Restricted LCS

The Longest Common Subsequence (LCS) of two or more strings is a fundamental well-studied problem which has a wide range of applications throughout computational sciences. When the common subsequence must contain one or more constraint strings as subsequences, the problem becomes the Constrained LCS (CLCS) problem. In this paper we consider the Restricted LCS (RLCS) problem, where our goal is finding a longest common subsequence between two or more strings that does not contain a given set of restriction strings as subsequences. First we show that in case of two input strings and an arbitrary number of restriction strings the RLCS problem is NP-hard. Afterwards, we present a dynamic programming solution for RLCS and we show that this algorithm implies that RLCS is in FPT when parameterized by the total length of the restriction strings. In the last part of this paper we present two approximation algorithms for the hard variants of the problem.