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dc.contributor.authorCharikar, Moses
dc.contributor.authorLeighton, Frank Thomson
dc.contributor.authorLi, Shi
dc.contributor.authorMoitra, Ankur
dc.date.accessioned2013-09-25T19:03:51Z
dc.date.available2013-09-25T19:03:51Z
dc.date.issued2010-10
dc.identifier.isbn978-1-4244-8525-3
dc.identifier.issn0272-5428
dc.identifier.issn1523-8288
dc.identifier.otherINSPEC Accession Number: 11703135
dc.identifier.urihttp://hdl.handle.net/1721.1/81177
dc.description.abstractThe notion of vertex sparsification (in particular cut-sparsification) is introduced in, where it was shown that for any graph G = (V, E) and any subset of k terminals K ⊂ V, there is a polynomial time algorithm to construct a graph H = (K, EH) on just the terminal set so that simultaneously for all cuts (A,K-A), the value of the minimum cut in G separating A from K-A is approximately the same as the value of the corresponding cut in H. Then approximation algorithms can be run directly on H as a proxy for running on G. We give the first super-constant lower bounds for how well a cut-sparsifier H can simultaneously approximate all minimum cuts in G. We prove a lower bound of Ω(log1/4 k) this is polynomially-related to the known upper bound of O(log k/log log k). Independently, a similar lower bound is given in. This is an exponential improvement on the Ω(log log k) bound given in which in fact was for a stronger vertex sparsification guarantee, and did not apply to cut sparsifiers. Despite this negative result, we show that for many natural optimization problems, we do not need to incur a multiplicative penalty for our reduction. Roughly, we show that any rounding algorithm which also works for the O-extension relaxation can be used to construct good vertex-sparsifiers for which the optimization problem is easy. Using this, we obtain optimal O(log k)-competitive Steiner oblivious routing schemes, which generalize the results in. We also demonstrate that for a wide range of graph packing problems (which includes maximum concurrent flow, maximum multiflow and multicast routing, among others, as a special case), the integrality gap of the linear program is always at most O(log k) times the integrality gap restricted to trees. Lastly, we use our ideas to give an efficient construction for vertex-sparsifiers that match the current best existential results - this was previously open. Our algorithm makes novel use of Earth-mover- - constraints.en_US
dc.description.sponsorshipNational Science Foundation (U.S.) (NSF award MSPA-MCS 0528414)en_US
dc.description.sponsorshipNational Science Foundation (U.S.) (NSF award CCF 0832797)en_US
dc.description.sponsorshipNational Science Foundation (U.S.) (NSF award AF 0916218)en_US
dc.description.sponsorshipHertz Foundation (Fellowship)en_US
dc.language.isoen_US
dc.publisherInstitute of Electrical and Electronics Engineersen_US
dc.relation.isversionofhttp://dx.doi.org/10.1109/FOCS.2010.32en_US
dc.rightsArticle is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.en_US
dc.sourceIEEEen_US
dc.titleVertex Sparsifiers and Abstract Rounding Algorithmsen_US
dc.typeArticleen_US
dc.identifier.citationCharikar, Moses, Tom Leighton, Shi Li, and Ankur Moitra. “Vertex Sparsifiers and Abstract Rounding Algorithms.” In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, 23-26 Oct. 2010, Las Vegas, NV. pp. 265-274. Institute of Electrical and Electronics Engineers, © 2010 IEEE.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Mathematicsen_US
dc.contributor.mitauthorLeighton, Frank Thomsonen_US
dc.contributor.mitauthorMoitra, Ankuren_US
dc.relation.journalProceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Scienceen_US
dc.eprint.versionFinal published versionen_US
dc.type.urihttp://purl.org/eprint/type/ConferencePaperen_US
eprint.statushttp://purl.org/eprint/status/NonPeerRevieweden_US
dspace.orderedauthorsCharikar, Moses; Leighton, Tom; Li, Shi; Moitra, Ankuren_US
dc.identifier.orcidhttps://orcid.org/0000-0001-7047-0495
dc.identifier.orcidhttps://orcid.org/0000-0002-1223-2015
mit.licensePUBLISHER_POLICYen_US


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