On a Positive Semideenite Relaxation of the Cut Polytope

We study the convex set L n deened by L n := fX j X = (x ij) positive semideenite n n matrix; x ii = 1 for all ig: We describe several geometric properties of L n. In particular, we show that L n has 2 n?1 vertices, which are its rank one matrices, corresponding to all bipartitions of the set f1; 2; : : : ; ng. Our main motivation for investigating the convex set L n comes from com-binatorial optimization, namely from approximating the max-cut problem. An important property of L n is that, due to the positive semideenite constraints , one can optimize over it in polynomial time. On the other hand, L n still inherits the diicult structure of the underlying combinatorial problem. In particular, it is NP-hard to decide whether the optimum of the problem is reached at a vertex. This result follows from the complete characterization of the matrices C of the form C = bb t for some vector b, for which the optimum of the above program is reached at a vertex.