Parameterized Complexity of Two-Interval Pattern Problem
A 2-interval is the union of two disjoint intervals on the real line. Two 2-intervals D₁ and D₂ are disjoint if their intersection is empty (i.e., no interval of D₁ intersects any interval of D₂). There can be three different relations between two disjoint 2-intervals; namely, preceding (<), nested (⊏) and crossing (≬). Two 2-intervals D₁ and D₂ are called R-comparable for some R∈{<,⊏,≬}, if either D₁RD₂ or D₂RD₁. 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 that it is W[1]-hard for both ℛ={⊏,≬} and ℛ={<,≬} (when parameterized by the size of an optimal solution). This answers the open question posed by Vialette [Encyclopedia of Algorithms, 2008].
Interval graphs
Two-interval pattern problem
Comparability
Multicoloured clique problem
Parameterized complexity
W[1]-hardness
Theory of computation~Computational geometry
Theory of computation~Parameterized complexity and exact algorithms
16:1-16:10
Regular Paper
The work is partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
Prosenjit
Bose
Prosenjit Bose
School of Computer Science, Carleton University, Ottawa, Canada
Saeed
Mehrabi
Saeed Mehrabi
School of Computer Science, Carleton University, Ottawa, Canada
Debajyoti
Mondal
Debajyoti Mondal
Department of Computer Science, University of Saskatchewan, Saskatoon, Canada
10.4230/LIPIcs.SWAT.2020.16
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Prosenjit Bose, Saeed Mehrabi, and Debajyoti Mondal
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