Graph Isomorphism Hard. Yeah, Right.
Complexity theorists talk about graph isomorphism as if it's a hard problem, since there's no known polynomial time algorithm. I'm extremely skeptical, since although I can't prove a particular algorithm is polynomial, there are ones which seem extremely hard to fool.
Feeding my skepticism is that the existing packages don't give any instances of isomorphism which they claim to be hard. They give examples of isomorphic graphs which are tricky to find mappings for, but no non-isomorphic graphs which are supposedly hard to identify as such. Since polynomial algorithms for determining isomorphism can generally be turned into polynomial ones for finding a mapping, (by being used as a method of pruning an exhaustive search), this seems tantamount to an admission that there are no hard instances.
The obvious algorithms for checking graph isomorphism all amount to finding some value for each node, then sorting those values, and comparing the sorted lists. Methods for coming up with these values inclued the following (all of these assume there are no separate islands in the graph, the case where there are islands is easy to deal with) -
- Calculate the minimum number of hops to each other node, and sort that list.
- Calculate the fraction of the time you'd spend at the given node if you did a neverending random walk around the graph along edges.
- Assume that you'll drop all other nodes with some probability. Calculate the value for that probability which makes the expected number of remaining reachable nodes equal to half the total. This can be computed via sampling, which requires some randomness and uncertainty, but is still polynomial.
All of these, of course, can be used jointly. Very symmetrical graphs can sometimes look homogenous to the above, but that can be fixed by for each node calculating the values for all other nodes if that node were removed and sorting the results. This can of course be done for multiple iterations, although each iteration multiplies the runtime by the number of nodes. Different iteration levels can, of course, be used jointly.
Given the above extremely powerful isomorphism checking techniques, I find it dubious that there are any hard instances of graph isomorphism. It looks rather like finding an odd perfect number - we have no proof that it doesn't exist, but the known constraints on it are quite ludicrous.
Note that for all of the above I'm talking about the problem of determining whether a mapping exists, or finding a single one, not the problem of finding all mappings. This is the problem which the interesting zero knowledge proof applies to, I'm distinctly uninterested in automorphisms.