Guruswami-Sinop Rounding without Higher Level Lasserre

Guruswami and Sinop [11] give a O(1/δ) 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, λr+1 ≥ Φ∗/(1 − δ). 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 l22 triangle inequality constraints. This results in a O(r/δ ) approximation guarantee for the non-uniform Sparsest Cut problem by adding only l22 triangle inequality constraints to the usual semidefinite program, provided that the same spectral condition λr+1 ≥ Φ∗/(1− δ) holds as above.