{"title":"运输网络参数的层次和硬度结果","authors":"Johannes Blum","doi":"10.4230/LIPIcs.IPEC.2019.4","DOIUrl":null,"url":null,"abstract":"The graph parameters highway dimension and skeleton dimension were introduced to capture the properties of transportation networks. As many important optimization problems like Travelling Salesperson, Steiner Tree or $k$-Center arise in such networks, it is worthwhile to study them on graphs of bounded highway or skeleton dimension. \nWe investigate the relationships between mentioned parameters and how they are related to other important graph parameters that have been applied successfully to various optimization problems. We show that the skeleton dimension is incomparable to any of the parameters distance to linear forest, bandwidth, treewidth and highway dimension and hence, it is worthwhile to study mentioned problems also on graphs of bounded skeleton dimension. Moreover, we prove that the skeleton dimension is upper bounded by the max leaf number and that for any graph on at least three vertices there are edge weights such that both parameters are equal. \nThen we show that computing the highway dimension according to most recent definition is NP-hard, which answers an open question stated by Feldmann et al. Finally we prove that on graphs $G=(V,E)$ of skeleton dimension $\\mathcal{O}(\\log^2 \\vert V \\vert)$ it is NP-hard to approximate the $k$-Center problem within a factor less than $2$.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"49 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"9","resultStr":"{\"title\":\"Hierarchy of Transportation Network Parameters and Hardness Results\",\"authors\":\"Johannes Blum\",\"doi\":\"10.4230/LIPIcs.IPEC.2019.4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The graph parameters highway dimension and skeleton dimension were introduced to capture the properties of transportation networks. As many important optimization problems like Travelling Salesperson, Steiner Tree or $k$-Center arise in such networks, it is worthwhile to study them on graphs of bounded highway or skeleton dimension. \\nWe investigate the relationships between mentioned parameters and how they are related to other important graph parameters that have been applied successfully to various optimization problems. We show that the skeleton dimension is incomparable to any of the parameters distance to linear forest, bandwidth, treewidth and highway dimension and hence, it is worthwhile to study mentioned problems also on graphs of bounded skeleton dimension. Moreover, we prove that the skeleton dimension is upper bounded by the max leaf number and that for any graph on at least three vertices there are edge weights such that both parameters are equal. \\nThen we show that computing the highway dimension according to most recent definition is NP-hard, which answers an open question stated by Feldmann et al. Finally we prove that on graphs $G=(V,E)$ of skeleton dimension $\\\\mathcal{O}(\\\\log^2 \\\\vert V \\\\vert)$ it is NP-hard to approximate the $k$-Center problem within a factor less than $2$.\",\"PeriodicalId\":137775,\"journal\":{\"name\":\"International Symposium on Parameterized and Exact Computation\",\"volume\":\"49 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-05-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"9\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Symposium on Parameterized and Exact Computation\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4230/LIPIcs.IPEC.2019.4\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Symposium on Parameterized and Exact Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4230/LIPIcs.IPEC.2019.4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 9
摘要
引入图形参数公路维数和骨架维数来捕捉交通网络的特性。由于在这类网络中出现了许多重要的优化问题,如旅行推销员、斯坦纳树或$k$-Center,因此在有界高速公路或骨架维数的图上进行研究是值得的。我们研究了上述参数之间的关系,以及它们如何与其他已成功应用于各种优化问题的重要图参数相关联。我们证明了骨架维数与线性森林距离、带宽、树宽和公路维数的任何参数都是不可比拟的,因此,在有界骨架维数的图上也值得研究上述问题。此外,我们证明了骨架维数的上界是最大叶数,并且对于任何至少有三个顶点的图,存在两个参数相等的边权。然后我们证明,根据最新定义计算高速公路维度是np困难的,这回答了Feldmann等人提出的一个悬而未决的问题。最后我们证明了在骨架维数为$ $ mathcal{O}(\log^2 \vert V \vert)$ $的图$G=(V,E)$ $上,在小于$ $2$的因子内逼近$k$-Center问题是np困难的。
Hierarchy of Transportation Network Parameters and Hardness Results
The graph parameters highway dimension and skeleton dimension were introduced to capture the properties of transportation networks. As many important optimization problems like Travelling Salesperson, Steiner Tree or $k$-Center arise in such networks, it is worthwhile to study them on graphs of bounded highway or skeleton dimension.
We investigate the relationships between mentioned parameters and how they are related to other important graph parameters that have been applied successfully to various optimization problems. We show that the skeleton dimension is incomparable to any of the parameters distance to linear forest, bandwidth, treewidth and highway dimension and hence, it is worthwhile to study mentioned problems also on graphs of bounded skeleton dimension. Moreover, we prove that the skeleton dimension is upper bounded by the max leaf number and that for any graph on at least three vertices there are edge weights such that both parameters are equal.
Then we show that computing the highway dimension according to most recent definition is NP-hard, which answers an open question stated by Feldmann et al. Finally we prove that on graphs $G=(V,E)$ of skeleton dimension $\mathcal{O}(\log^2 \vert V \vert)$ it is NP-hard to approximate the $k$-Center problem within a factor less than $2$.