# Testing approximate symmetry in the plane is NP-hard

## Abstract

Let *A* be a set of *n* points in the Euclidean plane. We would like to know its symmetry group. Since in a computer points can be represented by rational coordinates only, the problem in the original way degenerates to a trivial case. This is why we introduce the notion of *approximate* symmetry. In our model we consider a fixed tolerance factor *ɛ* and ask for the set *A*′ consisting of *n* points of the *ɛ*-neighborhoods of the points in *A*, each point of *A*′ belonging to a different point of *A*, with maximal symmetry group. A similar concept has been developed for approximate congruence recently and several efficient algorithms are known for that related problem. In this paper we obtain the surprising result that the decision problem for approximate symmetry is NP-hard.

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## References

- [AMWW88]Helmut Alt / Kurt Mehlhorn / Hubert Wagener / Emo Welzl:
*Congruence, Similarity, and Symmetries of Geometric Objects*, Discrete Comput. Geom. 3 pp.237–256, 1988Google Scholar - [Ata84]Mikhail J. Atallah:
*Checking Similarity of Planar Figures*, International J. Comp. Inf. Science 13 pp. 279–290, 1984Google Scholar - [Ata85]
- [Atk87]M. D. Atkinson:
*An Optimal Algorithm for Geometrical Congruence*, J. Algorithms 8 pp. 159–172, 1987Google Scholar - [DF86]M. E. Dyer / A. M. Frieze:
*Planar 3DM is NP-Complete*, J. Algorithms 7 pp. 174–184, 1986Google Scholar - [GJ79]Michael R. Garey / David S. Johnson:
*Computers and Intractability — A Guide to the Theory of NP-Completeness*, Freeman, 1979Google Scholar - [H86]P. T. Highnam:
*Optimal Algorithms for Finding the Symmetries of a Planar Point Set*, Information Processing Letters 22 pp. 219–222, 1986Google Scholar - [I88]Sebastian Iwanowski:
*Linear Time Algorithms for Testing Approximate Congruence in the Plane*, Proceedings of the Workshop on Graph Theoretic Concepts 88, Lecture Notes in Computer Science 344, pp. 213–228Google Scholar - [L82]David Lichtenstein:
*Planar Formulae and their Uses*, SIAM J. Comp. 11,2 pp.329–343, 1982Google Scholar - [RT86]Pierre Rosenstiehl / Robert Tarjan:
*Rectilinear Planar Layouts and Bipolar Orientations of Planar Graphs*, Discrete Comput. Geom. 1 pp.343–353, 1986Google Scholar