An LP Formulation and Approximation Algorithms for the Metric Labeling Problem

We consider approximation algorithms for the metric labeling problem. This problem was introduced in a paper by Kleinberg and Tardos [26], and captures many classification problems t hat arise in computer vision and related fields. They gave an approximation for the general case where is the number of labels, and a -approximation for the uniform metric case. (In fact, the bound for general metrics can be improved to by the work of Fakcheroenphol, Rao, and Talwar [16].) Subsequently, Gupta and Tardos [18] gave a -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 i mproves 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 LP is bounded by for a general -point metric and for the uniform metric thus matching the known ratios. We also develop an algorithm based on our LP that achieves a ratio of for the truncated linear metric improving the earlier known ratio of . Our algorithm uses the fact that the integrality gap of the LP is on a linear metric.