The Parameterized Complexity of Geometric Graph Isomorphism

Abstract

AbstractWe 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 \(\mathbb {Q}^2\). KeywordsGeometric IsomorphismShortest VectorColor Class SizeComputational Geometry LiteratureHypergraph IsomorphismThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.