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International Workshop on Approximation and Online Algorithms

WAOA 2016: Approximation and Online Algorithms pp 103–115Cite as

Vertex Sparsification in Trees

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10138))

Abstract

Given an unweighted tree \(T=(V,E)\) with terminals \(K \subset V\), we show how to obtain a 2-quality vertex flow and cut sparsifier H with \(V_H = K\). We prove that our result is essentially tight by providing a \(2-o(1)\) lower-bound on the quality of any cut sparsifier for stars.

In addition we give improved results for quasi-bipartite graphs. First, we show how to obtain a 2-quality flow sparsifier with \(V_H = K\) for such graphs. We then consider the other extreme and construct exact sparsifiers of size \(O(2^{k})\), when the input graph is unweighted.

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Notes

  1. 1.

    Alternatively, one can view this step as contracting an arbitrary child-edge of v.

  2. 2.

    Note that the dual requires that \(\delta _{st}\) is at most the length of the shortest s-t path. In our scenario this is always a 2-hop path. Hence, the above formulation is correct.

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Author information

Authors and Affiliations

  1. Faculty of Computer Science, University of Vienna, Vienna, Austria

    Gramoz Goranci

  2. Institut für Informatik, Technische Universität München, Garching, Germany

    Harald Räcke

Authors
  1. Gramoz Goranci
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  2. Harald Räcke
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Corresponding author

Correspondence to Gramoz Goranci .

Editor information

Editors and Affiliations

  1. Institut für Informatik, Christian-Albrechts-Universität, Kiel, Germany

    Klaus Jansen

  2. Istituto Dalle Molle di Studi sull’ Intelligenza Artificiale, Manno (Lugano), Switzerland

    Monaldo Mastrolilli

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Goranci, G., Räcke, H. (2017). Vertex Sparsification in Trees. In: Jansen, K., Mastrolilli, M. (eds) Approximation and Online Algorithms. WAOA 2016. Lecture Notes in Computer Science(), vol 10138. Springer, Cham. https://doi.org/10.1007/978-3-319-51741-4_9

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  • DOI: https://doi.org/10.1007/978-3-319-51741-4_9

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-51740-7

  • Online ISBN: 978-3-319-51741-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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