A Face Cover Perspective to ℓ₁ Embeddings of Planar Graphs

It was conjectured by Gupta et al. that every planar graph can be embedded into ℓ₁ with constant distortion. However, given an n-vertex weighted planar graph, the best upper bound on the distortion is only O(plog n), by Rao. In this article, we study the case where there is a set K of terminals, and the goal is to embed only the terminals into ℓ₁ with low distortion. In a seminal article, Okamura and Seymour showed that if all the terminals lie on a single face, they can be embedded isometrically into ℓ₁. The more general case, where the set of terminals can be covered by γ faces, was studied by Lee and Sidiropoulos and Chekuri et al. The state of the art is an upper bound of O(log γ) by Krauthgamer, Lee and Rika. Our contribution is a further improvement on the upper bound to O(plog γ). Since every planar graph has at most O(n) faces, any further improvement on this result will be a major breakthrough, directly improving upon Rao’s long standing upper bound. Moreover, it is well known that the flow-cut gap equals to the distortion of the best embedding into ℓ₁. Therefore, our result provides a polynomial time O(plog γ)-approximation to the sparsest cut problem on planar graphs, for the case where all the demand pairs can be covered by γ faces.