The hardness of metric labeling

The metric labeling problem is an elegant and powerful mathematical model capturing a wide range of classification problems. The input to the problem consists of a set of labels and a weighted graph. Additionally, a metric distance function on the labels is defined, and for each label and each vertex, an assignment cost is given. The goal is to find a minimum-cost assignment of the vertices to the labels. The cost of the solution consists of two parts: the assignment costs of the vertices and the separation costs of the edges (each edge pays its weight times the distance between the two labels to which its endpoints are assigned). Due to the simple structure and variety of the applications, the problem and its special cases (with various distance functions on the labels) have recently received much attention. Metric labeling has a known logarithmic approximation, and it has been an open question for several years whether a constant approximation exists. We refute this possibility and show that no constant approximation can be obtained for the problem unless P=NP, and we also show that the problem is Ω(√logn)-hard to approximate, unless NP has quasi-polynomial time algorithms.