[hal-00620366] New Results for the 2-Interval Pattern Problem

We present new results concerning the problem of finding a constrained pattern in a set of 2-intervals. Given a set of n 2-intervals D and a model R describing if two disjoint 2-intervals can be in precedence order ($<$), be allowed to nest ($\sqsubset$) and/or be allowed to cross $(\between$), we consider the problem of finding a maximum cardinality subset $\D' \subseteq \D$ of disjoint $2$-intervals such that any two $2$-intervals in $\D'$ agree with $R$. We improve the time complexity of the best known algorithm for $R = \{\sqsubset\}$ by giving an optimal O(n log n) time algorithm. Also, we give a graph-like relaxation for $R = \{\sqsubset,\between\}$ that is solvable in $O(n^2 \sqrt{n})$ time. Finally, we prove that the problem is NP-complete for $R = \{<,\between\}$ and in addition to that, we give a fixed-parameter tractability result based on the crossing structure of D.



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