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- Path: sparky!uunet!munnari.oz.au!manuel.anu.edu.au!tyl.anu.edu.au!not-for-mail
- From: bdm@cs.anu.edu.au (Brendan McKay)
- Newsgroups: comp.theory
- Subject: Re: randomized-hash graph-iso algorithm - more discussion
- Date: 1 Jan 1993 16:24:02 +1100
- Organization: Australian National University
- Lines: 120
- Distribution: inet
- Message-ID: <1i0khiINN5vh@tyl.anu.edu.au>
- References: <C0402I.L7p@research.canon.oz.au>
- NNTP-Posting-Host: 150.203.23.70
-
- naylor@research.canon.oz.au (William Naylor) writes:
- [...William's article heavily editted to save space...]
-
- > 2) bdm@cs.anu.edu.au (Brendan McKay) claims that P algorithms for
- > graph-iso are already known for
-
- > a) graphs with bounded vertex degree
- > b) graphs of bounded genus
-
- The key papers on this appeared in the Proceedings of the Foundations
- of Computer Science conferences, around 1980-3. The authors are
- Babai, Luks, Miller, Filotti, Meyer and a few others. A common
- generalisation of (a) and (b) can be found in
-
- G. Miller, Information and Control 56 (1983) 1-20.
-
- > 3) bdm@cs.anu.edu.au (Brendan McKay) claims that my algorithm does
- > not work properly on large classes of graphs which are of interest in
- > various mathematical work. If what he says is true, this
- > weakens my argument in favor of the Gismondi-Swart strategy.
-
- > Clearly, the algorithm I gave does not work on graphs where
- > all vertices have the same degree. This problem can be easily patched
- > (see below).
-
- It also doesn't work for many interesting graphs where the degrees are
- not all the same. For example, take a regular graph and subdivide
- each edge with a new vertex. It won't work for that either.
- Other more direct examples exist too; see below.
-
- > Can somebody out there point out some classes of graphs
- > for which my algorithm usually fails, which cannot be easily patched?
-
- I will send some by e-mail.
-
- > 4) My original argument still seems correct.
-
-
- >====================================================================
- > MY ORIGINAL POST
- >====================================================================
-
- [deleted by bdm]
-
- >====================================================================
- > DISCUSSION of a POST by bdm@cs.anu.edu.au (Brendan McKay)
- >====================================================================
-
- >In article 5857 of comp.theory, bdm@cs.anu.edu.au (Brendan McKay) wrote
- >a detailed response to my post. I have attempted to clarify several
- >points with him by e-mail, but I have received no reply. Therefore,
- >I must discuss his post without benefit of discussion with him.
-
- Sorry.
-
- >In article 5857 of comp.theory, bdm@cs.anu.edu.au (Brendan McKay) writes:
-
- >>>The relevance of my algorithm to the discussion of the paper of
- >>>Swart et. al. is this:
- >>>the existence of this successful randomized algorithm for graph isomorphism,
-
- >>There is no evidence for such an algorithm. Yours only works for
- >>restricted classes of graphs.
-
- >The original algorithm I presented fails to distinguish graphs of the
- >same order with vertices all having the same degree. This is
- >a very small fraction of all possible graphs. Anyway, this problem can
- >be patched (see below).
-
- >Does anybody out there see any classes of graphs that can fool my
- >(patched) algorithm?
-
- Yes, there are many. I will send you some by e-mail.
- The most natural examples are strongly-regular graphs and the
- incidence graphs of balanced designs. The latter are not
- necessarily regular. These are standard test-cases for
- isomorphism programs. I have some others, maybe even harder.
-
- >>Methods that are as strong or stronger date back to the 1960s.
- >>Modern approaches supplement invariant computations with backtracking.
- >>Several programs exist that can handle thousands of vertices
- >>(one is mine, enquiries welcome).
-
- >Could somebody post some references to these algorithms?
-
- One early algorithm was due to Corneil and Gotlieb. (Sorry, I'm
- at home and can't look up the reference.) Their algorithm is
- based on a conjecture later proved by false by Rudi Mathon.
- Even though it is deterministic, it is stronger than even your
- patched algorithm in the sense that if their's fails, yours
- will fail for every hash function. The reason is this:
-
- Take a graph G and a vertex v. Let V1,V2,...,Vk be the coarsest
- partition of the vertex set of G such that
- (i) V1 = {v}
- (ii) If C is any cell, and x,y are two vertices in the same
- cell (possibly C), then x and y are adjacent to the same
- number of vertices in C.
- Now define the "quotient matrix" M of order k*k:
- M[i,j] = number of y in Vj adjacent to each fixed x in Vi.
- (I didn't specify the order of the cells V2,...,Vk, but these
- can easily be put into a canonical order using M. Think about it.)
- Now, the value your patched algorithm gets for this particular v
- distingushed depends only on M and not on the finer structure
- of G. Any two graphs which have the same set of M's (one per v)
- will be indistinguisable by your method. As an example, any
- strongly regular graph was the same M for every vertex and
- that M only depends on the graph parameters.
-
- Advertisement: I distribute a deterministic algorithm called nauty.
- It computes canonical labellings and generators for the automorphism
- group. I think it's pretty good. It is free subject to weak
- conditions of use. For further information, send a note to me
- that includes your complete (paper and electronic) address.
-
- >My (unpatched) program can work on graphs with 50,000 vertices in 5-10 minutes.
-
- >Will Naylor net: naylor@research.canon.oz.au
-
- Brendan.
-