SDP integrality gaps with local ℓ₁-embeddability

We construct integrality gap instances for SDP relaxation of the MAXIMUM CUT and the SPARSEST CUT problems. If the triangle inequality constraints are added to the SDP, then the SDP vectors naturally define an n-point negative type metric where n is the number of vertices in the problem instance. Our gap-instances satisfy a stronger constraint that every sub-metric on t = O((log log log n)1/6) points is isometrically embeddable into ℓ₁. The local ℓ₁-embeddability constraints are implied when the basic SDP relaxation is augmented with t rounds of the Sherali-Adams LP-relaxation. For the MAXIMUM CUT problem, we obtain an optimal gap of α_GW ^-1 - ε, where αGW is the Goemans-Williamson constant [11] and ε > 0 is an arbitrarily small constant. For the SPARSEST C UT problem, we obtain a gap of Ω((log log log n) 1/13). The latter result can be rephrased as a construction of an n-point negative type metric such that every t-point sub-metric is isometrically ℓ₁ -embeddable, but embedding the whole metric into ℓ₁ incurs distortion Ω((log log log n) 1/13).