On the parameterized complexity of d-dimensional point set pattern matching

Deciding whether two n-point sets A, B ∈ R d 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° (d) -time, unless SNP C DTIME(2° (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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