{"title":"通过提升树上的边来解决有卡片数量限制的根叶距离阻断问题的总和","authors":"Xiao Li, Xiucui Guan, Qiao Zhang, Xinyi Yin, Panos M. Pardalos","doi":"10.1007/s10878-024-01230-x","DOIUrl":null,"url":null,"abstract":"<p>A network for the transportation of supplies can be described as a rooted tree with a weight of a degree of congestion for each edge. We take the sum of root-leaf distance (SRD) on a rooted tree as the whole degree of congestion of the tree. Hence, we consider the SRD interdiction problem on trees with cardinality constraint by upgrading edges, denoted by (<b>SDIPTC</b>). It aims to maximize the SRD by upgrading the weights of <i>N</i> critical edges such that the total upgrade cost under some measurement is upper-bounded by a given value. The relevant minimum cost problem (<b>MCSDIPTC</b>) aims to minimize the total upgrade cost on the premise that the SRD is lower-bounded by a given value. We develop two different norms including weighted <span>\\(l_\\infty \\)</span> norm and weighted bottleneck Hamming distance to measure the upgrade cost. We propose two binary search algorithms within O(<span>\\(n\\log n\\)</span>) time for the problems (<b>SDIPTC</b>) under the two norms, respectively. For problems (<b>MCSDIPTC</b>), we propose two binary search algorithms within O(<span>\\(N n^2\\)</span>) and O(<span>\\(n \\log n\\)</span>) under weighted <span>\\(l_\\infty \\)</span> norm and weighted bottleneck Hamming distance, respectively. These problems are solved through their subproblems (<b>SDIPT</b>) and (<b>MCSDIPT</b>), in which we ignore the cardinality constraint on the number of upgraded edges. Finally, we design numerical experiments to show the effectiveness of these algorithms.\n</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"17 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The sum of root-leaf distance interdiction problem with cardinality constraint by upgrading edges on trees\",\"authors\":\"Xiao Li, Xiucui Guan, Qiao Zhang, Xinyi Yin, Panos M. Pardalos\",\"doi\":\"10.1007/s10878-024-01230-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A network for the transportation of supplies can be described as a rooted tree with a weight of a degree of congestion for each edge. We take the sum of root-leaf distance (SRD) on a rooted tree as the whole degree of congestion of the tree. Hence, we consider the SRD interdiction problem on trees with cardinality constraint by upgrading edges, denoted by (<b>SDIPTC</b>). It aims to maximize the SRD by upgrading the weights of <i>N</i> critical edges such that the total upgrade cost under some measurement is upper-bounded by a given value. The relevant minimum cost problem (<b>MCSDIPTC</b>) aims to minimize the total upgrade cost on the premise that the SRD is lower-bounded by a given value. We develop two different norms including weighted <span>\\\\(l_\\\\infty \\\\)</span> norm and weighted bottleneck Hamming distance to measure the upgrade cost. We propose two binary search algorithms within O(<span>\\\\(n\\\\log n\\\\)</span>) time for the problems (<b>SDIPTC</b>) under the two norms, respectively. For problems (<b>MCSDIPTC</b>), we propose two binary search algorithms within O(<span>\\\\(N n^2\\\\)</span>) and O(<span>\\\\(n \\\\log n\\\\)</span>) under weighted <span>\\\\(l_\\\\infty \\\\)</span> norm and weighted bottleneck Hamming distance, respectively. These problems are solved through their subproblems (<b>SDIPT</b>) and (<b>MCSDIPT</b>), in which we ignore the cardinality constraint on the number of upgraded edges. Finally, we design numerical experiments to show the effectiveness of these algorithms.\\n</p>\",\"PeriodicalId\":50231,\"journal\":{\"name\":\"Journal of Combinatorial Optimization\",\"volume\":\"17 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-11-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Optimization\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10878-024-01230-x\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-024-01230-x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
The sum of root-leaf distance interdiction problem with cardinality constraint by upgrading edges on trees
A network for the transportation of supplies can be described as a rooted tree with a weight of a degree of congestion for each edge. We take the sum of root-leaf distance (SRD) on a rooted tree as the whole degree of congestion of the tree. Hence, we consider the SRD interdiction problem on trees with cardinality constraint by upgrading edges, denoted by (SDIPTC). It aims to maximize the SRD by upgrading the weights of N critical edges such that the total upgrade cost under some measurement is upper-bounded by a given value. The relevant minimum cost problem (MCSDIPTC) aims to minimize the total upgrade cost on the premise that the SRD is lower-bounded by a given value. We develop two different norms including weighted \(l_\infty \) norm and weighted bottleneck Hamming distance to measure the upgrade cost. We propose two binary search algorithms within O(\(n\log n\)) time for the problems (SDIPTC) under the two norms, respectively. For problems (MCSDIPTC), we propose two binary search algorithms within O(\(N n^2\)) and O(\(n \log n\)) under weighted \(l_\infty \) norm and weighted bottleneck Hamming distance, respectively. These problems are solved through their subproblems (SDIPT) and (MCSDIPT), in which we ignore the cardinality constraint on the number of upgraded edges. Finally, we design numerical experiments to show the effectiveness of these algorithms.
期刊介绍:
The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering.
The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.