
A Quasipolynomial Time Algorithm for Graph Isomorphism: The Details - Schiphol
http://jeremykun.com/2015/11/12/a-quasipolynomial-time-algorithm-for-graph-isomorphism-the-details/
======
s-phi-nl
Prof. Richard Lipton of Georgia Tech has an interesting series of blog posts
on this result. See:

[https://rjlipton.wordpress.com/2015/11/04/a-big-result-on-
gr...](https://rjlipton.wordpress.com/2015/11/04/a-big-result-on-graph-
isomorphism/)

[https://rjlipton.wordpress.com/2015/11/09/the-world-
series-o...](https://rjlipton.wordpress.com/2015/11/09/the-world-series-of-
complexity-theory/)

[https://rjlipton.wordpress.com/2015/11/11/a-fast-graph-
isomo...](https://rjlipton.wordpress.com/2015/11/11/a-fast-graph-isomorphism-
algorithm/)

I expect he will post more as he gets more information.

~~~
isomorphic
Conjecture from the first link: "Raising questions about other problems. This
a surprising result. Is a similar result for factoring around the corner?
[...] Placing factoring in this complexity class would be a huge difficulty
for cryptography."

If factoring is indeed in the quasi-polynomial class, the above may well be
the understatement of the decade.

~~~
moyix
Yeah. But as Scott Aaronson points out, GI has a very different feel to
factoring. A randomly chosen number will be hard to factor, but it's actually
quite hard to come up with examples of GI that are hard:

> But then again, in practice, graph isomorphism has already been “basically
> in P” for decades! If you have two large graphs for which you actually need
> to know whether they’re isomorphic, just download NAUTY and run it.

> This contrasts with the case of factoring, for which I’d personally say that
> it remains much less clear whether it should or shouldn’t be in P.

[http://www.scottaaronson.com/blog/?p=2521#comment-889824](http://www.scottaaronson.com/blog/?p=2521#comment-889824)

~~~
isomorphic
Well, we're resting at least a trillion-USD economy on that "much less clear";
probably more. Given that, bad actors are properly incentivized to work on
factoring, and we can take comfort that we haven't yet heard any evidence yet
of a (quasi-) polynomial solution. That is assuming someone couldn't keep
something like that a secret.

------
ebola1717
I learned linear algebra and spectral graph theory from Babai, and he also
taught a notoriously difficult combinatorics class. It was a rare privilege to
be taught by someone who was doing brilliant research and also cared deeply
about undergraduate students.

~~~
kevinalexbrown
I sat in on a couple of those classes. At the beginning of the class there
were 50-100 people registered. At the end, there were 6 (including graduate
students).

~~~
idorosen
50 -> ~10 students standing was my experience with laci's algorithms class.
He's still an excellent instructor, though. Willingness to do hours of
background and follow-up reading outside of class was a requirement if you
wanted to get a deep understanding of the material, and it pays off well. (He
was very approachable though, not at all the research professor stereotype.)

(HatLab, what a small world... :)

------
joewalker
In Our Time (BBC R4) did a recent episode called "P vs NP" [1] which talked
about graph isomorphism, and might serve as an introduction to this.

There's an MP3 download at the link. I don't think it's UK only.

[1]:
[http://www.bbc.co.uk/programmes/b06mtms8](http://www.bbc.co.uk/programmes/b06mtms8)

------
gone35
Very nice write-up, Jeremy! Too bad you won't be able to make it to the second
talk.

Also nice disclaimer at the end:

 _At the time of this writing, Babai’s work has not been peer reviewed, and my
understanding of his lectures has large gaps and may be faulty. Do not put
your life in danger based on information in this post._

~~~
msie
It really sucks that he can't make the second talk. It was a really good
write-up and finding someone who both communicates well and can understand the
material is tough.

------
eveningcoffee
_I also hear that some people use graph isomorphism to compare files, do
optical character recognition, and analyze social networks, but it seems
highly probable to me that GI is not the central workhorse (or even a main
workhorse) in these fields._

Can somebody in HN more familiar with this add a comment about this?

~~~
jules
Maybe there is somebody who uses GI to do those things, but that is not a
standard method and likely a bad idea. For comparing files the standard method
is just a diff based on Levenshtein distance, and the standard methods for
character recognition are various forms of machine learning (e.g. SVM, NN).

~~~
OJFord
Social networks sounds like a likely application though.

Consider "Graph A is Alice's network of friends; B is Bob's. Do they know
exactly and only the same people?"

That might be slightly contrived. But it could be used to indicate how
introverted a certain community is!

~~~
bmm6o
In your hypothetical case the vertices would be labeled by friends' names or
ids. So you only have to check whether the obvious isomorphism applies i.e.
that the graphs are equivalent as sets.

~~~
OJFord
Ah, yes that's true of course - I clearly wasn't thinking..

Well, then I can't even contrive an example! Still, it's interesting; I wish I
had the background to understand it better.

------
crb002
Lemmas aside, how is this different than only checking repeated prime cycles
in conjunction with one of the existing partition algorithms like NAUTY? Laci
personally rejected my note submitted to Information Processing Letters saying
the lower bound formula was "trivial",
[https://oeis.org/A186202](https://oeis.org/A186202)

------
yeukhon
For those who have really difficult time to understand this... I find
[http://news.sciencemag.org/math/2015/11/mathematician-
claims...](http://news.sciencemag.org/math/2015/11/mathematician-claims-
breakthrough-complexity-theory) opens a small door. I still have tons of
questions, probably due to my failure of completing my complexity class in
undergraduate.

------
lovboat
There are 140 comments in the blog of Scot Aaronson about this topic:
[http://www.scottaaronson.com/blog/?p=2521#comments](http://www.scottaaronson.com/blog/?p=2521#comments)

Personal Opinion: Perhaps the Complexity Hierarchy is going to collapse since
intuition is not so clear and perhaps the distance from P to NP is shrinking.

~~~
abetusk
I don't know why you would think this. There was strong empirical evidence (in
my opinion) for thinking Graph Isomorphism was easy. NAUTY (and SAUCY) were
good practical implementations that found solutions efficiently (except for
some harder class of graphs, maybe). GI was known to be in NP and co-NP which
is usually a red flag that the problem is not NP-Complete. For example, linear
programming, integer polynomial factorization and primality testing were all
in this NP and co-NP region (also discrete logarithm but that's still open).

Though extremely informative, there's no reason that GI is polynomial (or
pseudo polynomial) would give serious reason to believe that P is anywhere
near NP.

~~~
lmkg

      > NAUTY (and SAUCY) were good practical implementations 
      > that found solutions efficiently (except for some 
      > harder class of graphs, maybe).
    

There are good practical implementations of SAT solvers as well (except
degenerate cases), even though SAT is the canonical NP-Complete problem.

~~~
abetusk
Fair enough, though I think the problem space is substantially different
(though I didn't mention it above). It's much easier to make (random) 3-SAT
difficult instances whereas for GI you needed highly structured graphs.
Apparently Johnson graphs were one of the classes of graphs that caused people
problems in GI.

~~~
wolfgke
There are other NP-complete problems, where really hard instances are a lot
more rare.

------
j2kun
Update: a video of the first talk has been posted on Laci's website:
[http://people.cs.uchicago.edu/~laci/2015-11-10talk.mp4](http://people.cs.uchicago.edu/~laci/2015-11-10talk.mp4)

