Approximability of constrained LCS

Abstract

The problem Constrained Longest Common Subsequence is a natural extension to the classical problem Longest Common Subsequence, and has important applications to bioinformatics. Given k input sequences A1,...,A k and l constraint sequences B1,...,Bl, C-LCS(k, l) is the problem of finding a longest common subsequence of A 1,...,Ak that is also a common supersequence of B 1,...,Bl. Gotthilf et al. gave a polynomial-time algorithm that approximates C-LCS(k,1) within a factor √m̂|∑|, where m̂ is the length of the shortest input sequence and |∑| is the alphabet size. They asked whether there are better approximation algorithms and whether there exists a lower bound. In this paper, we answer their questions by showing that their approximation factor √m̂|∑| is in fact already very close to optimal although a small improvement is still possible: 1 For any computable function f and any ε > 0, there is no polynomial-time algorithm that approximates C-LCS(k,1) within a factor f(|∑|) ·m̂1/2-ε unless NP = P. Moreover, this holds even if the constraint sequence is unary. 2 There is a polynomial-time randomized algorithm that approximates C-LCS (k,1) within a factor |∑|·O(√ OPT·log log OPT/log OPT) with high probability, where OPT is the length of the optimal solution, OPT ≥ m̂. For the problem over an alphabet of arbitrary size, we show that 3. For any ε > 0, there is no polynomial-time algorithm that approximates C-LCS(k,1) within a factor m̂1-ε unless NP = SP. 4. There is a polynomial-time algorithm that approximates C-LCS(k, 1) within a factor O(m̂/log m̂). We also present some complementary results on exact and parameterized algorithms for C-LCS(k,l). © 2010 Springer-Verlag.