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:
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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.
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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
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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.
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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.
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Jiang, M. (2010). Approximability of Constrained LCS. In: Cheong, O., Chwa, KY., Park, K. (eds) Algorithms and Computation. ISAAC 2010. Lecture Notes in Computer Science, vol 6507. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17514-5_16
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DOI: https://doi.org/10.1007/978-3-642-17514-5_16
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