A. Given two weighted complete graphs, determine whether they are isomorphic.

Since we can just assign 1/0 weights, this is a superproblem of GI, though I suspect it may not be any harder. But what if we restrict it, as

B. Assume every weight is the square root of an integer, and the triangle inequality holds.

Does the problem get any easier then? Surely it must get easier if we consider the real problem that made me think of these problems in the first place, which is

C. Given two n-dimensional simplices with integer coordinates, determine if they are congruent.

Or more specifically,

D. Given two n-dimensional simplices with vertices taken from the unit n-cube, determine if they are congruent.

Which is really just a necessary condition for the real problem I want solved,

E. Given two simplices as in F, determine whether any of the (2^n * n!) automorphisms of the n-cube map one simplex onto the other.

If anyone has insight into solving any of these problems in less than n! time, I'd love to hear about it.

1) modular decomposition — which produces a tree-structure in poly-time

2) tree-isomorphism testing — which I think should be poly-time?

The tricksie bit might be general graph isomorphism between pairs of prime modules. But prime modules may be sufficiently “weird” — weird enough so that simpler tests, e.g. spectral analysis, can determine graph isomorphism in poly-time.

Of course, I am just talking out of my a** But I’ve always wanted to do SOMETHING with modular decompositions: it’s the “nicest” poly-time graph algorithm which nobody seems to implement or use.

A. Given two weighted complete graphs, determine whether they are isomorphic.

Since we can just assign 1/0 weights, this is a superproblem of GI, though I suspect it may not be any harder. But what if we restrict it, as

B. Assume every weight is the square root of an integer, and the triangle inequality holds.

Does the problem get any easier then? Surely it must get easier if we consider the real problem that made me think of these problems in the first place, which is

C. Given two n-dimensional simplices with integer coordinates, determine if they are congruent.

Or more specifically,

D. Given two n-dimensional simplices with vertices taken from the unit n-cube, determine if they are congruent.

Which is really just a necessary condition for the real problem I want solved,

E. Given two simplices as in F, determine whether any of the (2^n * n!) automorphisms of the n-cube map one simplex onto the other.

If anyone has insight into solving any of these problems in less than n! time, I’d love to hear about it.