Approximation Algorithms for the Metric Labeling Problem via a New Linear Programming Formulation

We consider approximation algorithms for the metric labeling problem. Informally speaking, we are given a weighted graph that specifies relations between pairs of objects drawn from a given set of objects. The goal is to find a minimum cost labeling of these objects where the cost of a labeling is determined by the pairwise relations between the objects and a distance function on labels; the distance function is assumed to be a metric. Each object also incurs an assignment cost that is label, and vertex dependent. The problem was introduced in a recent paper by Kleinberg and Tardos [19], 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. More recently, Gupta and Tardos [14] gave a 4-approximation for the truncated linear metric, a natural non-uniform metric motivated by practical applications to image restoration and visual correspondence. In this paper we introduce a new natural 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. Specifically, we show that the integrality gap of our LP is bounded by O(log k log log k) for general metric and 2 for the uniform metric thus matching the ratios in [19]. We also develop an algorithm based on our LP that achieves a ratio of 2 + √ 2 ≃ 3.414 for the truncated linear metric improving the ratio provided by [14]. Our algorithm uses the fact that the integrality gap of our LP is 1 on a linear metric. We believe that our formulation has the potential to provide improved approximation algorithms for the general case and other useful special cases. Finally, our formulation admits general non-metric distance functions. This leads to a non-trivial approximation guarantee for a non-metric case that arises in practice [20], namely the truncated quadratic distance function. We note here that there are non-metric distance functions for which no bounded approximation ratio is achievable.