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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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
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
Dates:	Date Event 2010 Published
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Status:	Not Peer Reviewed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:	Related item in WRAP
URI:	http://wrap.warwick.ac.uk/id/eprint/47422

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