Abstract
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|=m<|B|=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(mn o(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.
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Cabello, S., Giannopoulos, P., Knauer, C. (2006). On the Parameterized Complexity of d-Dimensional Point Set Pattern Matching. In: Bodlaender, H.L., Langston, M.A. (eds) Parameterized and Exact Computation. IWPEC 2006. Lecture Notes in Computer Science, vol 4169. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11847250_16
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DOI: https://doi.org/10.1007/11847250_16
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