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arXiv · 0910.2024

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

Jeff Cheeger·Bruce Kleiner·Assaf Naor
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Abstract

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 $L_1$ 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 $L_1$ when restricted to cosets of the center.

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Jeff Cheeger, Bruce Kleiner, Assaf Naor. 2009-11-18. A $(\log n)^{Ω(1)}$ integrality gap for the Sparsest Cut SDP. https://arxiv.org/abs/0910.2024

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