On the Graph Bisection Cut Polytope

Abstract

Given a graph $G=(V,E)$ with node weights $\varphi_v \in \mathbb{N}\cup\{0\}$, $v\in V$, and some number $F\in \mathbb{N}\cup\{0\}$, the convex hull of the incidence vectors of all cuts $\delta(S)$, $S\subseteq V$, with $\varphi(S)\le F$ and $\varphi(V\setminus S)\le F$ 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 [C. E. Ferreira et al., Math. Programming, 74 (1996), pp. 247-267] to be facet-defining. We extend these inequalities to a richer class by exploiting the fact that each cut intersects each cycle in an even number of edges. Finally, we present a new class of inequalities that are based on nonconnected substructures yielding nonlinear right-hand sides. We show that the supporting hyperplanes of the convex envelope of this nonlinear function correspond to the faces of the so-called cluster weight polytope, for which we give a complete description under certain conditions.