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 $\pi : \{ 1, \dots, m\} \to \{ 1, \dots, m\}$ to maximize the number of strings in $S$ that are substrings (subsequences) of $\pi(t) = t_{ \pi(1)}t_{ \pi(2)}...t_{ \pi(m)}$ (we call this ordering of the multiset, $\pi(t)$, 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 $n^{ 1-\epsilon}$, for any $\epsilon > 0$, 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.
Categories: cs.DS