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### Vertex sparsifiers : new results from old techniques

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Englert, Matthias, Gupta, A., Krauthgamer, Robert, Räcke, Harald, Talgam-Cohen, Inbal and Talwar, Kunal
(2010)
*Vertex sparsifiers : new results from old techniques.*
In: Serna, Maria and Shaltiel, Ronen and Jansen, Klaus and Rolim, José, (eds.)
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques.
Lecture Notes in Computer Science
(6302).
Springer Verlag, pp. 152-165.
ISBN 9783642153686

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Official URL: http://dx.doi.org/10.1007/978-3-642-15369-3_12

## 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.

Item Type: | Book Item |
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Subjects: | Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software |

Divisions: | Faculty of Science > Computer Science |

Library of Congress Subject Headings (LCSH): | Computer terminals, Graph algorithms |

Series Name: | Lecture Notes in Computer Science |

Publisher: | Springer Verlag |

ISBN: | 9783642153686 |

Book Title: | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques |

Editor: | Serna, Maria and Shaltiel, Ronen and Jansen, Klaus and Rolim, José |

Official Date: | 2010 |

Number: | 6302 |

Page Range: | pp. 152-165 |

Identification Number: | 10.1007/978-3-642-15369-3_12 |

Status: | Not Peer Reviewed |

Publication Status: | Published |

Description: | |

Funder: | Engineering and Physical Sciences Research Council (EPSRC), University of Warwick. Centre for Discrete Mathematics and Its Applications |

Version or Related Resource: | Conference paper. http://wrap.warwick.ac.uk/id/eprint/4700 |

Related URLs: | |

URI: | http://wrap.warwick.ac.uk/id/eprint/47422 |

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