Improved guarantees for vertex sparsification in planar graphs
Goranci G, Henzinger MH, Peng P. 2020. Improved guarantees for vertex sparsification in planar graphs. SIAM Journal on Discrete Mathematics. 34(1), 130β162.
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https://arxiv.org/abs/1702.01136
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Author
Goranci, Gramoz;
Henzinger, MonikaISTA ;
Peng, Pan
Abstract
Graph sparsification aims at compressing large graphs into smaller ones while 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. We focus on the following notions: (1) Given a digraph πΊ=(π,πΈ) and terminal vertices πΎβπ with |πΎ|=π, a (vertex) reachability sparsifier of πΊ is a digraph π»=(ππ»,πΈπ»), πΎβππ» that preserves all reachability information among terminal pairs. Let |ππ»| denote the size of π». In this work we introduce the notion of reachability-preserving minors (RPMs), i.e., we require π» to be a minor of πΊ. We show any directed graph πΊ admits an RPM π» of size π(π3), and if πΊ is planar, then the size of π» improves to π(π2logπ). We complement our upper bound by showing that there exists an infinite family of grids such that any RPM must have Ξ©(π2) vertices. (2) Given a weighted undirected graph πΊ=(π,πΈ) and terminal vertices πΎ with |πΎ|=π, an exact (vertex) cut sparsifier of πΊ is a graph π» with πΎβππ» that preserves the value of minimum cuts separating any bipartition of πΎ. We show that planar graphs with all the π terminals lying on the same face admit exact cut sparsifiers of size π(π2) that are also planar. Our result extends to flow and distance sparsifiers. It improves the previous best-known bound of π(π222π) for cut and flow sparsifiers by an exponential factor and matches an Ξ©(π2) lower-bound for this class of graphs.
Publishing Year
Date Published
2020-01-01
Journal Title
SIAM Journal on Discrete Mathematics
Volume
34
Issue
1
Page
130-162
ISSN
eISSN
IST-REx-ID
Cite this
Goranci G, Henzinger MH, Peng P. Improved guarantees for vertex sparsification in planar graphs. SIAM Journal on Discrete Mathematics. 2020;34(1):130-162. doi:10.1137/17m1163153
Goranci, G., Henzinger, M. H., & Peng, P. (2020). Improved guarantees for vertex sparsification in planar graphs. SIAM Journal on Discrete Mathematics. Society for Industrial & Applied Mathematics. https://doi.org/10.1137/17m1163153
Goranci, Gramoz, Monika H Henzinger, and Pan Peng. βImproved Guarantees for Vertex Sparsification in Planar Graphs.β SIAM Journal on Discrete Mathematics. Society for Industrial & Applied Mathematics, 2020. https://doi.org/10.1137/17m1163153.
G. Goranci, M. H. Henzinger, and P. Peng, βImproved guarantees for vertex sparsification in planar graphs,β SIAM Journal on Discrete Mathematics, vol. 34, no. 1. Society for Industrial & Applied Mathematics, pp. 130β162, 2020.
Goranci G, Henzinger MH, Peng P. 2020. Improved guarantees for vertex sparsification in planar graphs. SIAM Journal on Discrete Mathematics. 34(1), 130β162.
Goranci, Gramoz, et al. βImproved Guarantees for Vertex Sparsification in Planar Graphs.β SIAM Journal on Discrete Mathematics, vol. 34, no. 1, Society for Industrial & Applied Mathematics, 2020, pp. 130β62, doi:10.1137/17m1163153.
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arXiv 1702.01136