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Restricted Common Superstring and Restricted Common Supersequence

The {\em shortest common superstring} and the {\em shortest common supersequence} are two well studied problems having a wide range of applications. In this paper we consider both problems with resource constraints, denoted as the Restricted Common Superstring (shortly \textit{RCSstr}) problem and the Restricted Common Supersequence (shortly \textit{RCSseq}). In the \textit{RCSstr} (\textit{RCSseq}) problem we are given a set $S$ of $n$ strings, $s_1$, $s_2$, $\ldots$, $s_n$, and a multiset $t = \{t_1, t_2, \dots, t_m\}$, and the goal is to find a permutation $π: \{1, \dots, m\} \to \{1, \dots, m\}$ to maximize the number of strings in $S$ that are substrings (subsequences) of $π(t) = t_{π(1)}t_{π(2)}...t_{π(m)}$ (we call this ordering of the multiset, $π(t)$, a permutation of $t$). We first show that in its most general setting the \textit{RCSstr} problem is {\em NP-complete} and hard to approximate within a factor of $n^{1-ε}$, for any $ε> 0$, unless P = NP. Afterwards, we present two separate reductions to show that the \textit{RCSstr} problem remains NP-Hard even in the case where the elements of $t$ are drawn from a binary alphabet or for the case where all input strings are of length two. We then present some approximation results for several variants of the \textit{RCSstr} problem. In the second part of this paper, we turn to the \textit{RCSseq} problem, where we present some hardness results, tight lower bounds and approximation algorithms.