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 \(\pi :A \rightarrow B\) 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 result is 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. This is substantially faster than the previous best time bound of \(O^*(2^{O(k^4)})\) 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 \(O^*(k^{O(k)})\) time. We also briefly discuss the isomorphism problem for other \(l_p\) metrics. Specifically, we describe a deterministic polynomial-time algorithm for finite point sets in \(\mathbb {Q}^2\).
Similar content being viewed by others
Notes
To the best of our knowledge, this paper appears to be unknown in the Computational Geometry literature.
There is a standard reduction that reduces hypergraph isomorphism for n-vertex and m-edge hypergraphs to bipartite graph isomorphism on \(n+m\) vertices. However, the point sets thus obtained will be in \(\mathbb {Q}^{n+m}\) and m could be much larger than n. The aim is to obtain point sets in as low a dimension as possible.
References
Evdokimov, S.A., Ponomarenko, I.N.: On the geometric graph isomorphism problem. Pure Appl Algebra 117–118, 253–276 (1997)
Akutsu, T.: On determining the congruence of point sets in d dimensions. Comput Geom 9(4), 247–256 (1998)
Braß, P., Knauer, C.: Testing the congruence of d-dimensional point sets. Int. J. Comput. Geom. Appl. 12, 115–124 (2002)
Alt, H., Mehlhorn, K., Wagener, H., Welzl, E.: Congruence, similarity, and symmetries of geometric objects. Discrete Comput. Geom. 3(1), 237–256 (1988)
Papadimitriou, Christos H., Safra, S.: The complexity of low-distortion embeddings between point sets. In: Proceedings of the 16th Annual Symposium on Discrete Algorithms, pp. 112–118 (2005)
Micciancio, D., Voulgaris, P.: A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations. SIAM J. Comput. 42(3), 1364–1391 (2013)
Haviv, I., Regev, O.: On the lattice isomorphism problem. In: Proceedings of the 25th Annual Symposium on Discrete Algorithms, pp. 391–404 (2014)
Babai, László, Luks, E.M.: Canonical labeling of graphs. In: Proceedings of the 15th Annual Symposium on Theory of Computing, pp. 171–183 (1983)
Furst, M.L., Hopcroft, J.E., Luks, E.M.: Polynomial-time algorithms for permutation groups. In: Proceedings of the 21st Annual Symposium on Foundations of Computer Science Conference, pp. 36–41 (1980)
Schrijver, A.: Theory of Integer and Linear Programming. Wiley-Interscience series in discrete mathematics and optimization, Hoboken (1998)
Corbalan, A. G., Mazon, M., Recio, T.: About Voronoi diagrams for strictly convex distances. In: Proceedings of the 9th European Workshop on Computational Geometry, pp. 17–22 (1993)
Luks, E.M.: Hypergraph isomorphism and structural equivalence of Boolean functions. In: Proceedings of the 31st Annual Symposium on Theory of Computing, pp. 652–658 (1999)
Acknowledgments
We are grateful to the referees for their detailed and helpful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Arvind, V., Rattan, G. The Parameterized Complexity of Geometric Graph Isomorphism. Algorithmica 75, 258–276 (2016). https://doi.org/10.1007/s00453-015-0024-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-015-0024-8