With the work of Khot and Vishnoi (FOCS 2005) as a starting point, we obtain integrality gaps for certain strong SDP relaxations of unique games. Specifically, we exhibit a gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner products of up to $\exp(\Omega(\log\log~n)^{1/4})$ vectors. For stronger relaxations obtained from the basic semidefinite program by $R$ rounds of Sherali--Adams lift-and-project, we prove a unique games integrality gap for $R = \Omega(\log\log~n)^{1/4}$.By composing these SDP gaps with UGC-hardness reductions, the above results imply corresponding integrality gaps for every problem for which a UGC-based hardness is known. Consequently, this work implies that including any valid constraints on up to$\exp(\Omega(\log\log~n)^{1/4})$ vectors to natural semidefinite program, does not improve the approximation ratio for any problem in the following classes: constraint satisfaction problems, ordering constraint satisfaction problems and metric labeling problems over constant-size metrics. We obtain similar SDP integrality gaps for balanced separator, building on Devanur et al. (STOC 2006). We also exhibit, for explicit constants $\gamma, \delta ≫ 0$, an n-point negative-type metric which requires distortion $\Omega(\log\log n)^{\gamma}$ to embed into$\ell_1$, although all its subsets of size$\exp(\Omega(\log\log~n)^{\delta})$ embed isometrically into $\ell_1$.
['Prasad Raghavendra', 'David Steurer']
Integrality Gaps for Strong SDP Relaxations of UNIQUE GAMES