The Parameterized Complexity of Geometric Graph Isomorphism

Abstract

We study the parameterized complexity of Geometric Graph Isomorphism (Known as the Point Set Congruence problem in computational geometry): given two sets of n points A and B with rational coordinates in k-dimensional euclidean space, with k as the fixed parameter, the problem is to decide if there is a bijection $$\pi :A \rightarrow B$$ź:AźB such that for all $$x,y \in A$$x,yźA, $$\Vert x-y\Vert = \Vert \pi (x)-\pi (y)\Vert $$źx-yź=źź(x)-ź(y)ź, where $$\Vert \cdot \Vert $$ź·ź is the euclidean norm. Our main result is the following: We give a $$O^*(k^{O(k)})$$Oź(kO(k)) time (The $$O^*(\cdot )$$Oź(·) notation here, as usual, suppresses polynomial factors) FPT algorithm for Geometric Isomorphism. This is substantially faster than the previous best time bound of $$O^*(2^{O(k^4)})$$Oź(2O(k4)) for the problem (Evdokimov and Ponomarenko in Pure Appl Algebra 117---118:253---276, 1997). In fact, we show the stronger result that even canonical forms for finite point sets with rational coordinates can also be computed in $$O^*(k^{O(k)})$$Oź(kO(k)) time. We also briefly discuss the isomorphism problem for other $$l_p$$lp metrics. Specifically, we describe a deterministic polynomial-time algorithm for finite point sets in $$\mathbb {Q}^2$$Q2.