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√2 λ.
@InProceedings{abuaffash_et_al:LIPIcs.ESA.2020.1, author = {Abu-Affash, A. Karim and Bhore, Sujoy and Carmi, Paz and Mitchell, Joseph S. B.}, title = {{Planar Bichromatic Bottleneck Spanning Trees}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {1:1--1:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.1}, URN = {urn:nbn:de:0030-drops-128670}, doi = {10.4230/LIPIcs.ESA.2020.1}, annote = {Keywords: Approximation Algorithms, Bottleneck Spanning Tree, NP-Hardness} }
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