The intractability of computing the Hamming distance

Given a string x and a language L, the Hamming distance of x to L is the minimum Hamming distance of x to any string in L. The edit distance of a string to a language is analogously defined. First, we prove that there is a language in $AC^0$ such that both Hamming and edit distance to this language are hard to approximate; they cannot be approximated with factor $O(n^{(1/3)-ε})$, for any ε>0, unless P=NP (n denotes the length of the input string). Second, we show the parameterized intractability of computing the Hamming distance. We prove that for every t∈N there exists a language in $AC^0$ for which computing the Hamming distance is W[t]-hard. Moreover, there is a language in P for which computing the Hamming distance is WP-hard. Then we show that the problems of computing the Hamming distance and of computing the edit distance are in some sense equivalent by presenting approximation ratio preserving reductions from the former to the latter and vice versa. Finally, we define HamP to be the class of languages to which the Hamming distance can efficiently, i.e. in polynomial time, be computed. We show some properties of the class HamP. On the other hand, we give evidence that a characterization in terms of automata or formal languages might be difficult.