On the generalized constrained longest common subsequence problems

We investigate four variants of the longest common subsequence problem. Given two sequences X, Y and a constrained pattern P of lengths m, n, and rho, respectively, the generalized constrained longest common subsequence (GC-LCS) problems are to find a longest common subsequence of X and Y including (or excluding) P as a subsequence (or substring). We propose new dynamic programming algorithms for solving the GC-LCS problems in O(mn rho) time. We also consider the case where the number of constrained patterns is arbitrary.