MIT Libraries logoDSpace@MIT

MIT
View Item 
  • DSpace@MIT Home
  • MIT Open Access Articles
  • MIT Open Access Articles
  • View Item
  • DSpace@MIT Home
  • MIT Open Access Articles
  • MIT Open Access Articles
  • View Item
JavaScript is disabled for your browser. Some features of this site may not work without it.

An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations

Author(s)
Lee, Yin Tat; Orecchia, Lorenzo; Kelner, Jonathan Adam; Sidford, Aaron D.
Thumbnail
DownloadKelner_An almost.pdf (445.5Kb)
PUBLISHER_POLICY

Publisher Policy

Article 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.

Terms of use
Article 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.
Metadata
Show full item record
Abstract
In this paper, we introduce a new framework for approximately solving flow problems in capacitated, undirected graphs and apply it to provide asymptotically faster algorithms for the maximum s-t flow and maximum concurrent multicommodity flow problems. For graphs with n vertices and m edges, it allows us to find an ∊-approximate maximum s-t flow in time O(m[superscript 1+o(1)]∊[superscript −2]), improving on the previous best bound of Õ(mn[superscript 1 over 3]poly(∊[superscript −1])). Applying the same framework in the multicommodity setting solves a maximum concurrent multicommodity flow problem with k commodities in O(m[superscript 1+o(1)]∊[superscript −2]k[superscript 2]) time, improving on the existing bound of Õ(m[superscript 4 over 3]poly(k, ∊[superscript −1])). Our algorithms utilize several new technical tools that we believe may be of independent interest: We give a non-Euclidean generalization of gradient descent and provide bounds on its performance. Using this, we show how to reduce approximate maximum flow and maximum concurrent flow to oblivious routing. We define and provide an efficient construction of a new type of flow sparsifier. Previous sparsifier constructions approximately preserved the size of cuts and, by duality, the value of the maximum flows as well. However, they did not provide any direct way to route flows in the sparsifier G′ back in the original graph G, leading to a longstanding gap between the efficacy of sparsification on flow and cut problems. We ameliorate this by constructing a sparsifier G' that can be embedded (very efficiently) into G with low congestion, allowing one to transfer flows from G′ back to G. We give the first almost-linear-time construction of an O(m[superscript o(1)])-competitive oblivious routing scheme. No previous such algorithm ran in time better than [~ over Ω](mn). By reducing the running time to almost-linear, our work provides a powerful new primitive for constructing very fast graph algorithms. The interested reader is referred to the full version of the paper [8] for a more complete treatment of these results.
Date issued
2014
URI
http://hdl.handle.net/1721.1/92917
Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Department of Mathematics
Journal
Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms
Publisher
Society for Industrial and Applied Mathematics
Citation
Kelner, Jonathan A., Yin Tat Lee, Lorenzo Orecchia, and Aaron Sidford. “An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and Its Multicommodity Generalizations.” Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (December 18, 2013): 217–226. © 2014 Society for Industrial and Applied Mathematics
Version: Final published version
ISSN
1071-9040
2160-1445
1557-9468

Collections
  • MIT Open Access Articles

Browse

All of DSpaceCommunities & CollectionsBy Issue DateAuthorsTitlesSubjectsThis CollectionBy Issue DateAuthorsTitlesSubjects

My Account

Login

Statistics

OA StatisticsStatistics by CountryStatistics by Department
MIT Libraries
PrivacyPermissionsAccessibilityContact us
MIT
Content created by the MIT Libraries, CC BY-NC unless otherwise noted. Notify us about copyright concerns.