On the Parameterized Complexity of d-Dimensional Point Set Pattern Matching

Deciding whether two n-point sets A,B ∈ R 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)-time, unless SNP ⊂ DTIME(2). 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.