Integrality Gaps for Strong SDP Relaxations of U G

With the work of Khot and Vishnoi [18] as a starting point, we obtain integrality gaps for certain strong SDP relaxations of U G. Specifically, we exhibit a U G gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner products of up to exp(Ω(log log n)1/4) vectors. For a stronger relaxation obtained from the basic semidefinite program by R rounds of Sherali–Adams liftand-project, we prove a U G integrality gap for R = Ω(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(Ω(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 B S, building on [11]. We also exhibit, for explicit constants γ, δ > 0, an n-point negative-type metric which requires distortion Ω(log log n) to embed into `1, although all its subsets of size exp(Ω(log log n)) embed isometrically into `1. Keywords-semidefinite programming, approximation algorithms, unique games conjecture, hardness of approximation, SDP hierarchies, Sherali–Adams hierarchy, integrality gap construction