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

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(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.