A Linear Programming 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 [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.