A face cover perspective to ℓ_1 embeddings of planar graphs
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It was conjectured by Gupta et. al. [Combinatorica04] that every planar graph can be embedded into ℓ_1 with constant distortion. However, given an n-vertex weighted planar graph, the best upper bound is only O(√( n)) by Rao [SoCG99]. In this paper we study the terminated case, where there is a set K of terminals, and the goal is to embed only the terminals into ℓ_1 with low distortion. In a seminal paper, Okamura and Seymour [J.Comb.Theory81] showed that if all the terminals lie on a single face, they can be embedded isometrically into ℓ_1. More generally, suppose that the terminals could be covered by γ faces. In a recent paper Krauthgamer, Lee and Rika [SODA19] showed an upper bound of O(γ) on the distortion, improving previous results by Lee and Sidiropoulos [STOC09] and Chekuri et. al. [J.Comb.Theory13]. Our contribution is a further improvement of the upper bound to O(√(γ)). Note that since every planar graph has at most O(n) faces, any further improvement of this result, will imply an improvement upon Rao's long standing upper bound. It is well known that the flow-cut gap equals to the distortion of the best embedding into ℓ_1. In particular, our result provide a polynomial time O(√(γ))-approximation to the sparsest cut problem on planar graph, for the case where all the demand pairs can be covered by γ faces.
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