{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:44:11Z","timestamp":1787341451652,"version":"build-2736575974"},"reference-count":33,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2014,1]]},"abstract":"<jats:p>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 \u201csimple\u201d 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.<\/jats:p>","DOI":"10.1137\/130908440","type":"journal-article","created":{"date-parts":[[2014,7,3]],"date-time":"2014-07-03T11:55:43Z","timestamp":1404388543000},"page":"1239-1262","source":"Crossref","is-referenced-by-count":24,"title":["Vertex Sparsifiers: New Results from Old Techniques"],"prefix":"10.1137","volume":"43","author":[{"given":"Matthias","family":"Englert","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anupam","family":"Gupta","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Robert","family":"Krauthgamer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Harald","family":"R\u00e4cke","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Inbal","family":"Talgam-Cohen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kunal","family":"Talwar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2014,7,3]]},"reference":[{"key":"atypb1","unstructured":"R. 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Chuzhoy,\n                      On vertex sparsifiers with Steiner nodes\n                      , in Proceedings of the 44th ACM Symposium on Theory of Computing (STOC), ACM, New York, 2012, pp. 673-688.","DOI":"10.1145\/2213977.2214039"},{"key":"atypb10","first-page":"780","author":"Chuzhoy J.","year":"2012","journal-title":"New York"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1145\/347476.347478"},{"key":"atypb12","first-page":"257","author":"Fakcharoenphol J.","year":"2003","journal-title":"New York"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcss.2004.04.011"},{"key":"atypb14","first-page":"36","author":"Fakcharoenphol J.","year":"2003","journal-title":"Berlin"},{"key":"atypb15","first-page":"621","author":"Golovin D.","year":"2006","journal-title":"Philadelphia"},{"key":"atypb16","first-page":"220","author":"Gupta A.","year":"2001","journal-title":"New York"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/j.orl.2010.02.009"},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"A. Gupta, I. Newman, Y. Rabinovich, and A. Sinclair,\n                      Cuts, trees and $\\ell_1$-embeddings of graphs\n                      , Combinatorica, 24 (2004), pp. 233-269.","DOI":"10.1007\/s00493-004-0015-x"},{"key":"atypb19","first-page":"439","volume":"43","author":"Hoory S.","year":"2006","journal-title":"S.)"},{"key":"atypb20","first-page":"1029","author":"Kamma L.","year":"2014","journal-title":"Philadelphia"},{"key":"atypb21","unstructured":"R. Khandekar,\n                      Lagrangian Relaxation Based Algorithms for Convex Programming Problems\n                      , Ph.D. thesis, Indian Institute of Technology Delhi, New Delhi, 2004."},{"key":"atypb22","first-page":"682","author":"Klein P.","year":"1993","journal-title":"New York"},{"key":"atypb23","first-page":"495","author":"Lee J. 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Moitra,\n                      Approximation algorithms for multicommodity-type problems with guarantees independent of the graph size\n                      , in Proceedings of the 50th IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society, Los Alamitos, CA, 2009, pp. 3-12.","DOI":"10.1109\/FOCS.2009.28"},{"key":"atypb30","doi-asserted-by":"crossref","unstructured":"S. Nowozin and C. H. Lampert,\n                      Global connectivity potentials for random field models\n                      , in Proceedings of the 22nd IEEE Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, Piscataway, NJ, 2009, pp. 818-825.","DOI":"10.1109\/CVPR.2009.5206567"},{"key":"atypb31","first-page":"255","author":"R\u00e4cke H.","year":"2008","journal-title":"New York"},{"key":"atypb32","first-page":"211","author":"Rao S.","year":"1998","journal-title":"New York"},{"key":"atypb33","doi-asserted-by":"crossref","unstructured":"S. Vicente, V. Kolmogorov, and C. Rother,\n                      Graph cut based image segmentation with connectivity priors\n                      , in Proceedings of the 21st IEEE Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, Piscataway, NJ, 2008.","DOI":"10.1109\/CVPR.2008.4587440"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/130908440","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:20:58Z","timestamp":1787340058000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/130908440"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,1]]},"references-count":33,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2014,1]]}},"alternative-id":["10.1137\/130908440"],"URL":"https:\/\/doi.org\/10.1137\/130908440","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,1]]}}}