Unifying semidefinite and set-copositive relaxations of binary problems and randomization techniques

A reformulation of quadratically constrained binary programs as duals of set-copositive linear optimization problems is derived using either {0, 1}-formulations or {1, 1}-formulations. The latter representation allows an extension of the randomization technique by Goemans and Williamson. An application to the max-clique problem shows that the max-clique problem is equivalent to a linear program over the max-cut polytope with one additional linear constraint. This transformation allows the solution of a semidefinite relaxation of the max-clique problem with about the same computational e↵ort as the semidefinite relaxation of the max-cut problem – independent of the number of edges in the underlying graph. A numerical comparison of this approach to the standard Lovasz number concludes the talk.