Extracting constrained 2-interval subsets in 2-interval sets

2-interval sets were used in [S. Vialette, Pattern matching over 2-intervals sets, in: Proc. 13th Annual Symposium Combinatorial Pattern Matching, CPM 2002, in: Lecture Notes in Computer Science, vol. 2373, Springer-Verlag, 2002, pp. 53–63; S. Vialette, On the computational complexity of 2-interval pattern matching, Theoret. Comput. Sci. 312 (2–3) (2004) 223–249] for establishing a general representation for macroscopic describers of RNA secondary structures. In this context, we have a 2-interval for each legal local fold in a given RNA sequence, and a constrained pattern made of disjoint 2-intervals represents a putative RNA secondary structure. We focus here on the problem of extracting a constrained pattern in a set of 2-intervals. More precisely, given a set of 2-intervals D and a model R describing if two disjoint 2-intervals in a solution can be in precedence order (<), be allowed to nest (⊏) and/or be allowed to cross (≬), we consider the problem of finding a maximum cardinality subset D′⊆D of disjoint 2-intervals such that any two 2-intervals in D′ agree with R. The different combinations of restrictions on model R alter the computational complexity of the problem, and need to be examined separately.In this paper, we improve the time complexity of [S. Vialette, On the computational complexity of 2-interval pattern matching, Theoret. Comput. Sci. 312 (2–3) (2004) 223–249] for model R={⊏} by giving an optimal O(nlogn) time algorithm, where n is the cardinality of the 2-interval set D. We also give a graph-like relaxation for model R={⊏,≬} that is solvable in time. Finally, we prove that the considered problem is NP-complete for model R={<,≬} even for same-length intervals, and give a fixed-parameter tractability result based on the crossing structure of D.