A $(\log n{)}^{\mathrm{\Omega }(1)}$ 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 $(\log n{)}^{\mathrm{\Omega }(1)}$. This is achieved by exhibiting $n$-point metric spaces of negative type whose $L_1$ distortion is $(\log n{)}^{\mathrm{\Omega }(1)}$. Our result is based on quantitative bounds on the rate of degeneration of Lipschitz maps from the Heisenberg group to $L_1$ when restricted to cosets of the center.