skip to main content
10.1145/1132516.1132595acmconferencesArticle/Chapter ViewAbstractPublication PagesstocConference Proceedingsconference-collections
Article

On earthmover distance, metric labeling, and 0-extension

Published: 21 May 2006 Publication History

Abstract

We study the fundamental classification problems O-EXTENSION and METRIC LABELING. MINIMUM WEIGHT TRIANGULATION is closely related to partitioning problems in graph theory and to Lipschitz extensions in Banach spaces; its generalization METRIC LABELING is motivated by applications in computer vision. Researchers had proposed using earthmover metrics to get polynomial time-solvable relaxations for these problems. A conjecture that has attracted much attention recently is that the integrality ratio for these relaxations is constant.We prove
that the integrality ratio of the earthmover relaxation for METRIC LABELING is Ω(log n) (which is asymptotically tight), k being the number of labels, whereas the best previous lower bound on the integrality ratio was only constant;
that the integrality ratio of the earthmover relaxation for O-EXTENSION is Ω(√log k), k being the number of terminals (it was known to be O((log k)/log log k)), whereas the best previous lower bound was only constant;
that for no ε>0 is there a polynomial-time O((log n)1/4-ε)-approximation algorithm for O-EXTENSION, n being the number of vertices, unless NP ⊆ DTIME(npoly(log n)), whereas the strongest inapproximability result known before was only MAX SNP-hardness; and
that there is a polynomial-time approximation algorithm for O-EXTENSION with performance ratio O(√diam(d)), where diam(d) is the ratio of the largest to smallest nonzero distances in the terminal metric.

References

[1]
A. Archer, J. Fakcharoenphol, C. Harrelson, R. Krauthgamer, K. Talwar, and É. Tardos, Approximate classification via earthmover metrics, in Proc. SODA '04.
[2]
Y. Aumann and Y. Rabani, An O(log k) approximate min-cut max-flow theorem and approximation algorithm, SIAM J. Comput., 27(1):291--301, 1998.
[3]
Y. Bartal, On approximating arbitrary metrics by tree metrics, in Proc. STOC '98.
[4]
J. Bourgain, The metrical interpretation of superreflexivity in Banach spaces, Israel J. Math., 56(2):222--230, 1986.
[5]
G. Calinescu, H. J. Karloff, and Y. Rabani, An improved approximation algorithm for Multiway Cut, J. Comput. and Syst. Sci., 60(3):564--574, 2000 (preliminary version in STOC '98).
[6]
G. Calinescu, H. J. Karloff, and Y. Rabani, Approximation algorithms for the 0-Extension problem, SIAM J. Comput., 34(2):358--372, 2004 (preliminary version in SODA '01).
[7]
M. Charikar, private communication, 2000.
[8]
C. Chekuri, S. Khanna, J. Naor, and L. Zosin, Approximation algorithms for the Metric Labeling problem via a new linear programming formulation, to appear in SIAM J. on Discrete Math (preliminary version in SODA '01).
[9]
J. Chuzhoy and J. Naor, The hardness of Metric Labeling, in Proc. FOCS '04, 108--114.
[10]
J. Fakcharoenphol, C. Harrelson, S. Rao, and K. Talwar, An improved approximation algorithm for the 0-Extension Problem, in Proc. SODA '03, 342--352.
[11]
J. Fakcharoenphol, S. Rao, and K. Talwar, A tight bound on approximating arbitrary metrics by tree metrics, in Proc. STOC '03, 448--455.
[12]
A. Gupta and E. Tardos, A constant factor approximation algorithm for a class of classification problems, in Proc. STOC '00, pages 652--658.
[13]
W. B. Johnson, J. Lindenstrauss, and G. Schechtman, Extensions of Lipschitz maps into Banach space, Israel J. Math., 54(2):129--138, 1986.
[14]
A. V. Karzanov, Minimum 0-extension of graph metrics, Europ. J. Combinat., 19:71--101, 1998.
[15]
R. Krauthgamer, J. Lee, M. Mendel, and A. Naor, "Measured Descent: A New Embedding Method For Finite Metrics," Geometric and Functional Analysis 15 (4), 839--858, 2005. A preliminary version appeared in FOCS 2004.
[16]
S. Khot and A. Naor, Nonembeddability theorems via Fourier analysis, in Proc. FOCS '05.
[17]
J. Kleinberg and É. Tardos. Approximation algorithms for classification problems with pairwise relationships: Metric Labeling and Markov random fields, J. Assoc. Comput. Mach., 49:616--639, 2002 (preliminary version in FOCS '99).
[18]
J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Math. Invent., 160(1): 59--95, 2005.
[19]
N. Linial, E. London, and Y. Rabinovich, The geometry of graphs and some of its algorithmic applications, Combinatorica 15(2):215--245, 1995 (preliminary version in FOCS '94).

Cited By

View all
  • (2017)On the Maximum Rate of Networked Computation in a Capacitated NetworkIEEE/ACM Transactions on Networking10.1109/TNET.2017.269557825:4(2444-2458)Online publication date: 1-Aug-2017
  • (2013)Earth mover's distance based similarity search at scaleProceedings of the VLDB Endowment10.14778/2732240.27322497:4(313-324)Online publication date: 1-Dec-2013
  • (2008)Sdp gaps and ugc hardness for multiway cut, 0-extension, and metric labelingProceedings of the fortieth annual ACM symposium on Theory of computing10.1145/1374376.1374379(11-20)Online publication date: 17-May-2008

Index Terms

  1. On earthmover distance, metric labeling, and 0-extension

    Recommendations

    Comments

    Information & Contributors

    Information

    Published In

    cover image ACM Conferences
    STOC '06: Proceedings of the thirty-eighth annual ACM symposium on Theory of Computing
    May 2006
    786 pages
    ISBN:1595931341
    DOI:10.1145/1132516
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

    Sponsors

    Publisher

    Association for Computing Machinery

    New York, NY, United States

    Publication History

    Published: 21 May 2006

    Permissions

    Request permissions for this article.

    Check for updates

    Qualifiers

    • Article

    Conference

    STOC06
    Sponsor:
    STOC06: Symposium on Theory of Computing
    May 21 - 23, 2006
    WA, Seattle, USA

    Acceptance Rates

    Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

    Upcoming Conference

    STOC '25
    57th Annual ACM Symposium on Theory of Computing (STOC 2025)
    June 23 - 27, 2025
    Prague , Czech Republic

    Contributors

    Other Metrics

    Bibliometrics & Citations

    Bibliometrics

    Article Metrics

    • Downloads (Last 12 months)1
    • Downloads (Last 6 weeks)0
    Reflects downloads up to 14 Feb 2025

    Other Metrics

    Citations

    Cited By

    View all
    • (2017)On the Maximum Rate of Networked Computation in a Capacitated NetworkIEEE/ACM Transactions on Networking10.1109/TNET.2017.269557825:4(2444-2458)Online publication date: 1-Aug-2017
    • (2013)Earth mover's distance based similarity search at scaleProceedings of the VLDB Endowment10.14778/2732240.27322497:4(313-324)Online publication date: 1-Dec-2013
    • (2008)Sdp gaps and ugc hardness for multiway cut, 0-extension, and metric labelingProceedings of the fortieth annual ACM symposium on Theory of computing10.1145/1374376.1374379(11-20)Online publication date: 17-May-2008

    View Options

    Login options

    View options

    PDF

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader

    Figures

    Tables

    Media

    Share

    Share

    Share this Publication link

    Share on social media