Variants of constrained longest common subsequence

We consider a variant of the classical Longest Common Subsequence problem called Doubly-Constrained Longest Common Subsequence (DC-LCS). Given two strings s1 and s2 over an alphabet Σ, a set Cs of strings, and a function Co:Σ→N, the DC-LCS problem consists of finding the longest subsequence s of s1 and s2 such that s is a supersequence of all the strings in Cs and such that the number of occurrences in s of each symbol σ∈Σ is upper bounded by Co(σ). The DC-LCS problem provides a clear mathematical formulation of a sequence comparison problem in Computational Biology and generalizes two other constrained variants of the LCS problem that have been introduced previously in the literature: the Constrained LCS and the Repetition-Free LCS. We present two results for the DC-LCS problem. First, we illustrate a fixed-parameter algorithm where the parameter is the length of the solution which is also applicable to the more specialized problems. Second, we prove a parameterized hardness result for the Constrained LCS problem when the parameter is the number of the constraint strings (|Cs|) and the size of the alphabet Σ. This hardness result also implies the parameterized hardness of the DC-LCS problem (with the same parameters) and its NP-hardness when the size of the alphabet is constant.