The longest letter-duplicated subsequence and related problems

Motivated by computing duplication patterns in sequences, a new problem called the longest letter-duplicated subsequence (LLDS) is proposed. Given a sequence S of length n, a letter-duplicated subsequence is a subsequence of S in the form of \(x_1^{d_1}x_2^{d_2}\ldots x_k^{d_k}\) with \(x_i\in \Sigma \), \(x_j\ne x_{j+1}\) and \(d_i\ge 2\) for all i in [k] and j in \([k-1]\). A linear time algorithm for computing a longest letter-duplicated subsequence (LLDS) of S can be easily obtained. In this paper, we focus on two variants of this problem: (1) ‘all-appearance’ version, i.e., all letters in \(\Sigma \) must appear in the solution, and (2) the weighted version. For the former, we obtain dichotomous results: We prove that, when each letter appears in S at least 4 times, the problem and a relaxed version on feasibility testing (FT) are both NP-hard. The reduction is from \((3^+,1,2^-)\)-SAT, where all 3-clauses (i.e., containing 3 lals) are monotone (i.e., containing only positive literals) and all 2-clauses contain only negative literals. We then show that when each letter appears in S at most 3 times, then the problem admits an O(n) time algorithm. Finally, we consider the weighted version, where the weight of a block \(x_i^{d_i} (d_i\ge 2)\) could be any positive function which might not grow with \(d_i\). We give a non-trivial \(O(n^2)\) time dynamic programming algorithm for this version, i.e., computing an LD-subsequence of S whose weight is maximized.