 
Summary: Quadratic Forms on Graphs
Noga Alon
Konstantin Makarychev
Yury Makarychev
Assaf Naor §
Abstract
We introduce a new graph parameter, called the Grothendieck constant of a graph G = (V, E),
which is defined as the least constant K such that for every A : E R,
sup
f:V SV 1
{u,v}E
A(u, v) · f(u), f(v) K sup
:V {1,+1}
{u,v}E
A(u, v) · (u)(v).
The classical Grothendieck inequality corresponds to the case of bipartite graphs, but the case of
general graphs is shown to have various algorithmic applications. Indeed, our work is motivated
by the algorithmic problem of maximizing the quadratic form {u,v}E A(u, v)(u)(v) over all
: V {1, 1}, which arises in the study of correlation clustering and in the investigation of
the spin glass model. We give upper and lower estimates for the integrality gap of this program.
