# From Balanced Graph Partitioning to Balanced Metric Labeling

## Abstract

The Metric Labeling problem is an elegant and powerful mathematical model capturing a wide range of classification problems that arise in computer vision and related fields. In a typical classification problem, one wishes to assign labels to a set of objects to optimize some measure of the quality of the labeling. The metric labeling problem captures a broad range of classification problems where the quality of a labeling depends on the pairwise relations between the underlying set of objects, as described by 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). Metric labeling has many applications as well as rich connections to some well known problems in combinatorial optimization.

The *balanced metric labeling* problem has an additional constraint requiring that at most ℓ vertices can be assigned to each label, i.e., labels have *capacity*. We focus on the case where the given metric is uniform and note that it already captures various well-known balanced graph partitioning problems. We discuss (pseudo) approximation algorithms for the balanced metric labeling problem, and focus on several important techniques used for obtaining the algorithms. Spreading metrics have proved to be very useful for balanced graph partitioning and our starting point for balanced metric labeling is a linear programming formulation that combines an embedding of the graph in a simplex together with spreading metrics and additional constraints that strengthen the formulation. The approximation algorithm is based on a novel randomized rounding that uses both a randomized metric decomposition technique and a randomized label assignment technique. At the heart of our approach is the fact that only limited dependency is created between the labels assigned to different vertices, allowing us to bound the expected cost of the solution and the number of vertices assigned to each label, simultaneously.