Parameterized Complexity of Two-Interval Pattern Problem

A \emph{2-interval} is the union of two disjoint intervals on the real line. Two 2-intervals $D_1$ and $D_2$ are \emph{disjoint} if their intersection is empty (i.e., no interval of $D_1$ intersects any interval of $D_2$). There can be three different relations between two disjoint 2-intervals; namely, preceding ($<$), nested ($\sqsubset$) and crossing ($\between$). Two 2-intervals $D_1$ and $D_2$ are called \emph{$R$-comparable} for some $R\in\{<,\sqsubset,\between\}$, if either $D_1RD_2$ or $D_2RD_1$. A set $\mathcal{D}$ of disjoint 2-intervals is $\mathcal{R}$-comparable, for some $\mathcal{R}\subseteq\{<,\sqsubset,\between\}$ and $\mathcal{R}\neq\emptyset$, if every pair of 2-intervals in $\mathcal{R}$ are $R$-comparable for some $R\in\mathcal{R}$. Given a set of 2-intervals and some $\mathcal{R}\subseteq\{<,\sqsubset,\between\}$, the objective of the \emph{2-interval pattern problem} is to find a largest subset of 2-intervals that is $\mathcal{R}$-comparable. The 2-interval pattern problem is known to be $W[1]$-hard when $|\mathcal{R}|=3$ and $NP$-hard when $|\mathcal{R}|=2$ (except for $\mathcal{R}=\{<,\sqsubset\}$, which is solvable in quadratic time). In this paper, we fully settle the parameterized complexity of the problem by showing it to be $W[1]$-hard for both $\mathcal{R}=\{\sqsubset,\between\}$ and $\mathcal{R}=\{<,\between\}$ (when parameterized by the size of an optimal solution); this answers an open question posed by Vialette [Encyclopedia of Algorithms, 2008].