Approximating Sparsest Cut in Low Rank Graphs via Embeddings from Approximately Low Dimensional Spaces

We consider the problem of embedding a finite set of points {x1, . . . , xn} ∈ R that satisfy l2 triangle inequalities into l1, when the points are approximately low-dimensional. Goemans (unpublished, appears in [20]) showed that such points residing in exactly d dimensions can be embedded into l1 with distortion at most √ d. We prove the following robust analogue of this statement: if there exists a r-dimensional subspace Π such that the projections onto this subspace satisfy ∑ i,j∈[n] ‖Πxi −Πxj‖ 2 2 ≥ Ω(1) ∑ i,j∈[n] ‖xi − xj‖ 2 2, then there is an embedding of the points into l1 withO( √ r) average distortion. A consequence of this result is that the integrality gap of the well-knownGoemans-Linial SDP relaxation for the Uniform Sparsest Cut problem isO( √ r) on graphs G whose r-th smallest normalized eigenvalue of the Laplacian satisfies λr(G)/n ≥ Ω(1)ΦSDP (G). Our result improves upon the previously known bound of O(r) on the average distortion, and the integrality gap of the Goemans-Linial SDP under the same preconditions, proven in [7, 6]. The Hebrew University of Jerusalem, Israel. e-mail: [email protected]. The Hebrew University of Jerusalem, Israel. e-mail: [email protected]. Supported by an I-Core Algorithms Fellowship.