Congruence, similarity and symmetries of geometric objects
Discrete & Computational Geometry - ACM Symposium on Computational Geometry, Waterloo
On the complexity of some geometric problems in unbounded dimension
Journal of Symbolic Computation
Geometric pattern matching under Euclidean motion
Computational Geometry: Theory and Applications - Special issue: computational geometry, theory and applications
On determining the congruence of point sets in d dimensions
Computational Geometry: Theory and Applications
Linear Programming in Linear Time When the Dimension Is Fixed
Journal of the ACM (JACM)
Exact Point Pattern Matching and the Number of Congruent Triangles in a Three-Dimensional Pointset
ESA '00 Proceedings of the 8th Annual European Symposium on Algorithms
The Earth Mover's Distance under Transformation Sets
ICCV '99 Proceedings of the International Conference on Computer Vision-Volume 2 - Volume 2
The complexity of low-distortion embeddings between point sets
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Tight lower bounds for certain parameterized NP-hard problems
Information and Computation
Efficient approximation schemes for geometric problems?
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Matching point sets with respect to the earth mover’s distance
ESA'05 Proceedings of the 13th annual European conference on Algorithms
The complexity of geometric problems in high dimension
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
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Deciding whether two n-point sets A, B ∈ℝd are congruent is a fundamental problem in geometric pattern matching. When the dimension d is unbounded, the problem is equivalent to graph isomorphism and is conjectured to be in FPT. When |A|=mB|=n, the problem becomes that of deciding whether A is congruent to a subset of B and is known to be NP-complete. We show that point subset congruence, with d as a parameter, is W[1]-hard, and that it cannot be solved in O(mno(d))-time, unless SNP⊂DTIME(2o(n)). This shows that, unless FPT=W[1], the problem of finding an isometry of A that minimizes its directed Hausdorff distance, or its Earth Mover's Distance, to B, is not in FPT.