
An Annoying Open Problem  - wglb
http://rjlipton.wordpress.com/2011/10/08/an-annoying-open-problem/
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jmmcd
Is there a layman-readable proof or proof-sketch of the reduction from Group
Isomorphism to Graph Isomorphism? Is the method constructive, so that every
group reduces to a corresponding graph in an understandable way?

I have a somewhat cranky reason for asking. In evolutionary algorithms, a
_mutation_ operator is one which transforms an existing genome into a single
new one. One can study the behaviour of an operator by drawing an edge from
old to new genomes, for every possible genome, and studying the resulting
graph. A _crossover_ operator takes two parent genomes to produce a new one.
No-one really has a satisfactory method of studying crossover analogous to
that for mutation. The problem is we need edges to lead from pairs of nodes to
single nodes. So kind of like hyperedges but not exactly. I've been hoping for
a while that there is an answer in existing graph theory.

Crossover is also kind of like a group operation, but again not exactly. So
that's why the idea of mapping a group to a graph is interesting.

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tgflynn
You might want to look at the definition of Cayley graphs :
<http://en.m.wikipedia.org/wiki/Cayley_graph>

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gms
I was really surprised to learn that this is still an open problem.

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veyron
There are a lot of simple heuristics (like score sequences) which get close
but can't handle a lot of annoying corner cases.

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dr_rezzy
Zeke was my theory and algorithms prof in college. Kind of hard to wrap your
head around these sort of problems.

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mckoss
Naive question. Can the cycle structure of two groups be used to optimize
comparison, just as the Poles used in WWII to catalog all the possible rotor
configurations of an Enigma machine?

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anonymoushn
Yes, but this won't help you with the general case.

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tgflynn
Does anyone know of any good test cases for GI ?

I have a GI algorithm or heuristic but I don't know how to evaluate it
compared to the current algorithms.

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anonymoushn
Is graph isomorphism known to be in P or not in P? The article seems to
suggest that it is known not to be in P, but I haven't heard anything about
this.

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veyron
Ladner's theorem established that if NP != P, then there exists "NP-easy"
problems which are in NP but not in P.

It has been shown that if P != NP, then GI is in that valley.

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bdhe
_It has been shown that if P != NP, then GI is in that valley._

Could you point to a reference? Is this true even if P != NP, but NP = coNP?

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thesteamboat
This pdf (<http://www.cs.princeton.edu/theory/complexity/ipchap.pdf>) which is
a draft chapter from a book by Arora and Barak, shows that if GI is NP-
complete, then the polynomial hierarchy collapses to the second level.
Collapse to the first level is when NP=coNP. So this is slightly weaker even
than that.

