Relaxations of Combinatorial Problems Via Association Schemes

Semidefinite programming relaxations of combinatorial problems date back to the work of Lovász [17] from 1979, who proposed a semidefinite programming relaxation for the maximum stable set problem which now is known as the Lovász theta number. More recently, Goemans and Williamson [9] showed how to use semidefinite programming to provide an approximation algorithm for the maximum-cut problem; this algorithm achieves the best known approximation ratio for the problem, which is moreover conjectured to be the best possible ratio under the unique games conjecture, a complexity-theoretical assumption (cf. Khot, Kindler, Mossel, and O’Donnell [12]). The usual approach to obtaining semidefinite programming relaxations of combinatorial problems has been via binary variable reformulations. For example, the convex hull of the set {xx : x ∈ {−1, 1} } is approximated by the convex elliptope