A Quasipolynomial Time Algorithm for Graph Isomorphism: The Details | Hacker News Reader

A Quasipolynomial Time Algorithm for Graph Isomorphism: The Details

jeremykun.com

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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/2015/11/09/the-world-series-o...

https://rjlipton.wordpress.com/2015/11/11/a-fast-graph-isomo...

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

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.

If this really is the case (and I'm highly skeptical), switching from 4096 bit to megabit or gigabit keys would buy us atleast a few years.

But all previously encrypted messages would effectively become plain text to anyone with the foresight to save them.

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

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.

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.

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).

He has a fitting surname. Spanish speakers say "bye bye" as "babai".

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... :)

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

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.

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.

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?

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).

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!

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.

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.

Graph Isomorphism is indeed not used much in these field. Subgraph isomorphism (and variants of it) are very useful, and surprisingly (well, suprising to me), the techniques for solving graph isomorphism and subgraph isomorphism are entirely different.

On practical use of graph isomorphism is finding symmetries in programs -- (basically) build an AST, and then run graph isomorphism on that.

Also, the algorithms used for graph isomorphism are used (with modification) to solve a large range of group theory problems, including group intersection, so this might help with a large range of other group theory problems.

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

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

There are 140 comments in the blog of Scot Aaronson about this topic: 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.

This would mean that NP=Co-NP and that there is a polynomial certificate for every Co-NP problem.

That is what breaks P=NP for me and I just can't wrap my head around how that would be possible.

Your comment is a little hard to parse. We could live in a world where there are short proofs that an NP-Complete problem is unsolvable but those short proofs are hard to find (i.e. NP=co-NP but P!=NP) [1]. I think there are other consequences but that's an area I don't know that much about.

[1] http://cstheory.stackexchange.com/questions/8087/consequence...

Well, you've read my implication-arrow the wrong way. Since I don't believe that NP=co-NP (in any case), I can't believe that P=NP because Co-P=P. So if the collapse would happen (what the OP mentioned as a possibility), one implication would be NP=co-NP. I don't believe that so I can't believe in collapse.

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.

  > 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.

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.

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

I didn't think that GI was known to be in co-NP.

Scott Aaronson's answer on MO indicates it was known under "suitable popular derandomization assumptions" [1].

[1] http://mathoverflow.net/a/32312

Lol. The polynomial hierarchy would collapse if GI was NP-Complete, but all efforts to prove that have failed. If anything, the distance from P to NP is growing.

You can shrink an interval [a,b] by two methods [a,b'] with b'<b or [a',b] with a'> a, what you see is [a,b'] (NP-complete seem easier or lower in the difficulty scale), what I see is [a',b] that is P cover a lot more ground, I think P will cover all NP-complete problems and then the interval becomes a unique point.

If [P,NP-c] = [0,1] I see GI as 1/4, if GI is proven to be in P then I should say [P,NP-c] is now [1/4,1].

In order to reduce more that interval one should find a problem that people think has a probability of 0.5 to be in P and 0.5 to be in NP, if it is proven that problem is in P then the interval get reduced a lot more. Just trying to explain my perception of the intuitive complexity of algorithms.

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

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