有界树宽图的最优填充分解

Arnold Filtser, Tobias Friedrich, Davis Issac, Nikhil Kumar, Hung Le, Nadym Mallek, Ziena Zeif
{"title":"有界树宽图的最优填充分解","authors":"Arnold Filtser, Tobias Friedrich, Davis Issac, Nikhil Kumar, Hung Le, Nadym Mallek, Ziena Zeif","doi":"arxiv-2407.12230","DOIUrl":null,"url":null,"abstract":"A $(\\beta,\\delta,\\Delta)$-padded decomposition of an edge-weighted graph $G =\n(V,E,w)$ is a stochastic decomposition into clusters of diameter at most\n$\\Delta$ such that for every vertex $v\\in V$, the probability that\n$\\rm{ball}_G(v,\\gamma\\Delta)$ is entirely contained in the cluster containing\n$v$ is at least $e^{-\\beta\\gamma}$ for every $\\gamma \\in [0,\\delta]$. Padded\ndecompositions have been studied for decades and have found numerous\napplications, including metric embedding, multicommodity flow-cut gap, muticut,\nand zero extension problems, to name a few. In these applications, parameter\n$\\beta$, called the padding parameter, is the most important parameter since it\ndecides either the distortion or the approximation ratios. For general graphs\nwith $n$ vertices, $\\beta = \\Theta(\\log n)$. Klein, Plotkin, and Rao showed\nthat $K_r$-minor-free graphs have padding parameter $\\beta = O(r^3)$, which is\na significant improvement over general graphs when $r$ is a constant. A\nlong-standing conjecture is to construct a padded decomposition for\n$K_r$-minor-free graphs with padding parameter $\\beta = O(\\log r)$. Despite\ndecades of research, the best-known result is $\\beta = O(r)$, even for graphs\nwith treewidth at most $r$. In this work, we make significant progress toward\nthe aforementioned conjecture by showing that graphs with treewidth $\\rm{tw}$\nadmit a padded decomposition with padding parameter $O(\\log \\rm{tw})$, which is\ntight. As corollaries, we obtain an exponential improvement in dependency on\ntreewidth in a host of algorithmic applications: $O(\\sqrt{ \\log n \\cdot\n\\log(\\rm{tw})})$ flow-cut gap, max flow-min multicut ratio of\n$O(\\log(\\rm{tw}))$, an $O(\\log(\\rm{tw}))$ approximation for the 0-extension\nproblem, an $\\ell^{O(\\log n)}_\\infty$ embedding with distortion $O(\\log\n\\rm{tw})$, and an $O(\\log \\rm{tw})$ bound for integrality gap for the uniform\nsparsest cut.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"37 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Optimal Padded Decomposition For Bounded Treewidth Graphs\",\"authors\":\"Arnold Filtser, Tobias Friedrich, Davis Issac, Nikhil Kumar, Hung Le, Nadym Mallek, Ziena Zeif\",\"doi\":\"arxiv-2407.12230\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A $(\\\\beta,\\\\delta,\\\\Delta)$-padded decomposition of an edge-weighted graph $G =\\n(V,E,w)$ is a stochastic decomposition into clusters of diameter at most\\n$\\\\Delta$ such that for every vertex $v\\\\in V$, the probability that\\n$\\\\rm{ball}_G(v,\\\\gamma\\\\Delta)$ is entirely contained in the cluster containing\\n$v$ is at least $e^{-\\\\beta\\\\gamma}$ for every $\\\\gamma \\\\in [0,\\\\delta]$. Padded\\ndecompositions have been studied for decades and have found numerous\\napplications, including metric embedding, multicommodity flow-cut gap, muticut,\\nand zero extension problems, to name a few. In these applications, parameter\\n$\\\\beta$, called the padding parameter, is the most important parameter since it\\ndecides either the distortion or the approximation ratios. For general graphs\\nwith $n$ vertices, $\\\\beta = \\\\Theta(\\\\log n)$. Klein, Plotkin, and Rao showed\\nthat $K_r$-minor-free graphs have padding parameter $\\\\beta = O(r^3)$, which is\\na significant improvement over general graphs when $r$ is a constant. A\\nlong-standing conjecture is to construct a padded decomposition for\\n$K_r$-minor-free graphs with padding parameter $\\\\beta = O(\\\\log r)$. Despite\\ndecades of research, the best-known result is $\\\\beta = O(r)$, even for graphs\\nwith treewidth at most $r$. In this work, we make significant progress toward\\nthe aforementioned conjecture by showing that graphs with treewidth $\\\\rm{tw}$\\nadmit a padded decomposition with padding parameter $O(\\\\log \\\\rm{tw})$, which is\\ntight. As corollaries, we obtain an exponential improvement in dependency on\\ntreewidth in a host of algorithmic applications: $O(\\\\sqrt{ \\\\log n \\\\cdot\\n\\\\log(\\\\rm{tw})})$ flow-cut gap, max flow-min multicut ratio of\\n$O(\\\\log(\\\\rm{tw}))$, an $O(\\\\log(\\\\rm{tw}))$ approximation for the 0-extension\\nproblem, an $\\\\ell^{O(\\\\log n)}_\\\\infty$ embedding with distortion $O(\\\\log\\n\\\\rm{tw})$, and an $O(\\\\log \\\\rm{tw})$ bound for integrality gap for the uniform\\nsparsest cut.\",\"PeriodicalId\":501216,\"journal\":{\"name\":\"arXiv - CS - Discrete Mathematics\",\"volume\":\"37 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-07-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - CS - Discrete Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2407.12230\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.12230","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

边加权图 $G =(V,E,w)$ 的$(\beta,\delta,\Delta)$填充分解是将其随机分解为直径至多为$\Delta$的簇,这样对于 V$ 中的每个顶点$v、对于 [0,\delta]$ 中的每一个 $\gamma ,$rm{ball}_G(v,\gamma\Delta)$ 完全包含在包含 $v$ 的簇中的概率至少为 $e^{-\beta\gamma}$。数十年来,人们一直在研究填充分解,并发现了许多应用,其中包括度量嵌入、多商品流切隙、突变和零扩展问题等等。在这些应用中,参数$\beta$(称为填充参数)是最重要的参数,因为它决定了失真度或近似率。对于具有 $n$ 顶点的一般图形,$\beta = \Theta(\log n)$。Klein、Plotkin 和 Rao 发现,当 $r$ 是常数时,$K_r$-无最小图的填充参数为 $\beta = O(r^3)$,这比一般图有了显著的改进。一直以来的猜想是为$K_r$无主图构建一个填充分解,其填充参数为$\beta = O(\log r)$。经过几十年的研究,最著名的结果是 $\beta = O(r)$,即使对于树宽最多为 $r$ 的图也是如此。在这项工作中,我们证明了树宽为 $\rm{tw}$ 的图可以进行填充分解,填充参数为 $O(\log\rm{tw})$,这是正确的,从而在实现上述猜想方面取得了重大进展。作为推论,我们在大量算法应用中获得了对树宽依赖性的指数级改进:O(\sqrt{\log n \cdot\log(\rm{tw})})$ 流切间隙,最大流-最小多切比率为$O(\log(\rm{tw}))$,0-扩展问题的$O(\log(\rm{tw}))$ 近似值、一个失真度为 $O(\log\rm{tw})$ 的 $ell^{O(\log n)}_\infty$ 嵌入,以及一个统一最难切割的积分差距的 $O(\log\rm{tw})$ 约束。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Optimal Padded Decomposition For Bounded Treewidth Graphs
A $(\beta,\delta,\Delta)$-padded decomposition of an edge-weighted graph $G = (V,E,w)$ is a stochastic decomposition into clusters of diameter at most $\Delta$ such that for every vertex $v\in V$, the probability that $\rm{ball}_G(v,\gamma\Delta)$ is entirely contained in the cluster containing $v$ is at least $e^{-\beta\gamma}$ for every $\gamma \in [0,\delta]$. Padded decompositions have been studied for decades and have found numerous applications, including metric embedding, multicommodity flow-cut gap, muticut, and zero extension problems, to name a few. In these applications, parameter $\beta$, called the padding parameter, is the most important parameter since it decides either the distortion or the approximation ratios. For general graphs with $n$ vertices, $\beta = \Theta(\log n)$. Klein, Plotkin, and Rao showed that $K_r$-minor-free graphs have padding parameter $\beta = O(r^3)$, which is a significant improvement over general graphs when $r$ is a constant. A long-standing conjecture is to construct a padded decomposition for $K_r$-minor-free graphs with padding parameter $\beta = O(\log r)$. Despite decades of research, the best-known result is $\beta = O(r)$, even for graphs with treewidth at most $r$. In this work, we make significant progress toward the aforementioned conjecture by showing that graphs with treewidth $\rm{tw}$ admit a padded decomposition with padding parameter $O(\log \rm{tw})$, which is tight. As corollaries, we obtain an exponential improvement in dependency on treewidth in a host of algorithmic applications: $O(\sqrt{ \log n \cdot \log(\rm{tw})})$ flow-cut gap, max flow-min multicut ratio of $O(\log(\rm{tw}))$, an $O(\log(\rm{tw}))$ approximation for the 0-extension problem, an $\ell^{O(\log n)}_\infty$ embedding with distortion $O(\log \rm{tw})$, and an $O(\log \rm{tw})$ bound for integrality gap for the uniform sparsest cut.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信