Planar bichromatic bottleneck spanning trees

A. Karim Abu-Affash; Sujoy Bhore; Paz Carmi; Joseph S.B. Mitchell

doi:10.4230/LIPIcs.ESA.2020.1

Planar bichromatic bottleneck spanning trees

A. Karim Abu-Affash, Sujoy Bhore, Paz Carmi, Joseph S.B. Mitchell

Department of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

Given a set P of n red and blue points in the plane, a planar bichromatic spanning tree of P is a geometric spanning tree of P, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree T, such that the length of the longest edge in T is minimized. In this paper, we show that this problem is NP-hard for points in general position. Our main contribution is a polynomial-time (8√2)-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck λ can be converted to a planar bichromatic spanning tree of bottleneck at most 8√2λ.

Original language	English
Title of host publication	28th Annual European Symposium on Algorithms, ESA 2020
Editors	Fabrizio Grandoni, Grzegorz Herman, Peter Sanders
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)	9783959771627
DOIs	https://doi.org/10.4230/LIPIcs.ESA.2020.1
State	Published - 1 Aug 2020
Event	28th Annual European Symposium on Algorithms, ESA 2020 - Virtual, Pisa, Italy Duration: 7 Sep 2020 → 9 Sep 2020

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	173
ISSN (Print)	1868-8969

Conference

Conference	28th Annual European Symposium on Algorithms, ESA 2020
Country/Territory	Italy
City	Virtual, Pisa
Period	7/09/20 → 9/09/20

Keywords

Approximation Algorithms
Bottleneck Spanning Tree
NP-Hardness

ASJC Scopus subject areas

Software

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10.4230/LIPIcs.ESA.2020.1

Cite this

Abu-Affash, A. K., Bhore, S., Carmi, P., & Mitchell, J. S. B. (2020). Planar bichromatic bottleneck spanning trees. In F. Grandoni, G. Herman, & P. Sanders (Eds.), 28th Annual European Symposium on Algorithms, ESA 2020 Article 1 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 173). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ESA.2020.1

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title = "Planar bichromatic bottleneck spanning trees",

abstract = "Given a set P of n red and blue points in the plane, a planar bichromatic spanning tree of P is a geometric spanning tree of P, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree T, such that the length of the longest edge in T is minimized. In this paper, we show that this problem is NP-hard for points in general position. Our main contribution is a polynomial-time (8√2)-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck λ can be converted to a planar bichromatic spanning tree of bottleneck at most 8√2λ.",

keywords = "Approximation Algorithms, Bottleneck Spanning Tree, NP-Hardness",

author = "Abu-Affash, {A. Karim} and Sujoy Bhore and Paz Carmi and Mitchell, {Joseph S.B.}",

note = "Publisher Copyright: {\textcopyright} A. Karim Abu-Affash, Sujoy Bhore, Paz Carmi, and Joseph S. B. Mitchell; 28th Annual European Symposium on Algorithms, ESA 2020 ; Conference date: 07-09-2020 Through 09-09-2020",

year = "2020",

month = aug,

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doi = "10.4230/LIPIcs.ESA.2020.1",

language = "English",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

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booktitle = "28th Annual European Symposium on Algorithms, ESA 2020",

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Abu-Affash, AK, Bhore, S, Carmi, P & Mitchell, JSB 2020, Planar bichromatic bottleneck spanning trees. in F Grandoni, G Herman & P Sanders (eds), 28th Annual European Symposium on Algorithms, ESA 2020., 1, Leibniz International Proceedings in Informatics, LIPIcs, vol. 173, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 28th Annual European Symposium on Algorithms, ESA 2020, Virtual, Pisa, Italy, 7/09/20. https://doi.org/10.4230/LIPIcs.ESA.2020.1

Planar bichromatic bottleneck spanning trees. / Abu-Affash, A. Karim; Bhore, Sujoy; Carmi, Paz et al.
28th Annual European Symposium on Algorithms, ESA 2020. ed. / Fabrizio Grandoni; Grzegorz Herman; Peter Sanders. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2020. 1 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 173).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - Planar bichromatic bottleneck spanning trees

AU - Abu-Affash, A. Karim

AU - Bhore, Sujoy

AU - Carmi, Paz

AU - Mitchell, Joseph S.B.

PY - 2020/8/1

Y1 - 2020/8/1

N2 - Given a set P of n red and blue points in the plane, a planar bichromatic spanning tree of P is a geometric spanning tree of P, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree T, such that the length of the longest edge in T is minimized. In this paper, we show that this problem is NP-hard for points in general position. Our main contribution is a polynomial-time (8√2)-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck λ can be converted to a planar bichromatic spanning tree of bottleneck at most 8√2λ.

AB - Given a set P of n red and blue points in the plane, a planar bichromatic spanning tree of P is a geometric spanning tree of P, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree T, such that the length of the longest edge in T is minimized. In this paper, we show that this problem is NP-hard for points in general position. Our main contribution is a polynomial-time (8√2)-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck λ can be converted to a planar bichromatic spanning tree of bottleneck at most 8√2λ.

KW - Approximation Algorithms

KW - Bottleneck Spanning Tree

KW - NP-Hardness

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M3 - Conference contribution

AN - SCOPUS:85092536083

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A2 - Grandoni, Fabrizio

A2 - Herman, Grzegorz

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ER -

Planar bichromatic bottleneck spanning trees

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