Embedding Planar Graphs into Graphs of Treewidth $O(\log^{3} n)$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916464344170496 |
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| author | Chang, Hsien-Chih Cohen-Addad, Vincent Conroy, Jonathan Le, Hung Pilipczuk, Marcin Pilipczuk, Michał |
| author_facet | Chang, Hsien-Chih Cohen-Addad, Vincent Conroy, Jonathan Le, Hung Pilipczuk, Marcin Pilipczuk, Michał |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00216 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Embedding Planar Graphs into Graphs of Treewidth $O(\log^{3} n)$ Chang, Hsien-Chih Cohen-Addad, Vincent Conroy, Jonathan Le, Hung Pilipczuk, Marcin Pilipczuk, Michał Data Structures and Algorithms 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. |
| title | Embedding Planar Graphs into Graphs of Treewidth $O(\log^{3} n)$ |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2411.00216 |