On earthmover distance, Metric labeling, and 0-Extension

Howard Karloff, Subhash Khot, Aranyak Mehta, Yuval Rabani

Research output: Contribution to journalArticle

Abstract

We study the fundamental classification problem s 0-Extension and Metric Labeling. A generalization of Multiway Cut, 0-Extension 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 (1) that the integrality ratio of the earthmover relaxation for Metric Labeling is Ω (log k) (which is asymptotically tight), k being the number of labels, whereas the best previous lower bound on the integrality ratio was only constant; (2) that the integrality ratio of the earthmover relaxation for 0-Extension is Ω (√log k), k being the number of terminals (it was known to be O((logk)/log logk)), whereas the best previous lower bound was only constant; (3) that for no ∈ > 0 is there a polynomialtime O((logn) 1/4-∈)-approximation algorithm for 0-Extension, n being the number of vertices, unless NP⊆DTIME(n poly(log n)), whereas the strongest inapproximability result known before was only MAX SNP-hardness; and (4) that there is a polynomial-time approximation algorithm for 0-Extension with performance ratio O(√ diam(d)), where diam(d) is the ratio of the largest to smallest nonzero distances in the terminal metric.

Original languageEnglish (US)
Pages (from-to)371-387
Number of pages17
JournalSIAM Journal on Computing
Volume39
Issue number2
DOIs
StatePublished - 2009

Fingerprint

Distance Metric
Labeling
Integrality
Approximation algorithms
Metric
Polynomials
Graph theory
Banach spaces
Relaxation time
Computer vision
Labels
Approximation Algorithms
Hardness
Multiway Cut
Lower bound
Inapproximability
Classification Problems
Computer Vision
Polynomial-time Algorithm
Lipschitz

Keywords

  • 0-Extension
  • Earthmover distance
  • Metric Labeling
  • Multiway Cut

ASJC Scopus subject areas

  • Mathematics(all)
  • Computer Science(all)

Cite this

On earthmover distance, Metric labeling, and 0-Extension. / Karloff, Howard; Khot, Subhash; Mehta, Aranyak; Rabani, Yuval.

In: SIAM Journal on Computing, Vol. 39, No. 2, 2009, p. 371-387.

Research output: Contribution to journalArticle

Karloff, Howard ; Khot, Subhash ; Mehta, Aranyak ; Rabani, Yuval. / On earthmover distance, Metric labeling, and 0-Extension. In: SIAM Journal on Computing. 2009 ; Vol. 39, No. 2. pp. 371-387.
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