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Vertex sparsifiers : new results from old techniques
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Englert, Matthias, Gupta, Anupam, Krauthgamer, Robert, Räcke, Harald, Talgam-Cohen, Inbal and Talwar, Kunal (2014) Vertex sparsifiers : new results from old techniques. SIAM Journal on Computing, Volume 43 (Number 4). pp. 1239-1262. 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. 3--12] and Leighton and Moitra [Proceedings of the 42nd ACM Symposium on Theory of Computing, ACM, New York, 2010, pp. 47--56], 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 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: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software |
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| Divisions: | Faculty of Science, Engineering and Medicine > 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: | 0097-5397 | ||||||||
| Official Date: | 3 July 2014 | ||||||||
| Dates: |
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| Volume: | Volume 43 | ||||||||
| Number: | Number 4 | ||||||||
| Page Range: | pp. 1239-1262 | ||||||||
| 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 Tel-Aviv. Hermann Minkowski Minerva Center for Geometry | ||||||||
| Grant number: | EP/F043333/1 (EPSRC), CCF-0729022 (NSF), 452/08 (ISF) | ||||||||
| Embodied As: | 1 |
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