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arXiv · 2411.00216

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
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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.

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Hsien-Chih Chang, Vincent Cohen-Addad, Jonathan Conroy, Hung Le, Marcin Pilipczuk, Michał Pilipczuk. 2024-10-31. Embedding Planar Graphs into Graphs of Treewidth $O(\log^{3} n)$. https://arxiv.org/abs/2411.00216

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