 License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SWAT.2020.16
URN: urn:nbn:de:0030-drops-122630
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12263/
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### Parameterized Complexity of Two-Interval Pattern Problem

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### 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 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-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-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},
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