Vertex Sparsifiers: New Results from Old Techniques

  • Matthias Englert
  • Anupam Gupta
  • Robert Krauthgamer
  • Harald Räcke
  • Inbal Talgam-Cohen
  • Kunal Talwar
Conference paper

DOI: 10.1007/978-3-642-15369-3_12

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6302)
Cite this paper as:
Englert M., Gupta A., Krauthgamer R., Räcke H., Talgam-Cohen I., Talwar K. (2010) Vertex Sparsifiers: New Results from Old Techniques. In: Serna M., Shaltiel R., Jansen K., Rolim J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. Lecture Notes in Computer Science, vol 6302. Springer, Berlin, Heidelberg

Abstract

Given a capacitated graph G = (V,E) and a set of terminals K ⊆ V, how should we produce a graph H only on the terminals K so that every (multicommodity) flow between the terminals in G could be supported in H with low congestion, and vice versa? (Such a graph H is called a flow-sparsifier for G.) What if we want H to be a “simple” graph? What if we allow H to be a convex combination of simple graphs?

Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC 2010], we give efficient algorithms for constructing: (a) a flow-sparsifier H that maintains congestion up to a factor of \({\smash{O(\frac{\log k}{\log \log k})}}\), where k = |K|. (b) a convex combination of trees over the terminals K that maintains congestion up to a factor of O(logk). (c) for a planar graph G, a convex combination of planar graphs that maintains congestion up to a constant factor. This requires us to give a new algorithm for the 0-extension problem, the first one in which the preimages of each terminal are connected in G. Moreover, this result extends to minor-closed families of graphs.

Our bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Matthias Englert
    • 1
  • Anupam Gupta
    • 2
  • Robert Krauthgamer
    • 3
  • Harald Räcke
    • 1
  • Inbal Talgam-Cohen
    • 3
  • Kunal Talwar
    • 4
  1. 1.Department of Computer Science and DIMAPUniversity of WarwickCoventryUK
  2. 2.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA
  3. 3.Weizmann Institute of ScienceRehovotIsrael
  4. 4.Microsoft Research Silicon ValleyMountain ViewUSA

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