Size-constrained graph partitioning polytope. Part II: Non-trivial facets

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. This is the second part of the two papers addressing the study of the facial structure of the size-constrained graph partitioning polytope. All the definitions and notation of the first paper (i.e., “part I”) apply for this paper as well. Hence, we start numbering the sections of this paper from where we have left in the first paper. In this paper, we prove facetness for P lu of several classes of valid inequalities: 2partition inequalities, lower and upper general clique inequalities, cycle inequalities and the lower and upper 2-star inequalities. As already stated in the first paper, the equipartition polytope P(n) and the k-way equi-partition polytope Pk−way(n, k) are special cases of P . Certain theorems of this paper prove for the first time in the literature that some of the aforementioned valid inequalities are facet defining for these two polytopes as well. We highlight such results we obtain as corollaries to corresponding theorems. ∗Corresponding author: Universit Libre de Bruxelles, Boulevard du Triomphe, CP 210/01, 1050 Bruxelles, Belgique. E-mail:[email protected]