License: 
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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SWAT.2020.16
URN: urn:nbn:de:0030-drops-122630
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12263/
Bose, Prosenjit ;
Mehrabi, Saeed ;
Mondal, Debajyoti
Parameterized Complexity of Two-Interval Pattern Problem
Abstract
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 ? 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].
BibTeX - Entry
@InProceedings{bose_et_al:LIPIcs:2020:12263,
author = {Prosenjit Bose and Saeed Mehrabi and Debajyoti Mondal},
title = {{Parameterized Complexity of Two-Interval Pattern Problem}},
booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)},
pages = {16:1--16:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-150-4},
ISSN = {1868-8969},
year = {2020},
volume = {162},
editor = {Susanne Albers},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12263},
URN = {urn:nbn:de:0030-drops-122630},
doi = {10.4230/LIPIcs.SWAT.2020.16},
annote = {Keywords: Interval graphs, Two-interval pattern problem, Comparability, Multicoloured clique problem, Parameterized complexity, W[1]-hardness}
}
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Keywords: |
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Interval graphs, Two-interval pattern problem, Comparability, Multicoloured clique problem, Parameterized complexity, W[1]-hardness |
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Collection: |
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17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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12.06.2020 |
