Polyhedral Properties of the K-median Problem on a Tree

The polyhedral structure of the K-median problem on a tree is examined. Even for very small connected graphs, we show that additional constraints are needed to describe the integer polytope. A complete description is given of those trees for which an optimal integer LP solution is guaranteed to exist. We present a new and simpler demonstration that an LP characterization of the 2-median problem is complete. Also, we provide a simpler proof of the value of a tight worst case bound for the LP relaxation. A new class of valid inequalities is identified. These inequalities describe a subclass of facets for the K-median problem on a general graph. Also, we provide polyhedral descriptions for several types of trees. As part of this work, we summarize most known results for the K-median problem on a tree.