On the longest common rigid subsequence problem

The longest common subsequence problem (LCS) and the closest substring problem (CSP) are two models for finding common patterns in strings, and have been studied extensively. Though both LCS and CSP are NP-Hard, they exhibit very different behavior with respect to polynomial time approximation algorithms. While LCS is hard to approximate within n (delta) for some delta > 0, CSP admits a polynomial time approximation scheme. In this paper, we study the longest common rigid subsequence problem (LCRS). This problem shares similarity with both LCS and CSP and has an important application in motif finding in biological sequences. We show that it is NP-hard to approximate LCRS within ratio n (delta) , for some constant delta > 0, where n is the maximum string length. We also show that it is NP-Hard to approximate LCRS within ratio Omega(m), where m is the number of strings.