Restricted and Swap Common Superstring: A Parameterized View

Abstract

In several areas, in particular in bioinformatics and in AI planning, Shortest Common Superstring problem (SCS) and variants thereof have been successfully applied. In this paper we consider two variants of SCS recently introduced (Restricted Common Superstring, $\ensuremath{\text{\textsc{RCS}}}$) and (Swapped Common Superstring, $\ensuremath{\text{\textsc{SWCS}}}$). In $\ensuremath{\text{\textsc{RCS}}}$ we are given a set S of strings and a multiset, and we look for an ordering $\mathcal{M}_o$ of $\mathcal{M}$ such that the number of input strings which are substrings of $\mathcal{M}_o$ is maximized. In $\ensuremath{\text{\textsc{SWCS}}}$ we are given a set S of strings and a text $\mathcal{T}$, and we look for a swap ordering $\mathcal{T}_o$ of $\mathcal{T}$ (an ordering of $\mathcal{T}$ obtained by swapping only some pairs of adjacent characters) such that the number of input strings which are substrings of $\mathcal{T}_o$ is maximized. In this paper we investigate the parameterized complexity of the two problems. We give two fixed-parameter algorithms, where the parameter is the size of the solution, for $\ensuremath{\text{\textsc{SWCS}}}$ and $\ensuremath{\text{\textsc{$\ell$-RCS}}}$ (the $\ensuremath{\text{\textsc{RCS}}}$ problem restricted to strings of length bounded by a parameter ℓ). Furthermore, we complement these results by showing that $\ensuremath{\text{\textsc{SWCS}}}$ and $\ensuremath{\text{\textsc{$\ell$-RCS}}}$ do not admit a polynomial kernel.