On the Graph Bisection Cut Polytope

Given a graph G = (V, E) with node weights v N {0}, v V , and some number F N{0}, the convex hull of the incidence vectors of all cuts (S), S V with (S) F and (V \ S) F is called the bisection cut polytope. We study the facial structure of this polytope which shows up in many graph partitioning problems with applications in VLSI-design or frequency assignment. We give necessary and in some cases sufficient conditions for the knapsack tree inequalities introduced in [9] to be facet-defining. We extend these inequalities to a richer class by exploiting that each cut intersects each cycle in an even number of edges. Finally, we present a new class of inequalities that are based on non-connected substructures yielding non-linear right-hand sides. We show that the supporting hyperplanes of the convex envelope of this non-linear function correspond to the faces of the so-called cluster weight polytope, for which we give a complete description under certain conditions.

• Type: Journal
• Year: 2008
• Where: SIAMDM
• Authors: Michael Armbruster, Christoph Helmberg, Marzena Fügenschuh, Alexander Martin
• Keywords: Convex Hull, Knapsack Tree Inequalities, Polytope, SIAMDM 2008