On a positive semidefinite relaxation of the cut polytope

Abstract

We study the convex set L n defined by L n Z≔ { X| X = ( x ij ) a positive semidefinite n × n matrix, x ii = 1 for all i}. 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 {1, 2, …, n}. Our main motivation for investigating the convex set L n comes from combinatorial optimization, namely from approximating the max-cut problem. An important property of L n is that, due to the positive semidefinite constraints, one can optimize over it in polynomial time. On the other hand, L n 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 ∈ L n 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.