eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-12
16:1
16:10
10.4230/LIPIcs.SWAT.2020.16
article
Parameterized Complexity of Two-Interval Pattern Problem
Bose, Prosenjit
1
Mehrabi, Saeed
1
Mondal, Debajyoti
2
School of Computer Science, Carleton University, Ottawa, Canada
Department of Computer Science, University of Saskatchewan, Saskatoon, Canada
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].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol162-swat2020/LIPIcs.SWAT.2020.16/LIPIcs.SWAT.2020.16.pdf
Interval graphs
Two-interval pattern problem
Comparability
Multicoloured clique problem
Parameterized complexity
W[1]-hardness