The intractability of computing the Hamming distance

Abstract

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 AC0 such that both Hamming and edit distance to this language are hard to approximate; they cannot be approximated with a factor O(n1/3-∈), for any ∈ > 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 AC0 for which computing the Hamming distance is W[t]-hard. Moreover, there is a language in P for which computing the Hamming distance is W[P]-hard. Finally, we show that the problems of computing the Hamming distance and of computing the edit distance are in some sense equivalent by presenting reductions from the former to the latter and vice versa. © Springer-Verlag Berlin Heidelberg 2003.