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Vertex sparsifiers : new results from old techniques
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Englert, Matthias, Gupta, Anupam, Krauthgamer, Robert, Räcke, Harald, TalgamCohen, Inbal and Talwar, Kunal (2014) Vertex sparsifiers : new results from old techniques. SIAM Journal on Computing, Volume 43 (Number 4). pp. 12391262. doi:10.1137/130908440

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Official URL: http://dx.doi.org/10.1137/130908440
Abstract
Given a capacitated graph $G = (V,E)$ and a set of terminals $K \subseteq V$, 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 [Proceedings of the 50th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2009, pp. 312] and Leighton and Moitra [Proceedings of the 42nd ACM Symposium on Theory of Computing, ACM, New York, 2010, pp. 4756], we give efficient algorithms for constructing (a) a flow sparsifier $H$ that maintains congestion up to a factor of $O(\frac{\log k}{\log \log k})$, where $k = K$; (b) a convex combination of trees over the terminals $K$ that maintains congestion up to a factor of $O(\log k)$; (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 0extension problem, the first one in which the preimages of each terminal are connected in $G$. Moreover, this result extends to minorclosed families of graphs. Our bounds immediately imply improved approximation guarantees for several terminalbased cut and ordering problems.
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software 

Divisions:  Faculty of Science > Computer Science  
Library of Congress Subject Headings (LCSH):  Approximation algorithms, Computer algorithms, Graph theory  
Journal or Publication Title:  SIAM Journal on Computing  
Publisher:  Society for Industrial and Applied Mathematics  
ISSN:  00975397  
Official Date:  3 July 2014  
Dates: 


Volume:  Volume 43  
Number:  Number 4  
Page Range:  pp. 12391262  
DOI:  10.1137/130908440  
Status:  Peer Reviewed  
Publication Status:  Published  
Access rights to Published version:  Open Access  
Funder:  Engineering and Physical Sciences Research Council (EPSRC), University of Warwick. Centre for Discrete Mathematics and Its Applications, National Science Foundation (U.S.) (NSF), Alfred P. Sloan Foundation, Israel Science Foundation (ISF), Universiṭat TelAviv. Hermann Minkowski Minerva Center for Geometry  
Grant number:  EP/F043333/1 (EPSRC), CCF0729022 (NSF), 452/08 (ISF)  
Embodied As:  1 
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