A Linear Programming Formulation and Approximation Algorithms for the Metric Labeling Problem

Loading...
Thumbnail Image
Penn collection
Departmental Papers (CIS)
Degree type
Discipline
Subject
metric labeling
linear program
approximation algorithm
truncated linear metric
Funder
Grant number
License
Copyright date
Distributor
Related resources
Author
Chekuri, Chandra
Naor, Joseph
Zosin, Leonid
Contributor
Abstract

We consider approximation algorithms for the metric labeling problem. This problem was introduced in a paper by Kleinberg and Tardos [J. ACM, 49 (2002), pp. 616–630] and captures many classification problems that arise in computer vision and related fields. They gave an O(log k log log k) approximation for the general case, where k is the number of labels, and a 2-approximation for the uniform metric case. (In fact, the bound for general metrics can be improved to O(log k) by the work of Fakcheroenphol, Rao, and Talwar [Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 2003, pp. 448–455].) Subsequently, Gupta and Tardos [Proceedings of the 32nd Annual ACM Symposium on the Theory of Computing, 2000, pp. 652–658] gave a 4-approximation for the truncated linear metric, a metric motivated by practical applications to image restoration and visual correspondence. In this paper we introduce an integer programming formulation and show that the integrality gap of its linear relaxation either matches or improves the ratios known for several cases of the metric labeling problem studied until now, providing a unified approach to solving them. In particular, we show that the integrality gap of our linear programming (LP) formulation is bounded by O(log k) for a general k-point metric and 2 for the uniform metric, thus matching the known ratios. We also develop an algorithm based on our LP formulation that achieves a ratio of 2 + √2 ~= 3.414 for the truncated linear metric improving the earlier known ratio of 4. Our algorithm uses the fact that the integrality gap of the LP formulation is 1 on a linear metric.

Advisor
Date Range for Data Collection (Start Date)
Date Range for Data Collection (End Date)
Digital Object Identifier
Series name and number
Publication date
2005-01-01
Journal title
Volume number
Issue number
Publisher
Publisher DOI
Journal Issue
Comments
Copyright SIAM, 2005. Reprinted in SIAM Journal on Discrete Mathematics, Volume 18, Issue 3, 2005, pages 608-625.
Recommended citation
Collection