- Published 2008 in Theory of Computing

We design approximation algorithms for a number of fundamental optimization problems in metric spaces, namely computing separating and padded decompositions, sparse covers, and metric triangulations. Our work is the first to emphasize <i>relative guarantees</i>, that compare the produced solution to the optimal one for the input at hand. By contrast, the extensive previous work on these topics has sought <i>absolute</i> bounds that hold for every possible metric space (or for a family of metrics). While absolute bounds typically translate to relative ones, our algorithms provide significantly better relative guarantees, using a rather different algorithm. Our technical approach is to cast a number of metric clustering problems that have been well studied---but almost always as disparate problems---into a common modeling and algorithmic framework, which we call the <i>consistent labeling</i> problem. Having identified the common features of all of these problems, we provide a family of linear programming relaxations and simple randomized rounding procedures that achieve provably good approximation guarantees.

@article{Krauthgamer2008MetricCV,
title={Metric Clustering via Consistent Labeling},
author={Robert Krauthgamer and Tim Roughgarden},
journal={Theory of Computing},
year={2008},
volume={7},
pages={49-74}
}