A face cover perspective to $\ell_1$ embeddings of planar graphs

TL;DR

The paper proves that for a planar graph with terminal set $K$ covered by $\gamma$ faces, there exists a polynomial-time embedding of $K$ into $\ell_1$ with distortion $O\left(\sqrt{\log \gamma}\right)$. This is achieved via a novel Partial Shortest Path Decomposition (PSPD) that yields lower-level clusters where terminals lie on a single face (Okamura–Seymour structure) and combines uniformly and non-uniformly truncated embeddings—constructed through stochastic embeddings into trees—to control cross-cluster distortion. A key step is a Lipschitz extension mechanism that extends the terminal embedding to all vertices while preserving no contraction on $K$ and maintaining expansion $O\left(\sqrt{\log \gamma}\right)$. Consequently, the authors obtain a polynomial-time $O\left(\sqrt{\log \gamma}\right)$-approximation for the sparsest cut in planar graphs with face-cover demands, matching Rao’s bound up to the face-cover parameter and representing a significant advancement over prior $O(\log \gamma)$ guarantees. The work blends hierarchical graph decompositions, truncated embeddings, and constructive Lipschitz extension to push forward the algorithmic frontiers of planar graph embeddings into $\ell_1$.

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 $γ$ 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 $O(\sqrt{\logγ})$. 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 $O(\sqrt{\log γ})$-approximation to the sparsest cut problem on planar graphs, for the case where all the demand pairs can be covered by $γ$ faces.

Figures (5)

• Figure 1: The terminal vertices colored in red. The size of the face cover is $4$. The faces in the cover are encircled by a blue dashed lines.
• Figure 2: Illustration of a PSPD of depth $2$. Here $\mathcal{P}_{1}$ is the purple path $P_1$, while $\mathcal{P}_{2}$ is the two green paths $P_2$ and $P_3$. The reminder of the PSPD is the $4$ clusters $\mathcal{C}=\{C_1,C_2,C_3,C_4\}$ encircled by a dashed red line, while the boundary is $\mathcal{B}=P_1\cup P_2\cup P_3$.
• Figure 3: Both figures on the left and on the right illustrate two different instances where \ref{['lem:BoundedOStoL1']} can be applied. The boundary is $\mathcal{B}$ encircled by a red dashed line. The interior $\mathcal{I}$, which is the rest of $G$ vertices, is encircled by a blue dashed line. The induced graph on the interior $G[\mathcal{I}]$ contains a face $F$ (denoted by a purple line, while $F$ vertices are red). \ref{['lem:BoundedOStoL1']} constructs a Lipschitz (guarantee (2)) embedding of $F$ vertices into $\ell_1$, where the norm of the image of each vertex $v\in F$ equals to its distance to the boundary (guarantee (1)). Furthermore, for every $u,v\in F$ it holds that $\left\Vert {f}(v)-{f}(u)\right\Vert _{1}\ge\frac{\min\left\{ d_{G}(v,\mathcal{B}),d_{G}(u,\mathcal{B})\right\} }{12}$ (guarantee (3)).
• Figure 4: On the left side displayed a graph $G$. The terminals are colored in red. The face cover consist of the faces $F_1,\dots,F_6$, surrounded by blue dashed lines. For each face $F_i$ let $v_{F_i}$ (denoted $v_i$) be an arbitrary vertex on $F_i$. Define a weight function $\omega$ by adding a unit of weight to every $v_i$. In the illustration $v_1,v_2,v_3,v_4$ have weight $1$, $v_5$ has weight $2$, while all other vertices have weight $0$. The separator consists of shortest paths $P_1,P_2$ colored purple. On the right side we display the graph after removing all separator vertices. In each connected component $C$, a new face cover is defined by taking the outer face and adding a single face for every $v_i\in C$.
• Figure :

Theorems & Definitions (32)

• Theorem 1
• Corollary 1
• Corollary 2
• Definition 1: Partial Shortest Path Decomposition (PSPD)
• Theorem 2: Embedding using PSPD
• Theorem 3
• Theorem 4
• Corollary 3
• proof
• Lemma 1