Goranci, G., Henzinger, M. and Peng, P. orcid.org/0000-0003-2700-5699 (2017) Improved guarantees for Vertex Sparsification in planar graphs. In: 25th Annual European Symposium on Algorithms (ESA 2017). 25th Annual European Symposium on Algorithms (ESA 2017), 04-08 Sep 2017, Vienna, Austria. Leibniz International Proceedings in Informatics, 87 . Schloss Dagstuhl - Leibniz-Zentrum für Informatik , 44:1-44:14. ISBN 978-3-95977-049-1
Abstract
Graph Sparsification aims at compressing large graphs into smaller ones while (approximately) preserving important characteristics of the input graph. In this work we study Vertex Sparsifiers, i.e., sparsifiers whose goal is to reduce the number of vertices. Given a weighted graph G=(V,E), and a terminal set K with |K|=k, a quality-q vertex cut sparsifier of G is a graph H with K contained in V_H that preserves the value of minimum cuts separating any bipartition of K, up to a factor of q. We show that planar graphs with all the k terminals lying on the same face admit quality-1 vertex cut sparsifier of size O(k^2) that are also planar. Our result extends to vertex flow and distance sparsifiers. It improves the previous best known bound of O(k^2 2^(2k)) for cut and flow sparsifiers by an exponential factor, and matches an Omega(k^2) lower-bound for this class of graphs. We also study vertex reachability sparsifiers for directed graphs. Given a digraph G=(V,E) and a terminal set K, a vertex reachability sparsifier of G is a digraph H=(V_H,E_H), K contained in V_H that preserves all reachability information among terminal pairs. We introduce the notion of reachability-preserving minors, i.e., we require H to be a minor of G. Among others, for general planar digraphs, we construct reachability-preserving minors of size O(k^2 log^2 k). We complement our upper-bound by showing that there exists an infinite family of acyclic planar digraphs such that any reachability-preserving minor must have Omega(k^2) vertices.
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © Gramoz Goranci, Monika Henzinger, and Pan Peng; licensed under Creative Commons License CC-BY (https://creativecommons.org/licenses/by/3.0/). |
Keywords: | Vertex Sparsification; Graph Sparsification; Planar Graphs; Metric Embedding; Reachability |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 13 Jun 2019 15:17 |
Last Modified: | 13 Jun 2019 15:21 |
Status: | Published |
Publisher: | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |
Series Name: | Leibniz International Proceedings in Informatics |
Refereed: | Yes |
Identification Number: | 10.4230/LIPIcs.ESA.2017.44 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:147336 |