A face cover perspective to ℓ₁ embeddings of planar graphs

Abstract

It was conjectured by Gupta et al. [Combinatorica04] that every planar graph can be embedded into 1 with constant distortion. However, given an n-vertex weighted planar graph, the best upper bound on the distortion is only O( √ 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 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 1 . The more general case, where the set of terminals can be covered by γ 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 γ) by Krauthgamer, Lee and Rika [SODA19]. Our contribution is a further improvement on the upper bound to O(√ log 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 1 . Therefore, our result provides a polynomial time O( √ log γ)-approximation to the sparsest cut problem on planar graphs, for the case where all the demand pairs can be covered by γ faces.

Metric embedding via shortest path decompositions

“…The lower bound for p = 1 differs from the upper bound of O( √ k) by an O( √ log k) factor. Our techniques already have been useful for other embedding results, e.g., for embedding planar graphs with small face covers into 1 [Fil19]. We hope that our techniques will find further applications.…”

“…In a follow up paper, [Fil19] (the second author) generalized our definition of SPD to partial-SPD (allowing the lower level in the partition hierarchy to be general subgraph rather than only a shortest path). Given a weighted planar graph G = (V, E, w) with a subset of terminals K, a face cover is a subset of faces such that every terminal lies on some face from the cover.…”

“…Given a weighted planar graph G = (V, E, w) with a subset of terminals K, a face cover is a subset of faces such that every terminal lies on some face from the cover. Given a face cover of size γ, using our embedding result for SPD, [Fil19] shows that the terminal set K can be embedded into 1 with distortion O( √ log γ).…”

Low Treewidth Embeddings of Planar and Minor-Free Metrics

“…Different types of embeddings were studied for planar and minor free graphs. K r -minor-free graphs embed into p space with multiplicative distortion O r (log AFGN18,Fil20a], in particular, they embed into ∞ of dimension O r (log 2 n) with a constant multiplicative distortion. They also admit spanners with multiplicative distortion 1 + and Õr ( −3 ) lightness [BLW17].…”

Multicommodity Flows in Planar Graphs with Demands on Faces

“…[6] made progress towards this conjecture by showing an O(log h) bound on the flow-cut gap, where h is the number of faces having source/sink vertices. This was subsequently improved to O( √ log h) by Filster [4]. Seymour [11] showed that if the union of supply and demand graphs is planar, then the cut condition is also sufficient for routing demands.…”