Approximability of Constrained LCS

Given two input sequences A 1 and A 2 and one constraint sequence B, C-LCS is the problem of finding a longest common subsequence C of A 1 and A 2 that is also a supersequence of B: A 1 = cgagggt A 2 = cgggagt B = cat C = cgagt Constrained Longest Commmon Subsequence is a natural extension to the classical problem Longest Common Subsequence, and has application to computing the homology of two biological sequences with a specific or putative structure in common (Tsai, 2003). Notations For a sequence S, let S[i] denote the letter of S at position i, let S[i, j] denote the subsequence of S starting at position i and ending at position j (the subsequence is empty when i > j), and let |S| denote the length of S. For two sequences S and T , let S T denote the concatenation of S and T , and write S T if S is a subsequence of T. For a sequence R and a non-negative integer r, let R r denote a sequence consisting of r repetitions of R concatenated together. Generalization The problem C-LCS can be easily generalized to a problem C-LCS(k, l) for an arbitrary number k of input sequences and an arbitrary number l of constraint sequences (Gotthilf et al. Here the input size n is the total length of the k input sequences and the l constraint sequences. For C-LCS(2, 1), the most basic version of the problem C-LCS on two input sequences A 1 and A 2 and one constraint sequence B, there are dynamic programming algorithms running in for some related results. The problem C-LCS(k, l) becomes intractable, however, when either the number k of input sequences or the number l of constraint sequences is unbounded. An early result of Middendorf (1995) on consistent sequences of type (Super, Sub) implies that even if the input and constraint sequences are over a binary alphabet, it is already NP-hard to decide whether a given instance of C-LCS(2, l) has a valid solution; see also Middendorf and Manlove (2004). Recently, Gotthilf et al. (2008) showed that if the sequences are over an arbitrary alphabet, then even if all constraint sequences have length 1, it is again NP-hard to decide whether a given instance of C-LCS(2, l) has a valid solution.