Polynomial-time equivalences and refined algorithms for longest common subsequence variants

The problem of computing the longest common subsequence of two sequences ( LCS for short) is a classical and fundamental problem in computer science. In this article, we study four variants of LCS : the Repetition-Bounded Longest Common Subsequence problem ( RBLCS ), the Multiset-Restricted Common Subsequence problem ( MRCS ), the Two-Side-Filled Longest Common Subsequence problem ( 2FLCS ), and the One-Side-Filled Longest Common Subsequence problem ( 1FLCS ). Although the original LCS can be solved in polynomial time, all these four variants are known to be NP-hard. Recently, an exact, O ( 1 . 4422 5 n ) -time, dynamic programming (DP) based algorithm for RBLCS was proposed, where the two input sequences have lengths n and p o l y ( n ) . Here, we first establish that each of MRCS , 1FLCS , and 2FLCS is polynomially equivalent to RBLCS . Then, we design a refined DP-based algorithm for RBLCS that runs in O ( 1 . 4142 2 n ) time, which implies that MRCS , 1FLCS , and 2FLCS can also be solved in O ( 1 . 4142 2 n ) time. Finally, we give a polynomial-time 2-approximation algorithm for 2FLCS .