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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2023.17
URN: urn:nbn:de:0030-drops-186705
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18670/
Berger, Aaron ;
Kaufmann, Jenny ;
Vassilevska Williams, Virginia
Approximating Min-Diameter: Standard and Bichromatic
Abstract
The min-diameter of a directed graph G is a measure of the largest distance between nodes. It is equal to the maximum min-distance d_{min}(u,v) across all pairs u,v ∈ V(G), where d_{min}(u,v) = min(d(u,v), d(v,u)). Min-diameter approximation in directed graphs has attracted attention recently as an offshoot of the classical and well-studied diameter approximation problem.
Our work provides a 3/2-approximation algorithm for min-diameter in DAGs running in time O(m^{1.426} n^{0.288}), and a faster almost-3/2-approximation variant which runs in time O(m^{0.713} n). (An almost-α-approximation algorithm determines the min-diameter to within a multiplicative factor of α plus constant additive error.) This is the first known algorithm to solve 3/2-approximation for min-diameter in sparse DAGs in truly subquadratic time O(m^{2-ε}) for ε > 0; previously only a 2-approximation was known. By a conditional lower bound result of [Abboud et al, SODA 2016], a better than 3/2-approximation can't be achieved in truly subquadratic time under the Strong Exponential Time Hypothesis (SETH), so our result is conditionally tight. We additionally obtain a new conditional lower bound for min-diameter approximation in general directed graphs, showing that under SETH, one cannot achieve an approximation factor below 2 in truly subquadratic time.
Our work also presents the first study of approximating bichromatic min-diameter, which is the maximum min-distance between oppositely colored vertices in a 2-colored graph. We show that SETH implies that in DAGs, a better than 2 approximation cannot be achieved in truly subquadratic time, and that in general graphs, an approximation within a factor below 5/2 is similarly out of reach. We then obtain an O(m)-time algorithm which determines if bichromatic min-diameter is finite, and an almost-2-approximation algorithm for bichromatic min-diameter with runtime Õ(min(m^{4/3} n^{1/3}, m^{1/2} n^{3/2})).
BibTeX - Entry
@InProceedings{berger_et_al:LIPIcs.ESA.2023.17,
author = {Berger, Aaron and Kaufmann, Jenny and Vassilevska Williams, Virginia},
title = {{Approximating Min-Diameter: Standard and Bichromatic}},
booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)},
pages = {17:1--17:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-295-2},
ISSN = {1868-8969},
year = {2023},
volume = {274},
editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18670},
URN = {urn:nbn:de:0030-drops-186705},
doi = {10.4230/LIPIcs.ESA.2023.17},
annote = {Keywords: diameter, min distances, fine-grained, approximation algorithm}
}
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Keywords: |
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diameter, min distances, fine-grained, approximation algorithm |
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Collection: |
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31st Annual European Symposium on Algorithms (ESA 2023) |
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Issue Date: |
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2023 |
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Date of publication: |
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30.08.2023 |
