Parameterized Complexity of Two-Interval Pattern Problem
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A 2-interval is the union of two disjoint intervals on the real line. Two 2-intervals D_1 and D_2 are 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 (⊏) and crossing (≬). Two 2-intervals D_1 and D_2 are called R-comparable for some R∈{<,⊏,≬}, if either D_1RD_2 or D_2RD_1. A set 𝒟 of disjoint 2-intervals is ℛ-comparable, for some ℛ⊆{<,⊏,≬} and ℛ≠∅, if every pair of 2-intervals in ℛ are R-comparable for some R∈ℛ. Given a set of 2-intervals and some ℛ⊆{<,⊏,≬}, the objective of the 2-interval pattern problem is to find a largest subset of 2-intervals that is ℛ-comparable. The 2-interval pattern problem is known to be W[1]-hard when |ℛ|=3 and NP-hard when |ℛ|=2 (except for ℛ={<,⊏}, 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 ℛ={⊏,≬} and ℛ={<,≬} (when parameterized by the size of an optimal solution); this answers an open question posed by Vialette [Encyclopedia of Algorithms, 2008].
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