A $(\log n)^{\Omega(1)}$ integrality gap for the Sparsest Cut SDP

TL;DR

This paper demonstrates a significant integrality gap for the Sparsest Cut SDP, showing that the relaxation can be far from optimal due to complex metric space properties.

Contribution

It establishes a logarithmic lower bound on the integrality gap for the Sparsest Cut SDP using advanced metric space analysis.

Findings

Existence of n-point metric spaces with high L1 distortion

Quantitative bounds on Lipschitz map degeneration from the Heisenberg group

Demonstration of a logarithmic integrality gap for the SDP

Abstract

We show that the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap $(\log n)^{\Omega(1)}$. This is achieved by exhibiting $n$-point metric spaces of negative type whose $L_1$ distortion is $(\log n)^{\Omega(1)}$. 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.