The Lovasz theta function of a graph is a well-known upper bound on the stability number. It can be computed efficiently by solving a semidefinite program (SDP). Actually, one can solve either of two SDPs, one due to Lovasz and the other to Groetschel et al. The former SDP is often thought to be preferable computationally, since it has fewer variables and constraints. We derive some new results on these two equivalent SDPs. The surprising result is that, if we weaken the SDPs by aggregating constraints, or strengthen them by adding cutting planes, the equivalence breaks down. In particular, the Groetschel et al. scheme typically yields a stronger bound than the Lovasz one.
Now published as: L. Galli & A.N. Letchford (2017) On the Lovász theta function and some variants. Discr. Optim., 25, 159-174.