The facial structure of the clique partitioning polytope

The clique partitioning problem (CPP) can be formulated as follows. Given is a complete graph G = (V; E), with edge weights w ij 2 R for all fi; jg 2 E. A subset A E is called a clique partition if there is a partition of V into non-empty, disjoint sets V 1 S k p=1 ffi; jgji; j 2 V p g. The weight of such a clique partition A is deened as P fi;jg2A w ij. The problem is now to nd a clique partition of maximal weight. The clique partitioning polytope P is the convex hull of the incidence vectors of all clique partitions of G. In this paper we introduce several new classes of facet deening inequalities of the clique partitioning polytope, and we present procedures that combine facet deening inequalities into new ones. Finally, we characterize all facet deening inequalities with right hand side equal to 1 or 2.