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 A ₁,...,A _k and l constraint sequences B ₁,...,B _l , C-LCS(k, l) is the problem of finding a longest common subsequence of A ₁,...,A _k that is also a common supersequence of B ₁,...,B _l . Gotthilf et al. gave a polynomial-time algorithm that approximates C-LCS(k,1) within a factor $\sqrt{\hat m |\Sigma|}$, where $\hat m$ is the length of the shortest input sequence and |Σ| 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 $\sqrt{\hat m |\Sigma|}$ 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-LCSk,1 within a factor $f(|\Sigma|)\cdot \hat m^{1/2-\epsilon}$ 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 $|\Sigma| \cdot O(\sqrt{{\rm OPT}\cdot\log\log{\rm OPT}/\log{\rm OPT}})$ with high probability, where OPT is the length of the optimal solution, ${\rm OPT} \le \hat m$.

  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 $\hat m^{1-\epsilon}$ unless NP = SP.

• 4. There is a polynomial-time algorithm that approximates C-LCS(k, 1) within a factor $O(\hat m/\log\hat m)$.

We also present some complementary results on exact and parameterized algorithms for C-LCS(k,l).

Supported in part by NSF grant DBI-0743670.