Assume we want to refute an unsatisfiable CNF. We can interpret it as an integer program, thus a refutation can be done by applying Lovasz-Schrijver semidefinite cuts ($LS_{+}$ cuts) to its linear relaxation until the polytope is empty.
For a polytope $P$ we consider $N_{+}(P)$ to be the polytope where all possible $LS_{+}$ cuts have been further applied. We can always iterate the operator $N_{+}$ until the resulting polytope is empty. The number of iterations required to reach the empty polytope is called the rank of the polytope.
Since each iteration increases the number of facet of a polynomial factor, a non constant number of iterations is out of question.
In the case of Gomory-Chvátal cuts, the usual PHP formula requires rank $\Omega(\log n)$, but there exists a refutation of polynomial size. Such refutation essentially produces the empty polytope using only a small number of the facets from each iteration.
My question is: does there exist a similar example for $LS_{+}$ cuts, i.e. a case where an short refutation exists but the rank is large?
Update: The following paper shows (among other things) that a tautology with short refutation and linear rank would separate Tree-Like LS+ from general LS+ refutations.
Toniann Pitassi and Nathan Segerlind, "Exponential Lower Bounds and Integrality Gaps for Tree-like Lovasz-Schrijver Procedures", SODA, 2009.
