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

TL;DR

This paper improves the stochastic embedding of planar graphs into graphs with smaller treewidth, narrowing the gap between upper bounds and known lower bounds, and introduces new techniques for constructing low-treewidth embeddings.

Contribution

It presents a new stochastic embedding of planar graphs into graphs of treewidth $O(rac{1}{ ext{epsilon}} ext{log}^3 n)$, improving previous bounds and techniques.

Findings

Achieved a stochastic embedding with treewidth $O(rac{1}{ ext{epsilon}} ext{log}^3 n)$.

Streamlined embedding construction using a single hierarchy of clusters.

Provided an optimal analysis of contraction sequences for graphs.

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 $\Omega(\log n)$ 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.