A face cover perspective to \(\ell_{1}\) embeddings of planar graphs

Abstract

It was conjectured by Gupta et al. [Combinatorica04] that every planar graph can be embedded into \(\ell_{1}\) with constant distortion. However, given an \(n\) -vertex weighted planar graph, the best upper bound on the distortion is only \(O(\sqrt{\log n})\) , by Rao [SoCG99]. In this paper we study the case where there is a set \(K\) of terminals, and the goal is to embed only the terminals into \(\ell_{1}\) 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 \(\ell_{1}\) . The more general case, where the set of terminals can be covered by \(\gamma\) faces, was studied by Lee and Sidiropoulos [STOC09] and Chekuri et al. [J.Comb.Theory13]. The state of the art is an upper bound of \(O(\log\gamma)\) by Krauthgamer, Lee and Rika [SODA19]. Our contribution is a further improvement on the upper bound to \(O(\sqrt{\log\gamma})\) . 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 \(\ell_{1}\) . Therefore, our result provides a polynomial time \(O(\sqrt{\log\gamma})\) -approximation to the sparsest cut problem on planar graphs, for the case where all the demand pairs can be covered by \(\gamma\) faces.