On finding minimal, maximal, and consistent sequences over a binary alphabet
Theoretical Computer Science
On the Approximation of Shortest Common Supersequencesand Longest Common Subsequences
SIAM Journal on Computing
The Complexity of Some Problems on Subsequences and Supersequences
Journal of the ACM (JACM)
The constrained longest common subsequence problem
Information Processing Letters
A simple algorithm for the constrained sequence problems
Information Processing Letters
Combined super-/substring and super-/subsequence problems
Theoretical Computer Science
New efficient algorithms for the LCS and constrained LCS problems
Information Processing Letters
Constrained LCS: Hardness and Approximation
CPM '08 Proceedings of the 19th annual symposium on Combinatorial Pattern Matching
Variants of constrained longest common subsequence
Information Processing Letters
On the generalized constrained longest common subsequence problems
Journal of Combinatorial Optimization
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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 m@?|@S|, where m@? is the length of the shortest input sequence and |@S| 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 m@?|@S| is in fact already very close to optimal although a small improvement is still possible:1.For any computable function f and any @e0, there is no polynomial-time algorithm that approximates C-LCS(k,1) within a factor f(|@S|)@?m@?^1^/^2^-^@e 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 |@S|@?O(OPT@?loglogOPT/logOPT) with high probability, where OPT is the length of the optimal solution, OPT=0, there is no polynomial-time algorithm that approximates C-LCS(k,1) within a factor m@?^1^-^@e unless NP=P. 4.There is a polynomial-time algorithm that approximates C-LCS(k,1) within a factor O(m@?/logm@?). We also present some complementary results on exact and parameterized algorithms for C-LCS(k,1).