Computing the Grothendieck constant of some graph classes

Given a graph G = ([n], E) and w ∈ R , consider the integer program maxx∈{±1}n ∑ ij∈E wijxixj and its canonical semidefinite programming relaxation max ∑ ij∈E wijv T i vj , where the maximum is taken over all unit vectors vi ∈ R. The integrality gap of this relaxation is known as the Grothendieck constant κ(G) of G. We present a closed-form formula for the Grothendieck constant of K5-minor free graphs and derive that it is at most 3/2. Moreover, we show that κ(G) ≤ κ(Kk) if the cut polytope of G is defined by inequalities supported by at most k points. Lastly, since the Grothendieck constant of Kn grows as Θ(log n), it is interesting to identify instances with large gap. However this is not the case for the clique-web inequalities, a wide class of valid inequalities for the cut polytope, whose integrality ratio is shown to be bounded by 3.