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### Vertex sparsifiers : new results from old techniques

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Englert, Matthias, Gupta, Anupam (Researcher in Computer Science), Krauthgamer, Robert, Raecke, Harald, Talgam-Cohen, Inbal and Talwar, Kunal
(2010)
*Vertex sparsifiers : new results from old techniques.*
In: 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2010)/14th International Workshop on Randomization and Computation (RANDOM 2010), Univ Politecnica Catalunya (UPC), Barcelona, Spain, 01-03 Sep 2010

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## Abstract

Given a capacitated graph G = (V, E) and a set of terminals K subset of 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 O(log k/log log k), where k = vertical bar K vertical bar. (b) a convex combination of trees over the terminals K that maintains congestion up to a factor of O(log k). (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: | Conference Item (Paper) |
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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 |

Editor: | Serna, M and Shaltiel, R and Jansen, K and Rolim, J |

Official Date: | 2010 |

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

Status: | Peer Reviewed |

Publication Status: | Published |

Version or Related Resource: | Englert, Matthias, Gupta, A., Krauthgamer, Robert, Räcke, Harald, Talgam-Cohen, Inbal and Talwar, Kunal (2010) Vertex sparsifiers : new results from old techniques. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. Lecture Notes in Computer Science (6302). Springer Verlag, pp. 152-165. ISBN 9783642153686 http://wrap.warwick.ac.uk/id/eprint/47422 |

Conference Paper Type: | Paper |

Title of Event: | 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2010)/14th International Workshop on Randomization and Computation (RANDOM 2010) |

Type of Event: | Workshop |

Location of Event: | Univ Politecnica Catalunya (UPC), Barcelona, Spain |

Date(s) of Event: | 01-03 Sep 2010 |

Related URLs: | |

URI: | http://wrap.warwick.ac.uk/id/eprint/4700 |

Data sourced from Thomson Reuters' Web of Knowledge

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