Approximating optimal solution structure with edit distance and its applications

An alternative notion of approximation arising in cognitive psychology, bioinformatics and linguistics is that of computing a solution which is structurally close to an optimal one. That is, an approximate solution is considered good if its distance from an optimal solution is small, for a distance measure such as Hamming distance or edit distance. There has been a modicum of work on approximating solution structure according to an arbitrary solution distance function [HMvRW07,vRW12]; several papers addressed the complexity of approximating a witness to an NP problem to within bounded Hamming distance [GHLP99,KS99,FLN00,SY13]. Most of these results are strong lower bounds. In this paper, we extend Hamming distance inapproximability results of Sheldon and Young [SY13] to a setting where a solution is considered good if it is within a given edit distance bound from some optimal solution. We show several inapproximability results for this measure, in particular extending [SY13] n/2− n inapproximability ∀ > 0 with respect to Hamming distance to edit distance for SAT and VertexCover problems. Finally, we show that these lower bounds for both edit distance and Hamming distance apply to two practical problems: Longest Common Subsequence and Weighted Sentence Alignment.