Balanced metric labeling

We define the balanced metric labeling problem, a generalization of the metric labeling problem, in which each label has a capacity, i.e., at most l vertices can be assigned to it. The balanced metric labeling problem is a generalization of fundamental problems in the area of approximation algorithms, e.g., arrangements and balanced partitions of graphs. It is also motivated by resource limitations in certain practical scenarios. We focus on the case where the given metric is uniform and note that this case alone encompasses various well-known graph partitioning problems. We present the first (pseudo) approximation algorithm for this problem, achieving for any ε, 0 < ε < 1, an approximation factor of O((ln n)/ε), while assigning at most min {O(ln k)/1 - ε, l + 1| ( 1 + ε) l vertices to each label (k is the number of labels). Our approximation algorithm is based on a novel randomized rounding of 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. Our randomized rounding technique 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. We note that the number of vertices assigned to each label is bounded via a new inequality of Janson[15] for tail bounds of (partly) dependent random variables.