The Complexity of Geometric Graph Isomorphism

We study the complexity of Geometric Graph Isomorphism, in l2 and other lp metrics: given two sets of n points A,B ⊂ Q in k-dimensional euclidean space the problem is to decide if there is a bijection π : A→ B such that for all x, y ∈ A, d(x, y) = d(π(x), π(y)), where d is the distance given by the metric. Our results are the following: • We describe algorithms for isomorphism and canonization of point sets with running time kpoly(nM), where M upper bounds the binary encoding length of numbers in the input. This is faster than previous algorithms for the problem. • From a complexity-theoretic perspective, we show that the problem is in NP[O(k log k)]∩ coIP[O(k log k)], where O(k log k) respectively bounds the nondeterministic witness length in NP and message length in the 2-round IP protocol. • We also briefly discuss the isomorphism problem for other lp metrics. We describe a deterministic logspace algorithm for point sets in Q.