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The Parameterized Complexity of Geometric Graph Isomorphism

  • Vikraman Arvind
  • Gaurav RattanEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8894)

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 \(\mathbb {Q}^2\).

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Institute of Mathematical SciencesChennaiIndia

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