Size-constrained graph partitioning polytopes

We consider the problem of clustering a set of items into subsets whose sizes are bounded from above and below. We formulate the problem as a graph partitioning problem and propose an integer programming model for solving it. This formulation generalizes several well-known graph partitioning problems from the literature like the clique partitioning problem, the equi-partition problem and the k-way equi-partition problem. In this paper, we analyze the structure of the corresponding polytope and prove several results concerning the facial structure. Our analysis yields important results for the closely related equi-partition and k-way equi-partition polytopes as well.