Better Heuristic Algorithms for the Repetition Free LCS and Other Variants

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

“…The formal definition of the Multiset Restricted Common Subsequence problem (MRCS) is as follows [11]:…”

“…For MRCS, the following exact exponential-time algorithm and the polynomial-time approximation algorithm are proposed in [11]: 11]). There is an O(nm(t + 1) k )-time algorithm for MRCS on two sequences X and Y , and a multiset M, where t and k are the maximum multiplicity of M and the alphabet size |Σ|, respectively 1 .…”

“…

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 paper, we study four variants of LCS: the Repetition-Bounded Longest Common Subsequence problem (RBLCS) [2], the Multiset-Restricted Common Subsequence problem (MRCS) [11], the Two-Side-Filled Longest Common Subsequence problem (2FLCS), and the One-Side-Filled Longest Common Subsequence problem (1FLCS) [5,6]. Although the original LCS can be solved in polynomial time, all these four variants are known to be NP-hard.

…”

“…We remark that the time complexity shown in Theorem 3 of[11] is O(nmt k ), but the correct one must be O(nm(t + 1) k ) because the algorithm has to store t + 1 values from 0 through t for the maximum multiplicity. As described before, if M = Σ, i.e., t = 1, then MRCS is essentially equivalent to RFLCS and thus MRCS is NP-hard.…”