The longest subsequence-duplicated subsequence and related problems

Motivated by computing duplication patterns in sequences, a new fundamental problem called the longest subsequence-duplicated subsequence (LSDS) is proposed. Given a sequence S of length n, a subsequence-duplicated subsequence is a subsequence of S in the form of $x_1^{d_1}{x}_2^{d_2}\cdots x_k^{d_k}$ with $x_i$ being a subsequence of S, $x_j\ne x_{j+1}$ and $d_i\ge 2$ for all i in $\left[k\right]$ and j in $[k-1]$. We first present an $O\left(n^6\right)$ time algorithm to compute the longest cubic subsequences of all the $O\left(n^2\right)$ substrings of S, improving the trivial $O\left(n^7\right)$ bound. Then, an $O\left(n^6\right)$ time algorithm for computing the longest subsequence-duplicated subsequence (LSDS) of S is obtained. Finally we focus on two variants of this problem. We first consider the constrained version when Σ is unbounded, each letter appears in S at most d times and all the letters in Σ must appear in the solution. We show that the problem is NP-hard for $d=4$, via a reduction from a special version of SAT (which is obtained from 3-COLORING). We then show that when each letter appears in S at most $d=3$ times, then the problem is solvable in $O\left(n^4\right)$ time.