Bose, Prosenjit ;
Mehrabi, Saeed ;
Mondal, Debajyoti
Parameterized Complexity of TwoInterval Pattern Problem
Abstract
A 2interval is the union of two disjoint intervals on the real line. Two 2intervals 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 2intervals; namely, preceding (<), nested (⊏) and crossing (≬). Two 2intervals D₁ and D₂ are called Rcomparable for some R∈{<,⊏,≬}, if either D₁RD₂ or D₂RD₁. A set 𝒟 of disjoint 2intervals is ℛcomparable, for some ℛ⊆{<,⊏,≬} and ℛ≠∅, if every pair of 2intervals in ℛ are Rcomparable for some R∈ℛ. Given a set of 2intervals and some ℛ⊆{<,⊏,≬}, the objective of the {2interval pattern problem} is to find a largest subset of 2intervals that is ℛcomparable.
The 2interval pattern problem is known to be W[1]hard when ℛ=3 and NPhard 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 TwoInterval Pattern Problem}},
booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)},
pages = {16:116:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771504},
ISSN = {18688969},
year = {2020},
volume = {162},
editor = {Susanne Albers},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12263},
URN = {urn:nbn:de:0030drops122630},
doi = {10.4230/LIPIcs.SWAT.2020.16},
annote = {Keywords: Interval graphs, Twointerval pattern problem, Comparability, Multicoloured clique problem, Parameterized complexity, W[1]hardness}
}
12.06.2020
Keywords: 

Interval graphs, Twointerval pattern problem, Comparability, Multicoloured clique problem, Parameterized complexity, W[1]hardness 
Seminar: 

17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

Issue date: 

2020 
Date of publication: 

12.06.2020 