非一致超图匹配的更简单和更强大的方法和fredi, Kahn和Seymour猜想

Georg Anegg, Haris Angelidakis, R. Zenklusen
{"title":"非一致超图匹配的更简单和更强大的方法和fredi, Kahn和Seymour猜想","authors":"Georg Anegg, Haris Angelidakis, R. Zenklusen","doi":"10.1137/1.9781611976496.22","DOIUrl":null,"url":null,"abstract":"A well-known conjecture of Furedi, Kahn, and Seymour (1993) on non-uniform hypergraph matching states that for any hypergraph with edge weights $w$, there exists a matching $M$ such that the inequality $\\sum_{e\\in M} g(e) w(e) \\geq \\mathrm{OPT}_{\\mathrm{LP}}$ holds with $g(e)=|e|-1+\\frac{1}{|e|}$, where $\\mathrm{OPT}_{\\mathrm{LP}}$ denotes the optimal value of the canonical LP relaxation. \nWhile the conjecture remains open, the strongest result towards it was very recently obtained by Brubach, Sankararaman, Srinivasan, and Xu (2020)---building on and strengthening prior work by Bansal, Gupta, Li, Mestre, Nagarajan, and Rudra (2012)---showing that the aforementioned inequality holds with $g(e)=|e|+O(|e|\\exp(-|e|))$. \nActually, their method works in a more general sampling setting, where, given a point $x$ of the canonical LP relaxation, the task is to efficiently sample a matching $M$ containing each edge $e$ with probability at least $\\frac{x(e)}{g(e)}$. \nWe present simpler and easy-to-analyze procedures leading to improved results. More precisely, for any solution $x$ to the canonical LP, we introduce a simple algorithm based on exponential clocks for Brubach et al.'s sampling setting achieving $g(e)=|e|-(|e|-1)x(e)$. \nApart from the slight improvement in $g$, our technique may open up new ways to attack the original conjecture. \nMoreover, we provide a short and arguably elegant analysis showing that a natural greedy approach for the original setting of the conjecture shows the inequality for the same $g(e)=|e|-(|e|-1)x(e)$ even for the more general hypergraph $b$-matching problem.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"124 1 1","pages":"196-203"},"PeriodicalIF":0.0000,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"Simpler and Stronger Approaches for Non-Uniform Hypergraph Matching and the Füredi, Kahn, and Seymour Conjecture\",\"authors\":\"Georg Anegg, Haris Angelidakis, R. Zenklusen\",\"doi\":\"10.1137/1.9781611976496.22\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A well-known conjecture of Furedi, Kahn, and Seymour (1993) on non-uniform hypergraph matching states that for any hypergraph with edge weights $w$, there exists a matching $M$ such that the inequality $\\\\sum_{e\\\\in M} g(e) w(e) \\\\geq \\\\mathrm{OPT}_{\\\\mathrm{LP}}$ holds with $g(e)=|e|-1+\\\\frac{1}{|e|}$, where $\\\\mathrm{OPT}_{\\\\mathrm{LP}}$ denotes the optimal value of the canonical LP relaxation. \\nWhile the conjecture remains open, the strongest result towards it was very recently obtained by Brubach, Sankararaman, Srinivasan, and Xu (2020)---building on and strengthening prior work by Bansal, Gupta, Li, Mestre, Nagarajan, and Rudra (2012)---showing that the aforementioned inequality holds with $g(e)=|e|+O(|e|\\\\exp(-|e|))$. \\nActually, their method works in a more general sampling setting, where, given a point $x$ of the canonical LP relaxation, the task is to efficiently sample a matching $M$ containing each edge $e$ with probability at least $\\\\frac{x(e)}{g(e)}$. \\nWe present simpler and easy-to-analyze procedures leading to improved results. More precisely, for any solution $x$ to the canonical LP, we introduce a simple algorithm based on exponential clocks for Brubach et al.'s sampling setting achieving $g(e)=|e|-(|e|-1)x(e)$. \\nApart from the slight improvement in $g$, our technique may open up new ways to attack the original conjecture. \\nMoreover, we provide a short and arguably elegant analysis showing that a natural greedy approach for the original setting of the conjecture shows the inequality for the same $g(e)=|e|-(|e|-1)x(e)$ even for the more general hypergraph $b$-matching problem.\",\"PeriodicalId\":93491,\"journal\":{\"name\":\"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)\",\"volume\":\"124 1 1\",\"pages\":\"196-203\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-09-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1137/1.9781611976496.22\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1137/1.9781611976496.22","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 4

摘要

Furedi, Kahn, and Seymour(1993)关于非均匀超图匹配的一个著名猜想指出,对于任何边权为$w$的超图,存在一个匹配$M$,使得不等式$\sum_{e\in M} g(e) w(e) \geq \mathrm{OPT}_{\mathrm{LP}}$与$g(e)=|e|-1+\frac{1}{|e|}$成立,其中$\mathrm{OPT}_{\mathrm{LP}}$表示正则LP松弛的最优值。虽然这一猜想仍然是开放的,但Brubach、Sankararaman、Srinivasan和Xu(2020)最近获得了最有力的结果——基于并加强了Bansal、Gupta、Li、Mestre、Nagarajan和Rudra(2012)的先前工作——表明上述不平等适用于$g(e)=|e|+O(|e|\exp(-|e|))$。实际上,他们的方法适用于更一般的采样设置,其中,给定一个正则LP松弛的点$x$,任务是以至少$\frac{x(e)}{g(e)}$的概率有效地采样包含每个边$e$的匹配$M$。我们提供更简单和易于分析的程序,从而改善结果。更准确地说,对于规范LP的任何解$x$,我们引入了一个基于指数时钟的简单算法,用于Brubach等人的采样设置,实现$g(e)=|e|-(|e|-1)x(e)$。除了对$g$的轻微改进之外,我们的技术可能会开辟新的方法来攻击原始猜想。此外,我们提供了一个简短而优雅的分析,表明对猜想的原始设置的自然贪婪方法显示了相同$g(e)=|e|-(|e|-1)x(e)$的不等式,甚至对于更一般的超图$b$匹配问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
Simpler and Stronger Approaches for Non-Uniform Hypergraph Matching and the Füredi, Kahn, and Seymour Conjecture
A well-known conjecture of Furedi, Kahn, and Seymour (1993) on non-uniform hypergraph matching states that for any hypergraph with edge weights $w$, there exists a matching $M$ such that the inequality $\sum_{e\in M} g(e) w(e) \geq \mathrm{OPT}_{\mathrm{LP}}$ holds with $g(e)=|e|-1+\frac{1}{|e|}$, where $\mathrm{OPT}_{\mathrm{LP}}$ denotes the optimal value of the canonical LP relaxation. While the conjecture remains open, the strongest result towards it was very recently obtained by Brubach, Sankararaman, Srinivasan, and Xu (2020)---building on and strengthening prior work by Bansal, Gupta, Li, Mestre, Nagarajan, and Rudra (2012)---showing that the aforementioned inequality holds with $g(e)=|e|+O(|e|\exp(-|e|))$. Actually, their method works in a more general sampling setting, where, given a point $x$ of the canonical LP relaxation, the task is to efficiently sample a matching $M$ containing each edge $e$ with probability at least $\frac{x(e)}{g(e)}$. We present simpler and easy-to-analyze procedures leading to improved results. More precisely, for any solution $x$ to the canonical LP, we introduce a simple algorithm based on exponential clocks for Brubach et al.'s sampling setting achieving $g(e)=|e|-(|e|-1)x(e)$. Apart from the slight improvement in $g$, our technique may open up new ways to attack the original conjecture. Moreover, we provide a short and arguably elegant analysis showing that a natural greedy approach for the original setting of the conjecture shows the inequality for the same $g(e)=|e|-(|e|-1)x(e)$ even for the more general hypergraph $b$-matching problem.
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
期刊最新文献
Revisiting Garg's 2-Approximation Algorithm for the k-MST Problem in Graphs Min-Max Optimization Made Simple: Approximating the Proximal Point Method via Contraction Maps Simpler and faster algorithms for detours in planar digraphs A Simple Optimal Algorithm for the 2-Arm Bandit Problem A Simple Algorithm for Submodular Minimum Linear Ordering
×
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
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 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 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1