# Approximating LCS and Alignment Distance over Multiple Sequences

We study the problem of aligning multiple sequences with the goal of finding an alignment that either maximizes the number of aligned symbols (the longest common subsequence (LCS)), or minimizes the number of unaligned symbols (the alignment distance (AD)). Multiple sequence alignment is a well-studied problem in bioinformatics and is used to identify regions of similarity among DNA, RNA, or protein sequences to detect functional, structural, or evolutionary relationships among them. It is known that exact computation of LCS or AD of m sequences each of length n requires Θ(n^m) time unless the Strong Exponential Time Hypothesis is false. In this paper, we provide several results to approximate LCS and AD of multiple sequences. If the LCS of m sequences each of length n is λ n for some λ∈ [0,1], then in Õ_m(n^⌊m/2⌋+1) time, we can return a common subsequence of length at least λ^2 n/2+ϵ for any arbitrary constant ϵ >0. It is possible to approximate the AD within a factor of two in time Õ_m(n^⌈m/2⌉). However, going below-2 approximation requires breaking the triangle inequality barrier which is a major challenge in this area. No such algorithm with a running time of O(n^α m) for any α < 1 is known. If the AD is θ n, then we design an algorithm that approximates the AD within an approximation factor of (2-3θ/16+ϵ) in Õ_m(n^⌊m/2⌋+2) time. Thus, if θ is a constant, we get a below-two approximation in Õ_m(n^⌊m/2⌋+2) time. Moreover, we show if just one out of m sequences is (p,B)-pseudorandom then, we get a below-2 approximation in Õ_m(nB^m-1+n^⌊m/2⌋+3) time irrespective of θ.

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