Testing approximate symmetry in the plane is NP-hard
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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