On a positive semidefinite relaxation of the cut polytope Original Research Article

We study the convex set imagen defined by imagen Zcolon, equals {XX = (xij) a positive semidefinite n × n matrix, xii = 1 for all i}. We describe several geometric properties of imagen. In particular, we show that imagen has 2n − 1 vertices, which are its rank one matrices, corresponding to all bipartitions of the set {1, 2, …, n}. Our main motivation for investigating the convex set imagen comes from combinatorial optimization, namely from approximating the max-cut problem. An important property of imagen is that, due to the positive semidefinite constraints, one can optimize over it in polynomial time. On the other hand, imagen still inherits the difficult structure of the underlying combinatorial problem. In particular, it is NP-hard to decide whether the optimum of the problem min Tr(CX), X set membership, variant Ln is reached at a vertex. This result follows from the complete characterization of the matrices C of the form C = bbt for some vector b, for which the optimum of the above program is reached at a vertex.