A $(log n)^{Omega(1)}$ Integrality Gap for the Sparsest Cut SDP

We show that the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap (log n)^Ω(1). This is achieved by exhibiting n-point metric spaces of negative type whose L₁ distortion is (log n)^Ω(1). Our result is based on quantitative bounds on the rate of degeneration of Lipschitz maps from the Heisenberg group to L₁ when restricted to cosets of the center.