Nonpolyhedral Relaxations of Graph-Bisection Problems

We study the problem of nding the minimum bisection of a graph into two parts of prescribed sizes. We formulate two lower bounds on the problem by relaxing nodeand edgeincidence vectors of cuts. We prove that both relaxations provide the same bound. The main fact we prove is that the duality between the relaxed edgeand nodevectors preserves very natural cardinality constraints on cuts. We present an analogous result also for the max-cut problem, and show a relation between the edge relaxation and some other optimality criteria studied before. Finally, we brie y mention possible applications for a practical computational approach.