The Longest Common Subsequence Problem with Crossing-Free Arc-Annotated Sequences

An arc-annotated sequence is a sequence, over a given alphabet, with additional structure described by a set of arcs, each arc joining a pair of positions in the sequence. As a natural extension of the longest common subsequence problem, Evans introduced the LONGEST ARC-PRESERVING COMMON SUBSEQUENCE (LAPCS) problem as a framework for studying the similarity of arc-annotated sequences. This problem has been studied extensively in the literature due to its potential application for RNA structure comparison, but also because it has a compact definition. In this paper, we focus on the nested case where no two arcs are allowed to cross because it is widely considered the most important variant in practice. Our contributions are three folds: (i) we revisit the nice NP-hardness proof of Lin et al. for LAPCS(NESTED, NESTED), (ii) we improve the running time of the FPT algorithm of Alber et al. from O(3.31(k1)+(k2)n) to O(3(k1+k2)n), where resp. k(1) and k(2) deletions from resp. the first and second sequence are needed to obtain an arc-preserving common subsequence, and (iii) we show that LAPCS(STEM, STEM) is NP-complete for constant alphabet size.