Constrained LCS: Hardness and Approximation

Abstract

The problem of finding the longest common subsequence (LCS) of two given strings A ₁ and A ₂ is a well-studied problem. The constrained longest common subsequence (C-LCS) for three strings A ₁, A ₂ and B ₁ is the longest common subsequence of A ₁ and A ₂ that contains B ₁ as a subsequence. The fastest algorithm solving the C-LCS problem has a time complexity of O(m ₁ m ₂ n ₁) where m ₁, m ₂ and n ₁ are the lengths of A ₁, A ₂ and B ₁ respectively. In this paper we consider two general variants of the C-LCS problem. First we show that in case of two input strings and an arbitrary number of constraint strings, it is NP-hard to approximate the C-LCS problem. Moreover, it is easy to see that in case of an arbitrary number of input strings and a single constraint, the problem of finding the constrained longest common subsequence is NP-hard. Therefore, we propose a linear time approximation algorithm for this variant, our algorithm yields a $1 / \sqrt{m_{min}|\Sigma|}$ approximation factor, where m _min is the length of the shortest input string and |Σ| is the size of the alphabet.