Guruswami-Sinop Rounding without Higher Level Lasserre

TL;DR

This paper improves approximation guarantees for the non-uniform Sparsest Cut problem by replacing complex Lasserre constraints with simpler $ ext{l}_2^2$ triangle inequalities, maintaining spectral conditions.

Contribution

It demonstrates that similar rounding techniques can be achieved using only $ ext{l}_2^2$ triangle inequality constraints instead of higher-level Lasserre constraints.

Findings

Achieves $O(r/ ext{delta}^2)$ approximation with $ ext{l}_2^2$ constraints.

Maintains spectral condition $ ext{lambda}_{r+1} \

Results in simpler SDP formulations for Sparsest Cut approximation.

Abstract

Guruswami and Sinop give a $O(1/\delta)$ approximation guarantee for the non-uniform Sparsest Cut problem by solving $O(r)$-level Lasserre semidefinite constraints, provided that the generalized eigenvalues of the Laplacians of the cost and demand graphs satisfy a certain spectral condition, namely, $\lambda_{r+1} \geq \Phi^{*}/(1-\delta)$. Their key idea is a rounding technique that first maps a vector-valued solution to $[0, 1]$ using appropriately scaled projections onto Lasserre vectors. In this paper, we show that similar projections and analysis can be obtained using only $\ell_{2}^{2}$ triangle inequality constraints. This results in a $O(r/\delta^{2})$ approximation guarantee for the non-uniform Sparsest Cut problem by adding only $\ell_{2}^{2}$ triangle inequality constraints to the usual semidefinite program, provided that the same spectral condition, $\lambda_{r+1} \geq \Phi^{*}/(1-\delta)$, holds.