### 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 language | English (US) |
---|---|

Pages (from-to) | 371-387 |

Number of pages | 17 |

Journal | SIAM Journal on Computing |

Volume | 39 |

Issue number | 2 |

DOIs | |

State | Published - 2009 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Computer Science(all)

### Cite this

*SIAM Journal on Computing*,

*39*(2), 371-387. https://doi.org/10.1137/070685671

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

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 39, no. 2, pp. 371-387. https://doi.org/10.1137/070685671

}

TY - JOUR

T1 - On earthmover distance, Metric labeling, and 0-Extension

AU - Karloff, Howard

AU - Khot, Subhash

AU - Mehta, Aranyak

AU - Rabani, Yuval

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - 0-Extension

KW - Earthmover distance

KW - Metric Labeling

KW - Multiway Cut

UR - http://www.scopus.com/inward/record.url?scp=67650112693&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67650112693&partnerID=8YFLogxK

U2 - 10.1137/070685671

DO - 10.1137/070685671

M3 - Article

AN - SCOPUS:67650112693

VL - 39

SP - 371

EP - 387

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 2

ER -