Vertex Sparsification in Trees

Conference paper

DOI: 10.1007/978-3-319-51741-4_9

Part of the Lecture Notes in Computer Science book series (LNCS, volume 10138)
Cite this paper as:
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

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.

Keywords

Graph sparsification Vertex flow sparsifiers Trees 

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Faculty of Computer ScienceUniversity of ViennaViennaAustria
  2. 2.Institut für InformatikTechnische Universität MünchenGarchingGermany

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