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Embedding Planar Graphs into Graphs of Treewidth $O(\log^{3} n)$

Hsien-Chih Chang, Vincent Cohen-Addad, Jonathan Conroy, Hung Le, Marcin Pilipczuk, Michał Pilipczuk

TL;DR

This work substantially narrows the gap between the treewidth of their embedding and the treewidth lower bound of $\Omega(\log n)$ shown by Carroll and Goel [CG04], and gives an optimal treewidth bound for graphs admitting a contraction sequence.

Abstract

Cohen-Addad, Le, Pilipczuk, and Pilipczuk [CLPP23] recently constructed a stochastic embedding with expected $1+\varepsilon$ distortion of $n$-vertex planar graphs (with polynomial aspect ratio) into graphs of treewidth $O(\varepsilon^{-1}\log^{13} n)$. Their embedding is the first to achieve polylogarithmic treewidth. However, there remains a large gap between the treewidth of their embedding and the treewidth lower bound of $Ω(\log n)$ shown by Carroll and Goel [CG04]. In this work, we substantially narrow the gap by constructing a stochastic embedding with treewidth $O(\varepsilon^{-1}\log^{3} n)$. We obtain our embedding by improving various steps in the CLPP construction. First, we streamline their embedding construction by showing that one can construct a low-treewidth embedding for any graph from (i) a stochastic hierarchy of clusters and (ii) a stochastic balanced cut. We shave off some logarithmic factors in this step by using a single hierarchy of clusters. Next, we construct a stochastic hierarchy of clusters with optimal separating probability and hop bound based on shortcut partition [CCLMST23, CCLMST24]. Finally, we construct a stochastic balanced cut with an improved trade-off between the cut size and the number of cuts. This is done by a new analysis of the contraction sequence introduced by [CLPP23]; our analysis gives an optimal treewidth bound for graphs admitting a contraction sequence.

Embedding Planar Graphs into Graphs of Treewidth $O(\log^{3} n)$

TL;DR

This work substantially narrows the gap between the treewidth of their embedding and the treewidth lower bound of shown by Carroll and Goel [CG04], and gives an optimal treewidth bound for graphs admitting a contraction sequence.

Abstract

Cohen-Addad, Le, Pilipczuk, and Pilipczuk [CLPP23] recently constructed a stochastic embedding with expected distortion of -vertex planar graphs (with polynomial aspect ratio) into graphs of treewidth . Their embedding is the first to achieve polylogarithmic treewidth. However, there remains a large gap between the treewidth of their embedding and the treewidth lower bound of shown by Carroll and Goel [CG04]. In this work, we substantially narrow the gap by constructing a stochastic embedding with treewidth . We obtain our embedding by improving various steps in the CLPP construction. First, we streamline their embedding construction by showing that one can construct a low-treewidth embedding for any graph from (i) a stochastic hierarchy of clusters and (ii) a stochastic balanced cut. We shave off some logarithmic factors in this step by using a single hierarchy of clusters. Next, we construct a stochastic hierarchy of clusters with optimal separating probability and hop bound based on shortcut partition [CCLMST23, CCLMST24]. Finally, we construct a stochastic balanced cut with an improved trade-off between the cut size and the number of cuts. This is done by a new analysis of the contraction sequence introduced by [CLPP23]; our analysis gives an optimal treewidth bound for graphs admitting a contraction sequence.

Paper Structure

This paper contains 24 sections, 20 theorems, 37 equations, 9 figures.

Table of Contents

  1. Introduction
  2. A brief overview of the CLPP embedding.
  3. Our ideas.
  4. Technical Overview
  5. Preliminaries
  6. The Embedding Framework: Proof of Theorem \ref{['thm:embedding']}
  7. $\textsc{Embed}_{\mathbb{C}}(T, P, {\partial\!} T, {\partial\!} \mathcal{C})$:
  8. $\textsc{Embed}(G)$:
  9. Bounding the treewidth
  10. Bounding the expected distortion
  11. Stochastic Balanced Cuts: Proof of Theorem \ref{['thm:balanced-cuts']}
  12. Separating Distribution of Clustering Chains: Proof of Theorem \ref{['thm:beta-separating']}
  13. Construction of stochastic buffered cop decomposition
  14. $\textsc{BuildTree}(\mathcal{S}, H)$:
  15. $\textsc{GrowBuffer}(\mathcal{S}, \mathcal{X}, H)$:
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $G$ be an $n$-vertex apex-minor-free graph with aspect ratio $\Phi$. For any parameter $\varepsilon \in (0,1)$, $G$ can be stochastically embedded with expected distortion $1+\varepsilon$ into a distribution of graphs of treewidth $O(\varepsilon^{-1}\cdot \log^2 \Phi \cdot \log(n\Phi))$.

Figures (9)

  • Figure 1: (a) A clustering chain $\mathbb{C} = \{\mathcal{C}_0,\mathcal{C}_1,\mathcal{C}_2,\mathcal{C}_3\}$. (b) Viewing $\mathbb{C}$ as a hierarchy represented by a tree. (c) Balanced cut $\mathcal{F} = \{A,B\}$ respects $\mathbb{C}$ since the cut contains two clusters $A$ and $B$ (at different levels) of $\mathbb{C}$; removing this cut results in connected components of size at most $2n/3$ each. If $\mathcal{X} = \{X_1,X_2,X_3\}$ as in (b), then the cut $\mathcal{F}$ in (c) conforms $\mathcal{X}$.
  • Figure 2: The embedding algorithm.
  • Figure 3: An embedding of $G$ obtained by calling $\textsc{Embed}_{\mathbb{C}}(V, G, \varnothing, \varnothing)$: the algorithm will call $\textsc{Cut}_{\mathbb{C}}(V, G, \varnothing)$ to sample a cut $\mathcal{F}$; assume that $\mathcal{F} = \{A,B\}$ as in the figure. Then, the algorithm will recursively embed the terminals in components of $G\setminus \mathcal{F}$, which are $H_1,H_2$. Note that the graphs we recurse on to embed terminals in $H_1$ and $H_2$ are $G$. Terminals are filled vertices, while non-terminals are hollow vertices. The algorithm also recursively embeds clusters in $\mathcal{F}$, which are $A$ and $B$. The final tree decomposition $\hat{\mathcal{T}}$ is obtained by connecting four tree decompositions from four recursive calls via the root bag $\{v_A,v_B\}$, which are representative vertices of $A$ and $B$, respectively. Dashed edges are new edges added during the embedding process.
  • Figure 4: Three cases in the analysis of the distortion overhead $D$. Filled vertices are terminals. The gray clusters form ${\partial\!} \mathcal{C}$. The yellow clusters form $\mathcal{F}$.
  • Figure 5: (a) An $(a,b,c)$ contraction sequence with $a = 7, b= 3, c=1$. (b) A tree $T$ associated with the contraction sequence in the proof of \ref{['lem:tw']}. $M$ contains exactly all the nodes at level 2; $(2,x^2_2)$ is minimally heavy and the only node in $M_1$; the rest of nodes at level 2 are in $M_2$.
  • ...and 4 more figures

Theorems & Definitions (71)

  • Conjecture 1.1
  • Theorem 1.2
  • Definition 2.1: Clustering Chain
  • Definition 2.2: $\beta$-Separating Distribution of Clustering Chains with Hop Guarantee
  • Theorem 2.3
  • Definition 2.4: $(\tau,\psi)$-Stochastic Balanced Cuts
  • Theorem 2.5
  • Theorem 2.6
  • proof : of \ref{['thm:main']}
  • Lemma 3.1: Lemma 2.1 in CLPP23
  • ...and 61 more