The Parameterized Complexity of Geometric Graph Isomorphism

Abstract

We study the parameterized complexity of Geometric Graph Isomorphism (It is known as Point Set Congruence problem in computational geometry): given two sets of $n$ points $A, B\subset \mathbb {Q}^k$ in $k$-dimensional euclidean space, with $k$ as the fixed parameter, the problem is to decide if there is a bijection $\pi :A \rightarrow B$ such that for all $x,y \in A$, $\Vert x-y\Vert = \Vert \pi (x)-\pi (y)\Vert$, where $\Vert \cdot \Vert$ is the euclidean norm. Our main results are the following:

• We give a $O^*(k^{O(k)})$ time (The $O^*(\cdot )$ notation here, as usual, suppresses polynomial factors) FPT algorithm for Geometric Isomorphism. In fact, we show the stronger result that canonical forms for finite point sets in $\mathbb {Q}^k$ can also be computed in $O^*(k^{O(k)})$ time. This is substantially faster than the previous best time bound of $O^*(2^{O(k^4)})$ for the problem [1].

• We also briefly discuss the isomorphism problem for other $l_p$ metrics. We describe a deterministic polynomial-time algorithm for finite point sets in $\mathbb {Q}^2$.