A Note on Polyhedral Relaxations for the Maximum Cut Problem

We consider three well-studied polyhedral relaxations for the maximum cut problem: the metric polytope of the complete graph, the metric polytope of a general graph, and the relaxation of the bipartite subgraph polytope. The metric polytope of the complete graph can be described with a polynomial number of inequalities, while the latter two may require exponentially many constraints. We give an alternate proof of a theorem of Barahona that states that the metric polytope of a general graph is a projection of the metric polytope of the complete graph. We then give an alternate proof of a theorem of Poljak that states that for any non-negative cost function, the optimal objective value over the relaxation of the bipartite subgraph polytope equals the optimal objective value over the metric polytope. Both proofs are based on the same technique: the separation oracle for the metric polytope of a general graph due to Barahona and Mahjoub. These proofs yield a simple, combinatorial method for proving that three wellstudied polyhedral upper bounds on the value of the maximum cut are the same for graphs with non-negative edge weights.