelt/hall_of_fame.php> (see Top
Community Masterminds section). Is it self-promotion? ;-)

Yes, I am not native English speaker, so I would be very thankful if somebody
will mark unclear statements in the paper and will suggest his/her version.
Very important: this is preprint only! It is not final result, the
investigation is under the progress. During discussion in other web-sites and
via emails: one minor bug in the proof and one bug in program had been found.
The program bug was fixed, also some time later I will upload the second
version of the paper to arXiv.

All the best, \-- Michael.

------
jfoutz
This makes me nervous: Further, without loss of generality of the task, we
will consider undirected connected graphs without loops.

Tree isomorphism is polynomial. It's not obvious to me how you cut the loops
out of two graphs and ensure you end up with similar trees. Quick scan didn't
show that in the paper. I guess i'll have to read in more detail after dinner.

~~~
leif
Loops as in "edges from vertex a to vertex a", not cycles, were discounted.

That's fine because you can ignore them until the end, and then just check
your vertices for any loops, in O(n) time.

~~~
jfoutz
ah! that makes a whole lot more sense. thanks.

------
pmiller2
I'm not buying this until it's peer reviewed by someone like Brendan McKay.

------
amichail
Check out the acknowledgments.

~~~
albertsun
What about them?

~~~
amichail
At least two of the people acknowledged are world famous algorithms
researchers.

But it's not clear how much of the paper they looked at and whether they think
it's correct.

~~~
_delirium
Yeah, I'd be curious what the opinions of other researchers in this area are.
There have been a number of debunked polynomial time algorithms for GI, so
it's one of those things people are always skeptical of; but that of course
doesn't mean this one is wrong too.

The author of the paper recently posted a blog comment in which he says he
submitted to a few journals, and complains that "referees did not find any
fatal error in my proof, but all the journals rejected this paper":
[http://rjlipton.wordpress.com/2009/05/04/an-approach-to-
grap...](http://rjlipton.wordpress.com/2009/05/04/an-approach-to-graph-
isomorphism/#comment-3156)

I'd take that as evidence that the referees at least didn't think it was a
successful proof. The comment is attached to an old blog post about the
problem from a well-known theoretical computer scientist (R.J. Lipton), but
since it was posted about a year after the initial post, it hasn't gotten any
replies.

