{"title":"关于多源瓶颈路径问题的高效算法","authors":"Kirill V. Kaymakov, Dmitry S. Malyshev","doi":"10.1007/s11590-024-02113-0","DOIUrl":null,"url":null,"abstract":"<p>For given edge-capacitated connected graph and two its vertices <i>s</i> and <i>t</i>, the bottleneck (or <span>\\(\\max \\min \\)</span>) path problem is to find the maximum value of path-minimum edge capacities among all paths, connecting <i>s</i> and <i>t</i>. It can be generalized by finding the bottleneck values between <i>s</i> and all possible <i>t</i>. These problems arise as subproblems in the known maximum flow problem, having applications in many real-life tasks. For any graph with <i>n</i> vertices and <i>m</i> edges, they can be solved in <i>O</i>(<i>m</i>) and <i>O</i>(<i>t</i>(<i>m</i>, <i>n</i>)) times, respectively, where <span>\\(t(m,n)=\\min (m+n\\log (n),m\\alpha (m,n))\\)</span> and <span>\\(\\alpha (\\cdot ,\\cdot )\\)</span> is the inverse Ackermann function. In this paper, we generalize of the bottleneck path problems by considering their versions with <i>k</i> sources. For the first of them, where <i>k</i> pairs of sources and targets are (offline or online) given, we present an <span>\\(O((m+k)\\log (n))\\)</span>-time randomized and an <span>\\(O(m+(n+k)\\log (n))\\)</span>-time deterministic algorithms for the offline and online versions, respectively. For the second one, where the bottleneck values are found between <i>k</i> sources and all targets, we present an <span>\\(O(t(m,n)+kn)\\)</span>-time offline/online algorithm.</p>","PeriodicalId":49720,"journal":{"name":"Optimization Letters","volume":"57 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On efficient algorithms for bottleneck path problems with many sources\",\"authors\":\"Kirill V. Kaymakov, Dmitry S. Malyshev\",\"doi\":\"10.1007/s11590-024-02113-0\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For given edge-capacitated connected graph and two its vertices <i>s</i> and <i>t</i>, the bottleneck (or <span>\\\\(\\\\max \\\\min \\\\)</span>) path problem is to find the maximum value of path-minimum edge capacities among all paths, connecting <i>s</i> and <i>t</i>. It can be generalized by finding the bottleneck values between <i>s</i> and all possible <i>t</i>. These problems arise as subproblems in the known maximum flow problem, having applications in many real-life tasks. For any graph with <i>n</i> vertices and <i>m</i> edges, they can be solved in <i>O</i>(<i>m</i>) and <i>O</i>(<i>t</i>(<i>m</i>, <i>n</i>)) times, respectively, where <span>\\\\(t(m,n)=\\\\min (m+n\\\\log (n),m\\\\alpha (m,n))\\\\)</span> and <span>\\\\(\\\\alpha (\\\\cdot ,\\\\cdot )\\\\)</span> is the inverse Ackermann function. In this paper, we generalize of the bottleneck path problems by considering their versions with <i>k</i> sources. For the first of them, where <i>k</i> pairs of sources and targets are (offline or online) given, we present an <span>\\\\(O((m+k)\\\\log (n))\\\\)</span>-time randomized and an <span>\\\\(O(m+(n+k)\\\\log (n))\\\\)</span>-time deterministic algorithms for the offline and online versions, respectively. For the second one, where the bottleneck values are found between <i>k</i> sources and all targets, we present an <span>\\\\(O(t(m,n)+kn)\\\\)</span>-time offline/online algorithm.</p>\",\"PeriodicalId\":49720,\"journal\":{\"name\":\"Optimization Letters\",\"volume\":\"57 1\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-04-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Optimization Letters\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s11590-024-02113-0\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Optimization Letters","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11590-024-02113-0","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
对于给定的有边容量的连通图及其两个顶点 s 和 t,瓶颈(或 \(\max \min \))路径问题是在连接 s 和 t 的所有路径中找到路径最小边容量的最大值。对于任何有 n 个顶点和 m 条边的图,它们可以分别在 O(m) 和 O(t(m, n)) 次内求解,其中(t(m,n)=\min (m+n\log (n),m\alpha (m,n)))和(\alpha (\cdot ,\cdot))是反阿克曼函数。在本文中,我们通过考虑有 k 个来源的瓶颈路径问题来概括这些问题。对于其中的第一个版本,即 k 对来源和目标是(离线或在线)给定的,我们为离线和在线版本分别提出了一个(O((m+k)\log (n))-time 随机算法和一个(O(m+(n+k)\log (n))-time 确定性算法。对于第二种算法,即在 k 个来源和所有目标之间找到瓶颈值,我们提出了一种离线/在线算法(O(t(m,n)+kn)\t(m,n)+kn)-time)。
On efficient algorithms for bottleneck path problems with many sources
For given edge-capacitated connected graph and two its vertices s and t, the bottleneck (or \(\max \min \)) path problem is to find the maximum value of path-minimum edge capacities among all paths, connecting s and t. It can be generalized by finding the bottleneck values between s and all possible t. These problems arise as subproblems in the known maximum flow problem, having applications in many real-life tasks. For any graph with n vertices and m edges, they can be solved in O(m) and O(t(m, n)) times, respectively, where \(t(m,n)=\min (m+n\log (n),m\alpha (m,n))\) and \(\alpha (\cdot ,\cdot )\) is the inverse Ackermann function. In this paper, we generalize of the bottleneck path problems by considering their versions with k sources. For the first of them, where k pairs of sources and targets are (offline or online) given, we present an \(O((m+k)\log (n))\)-time randomized and an \(O(m+(n+k)\log (n))\)-time deterministic algorithms for the offline and online versions, respectively. For the second one, where the bottleneck values are found between k sources and all targets, we present an \(O(t(m,n)+kn)\)-time offline/online algorithm.
期刊介绍:
Optimization Letters is an international journal covering all aspects of optimization, including theory, algorithms, computational studies, and applications, and providing an outlet for rapid publication of short communications in the field. Originality, significance, quality and clarity are the essential criteria for choosing the material to be published.
Optimization Letters has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time one of the most striking trends in optimization is the constantly increasing interdisciplinary nature of the field.
Optimization Letters aims to communicate in a timely fashion all recent developments in optimization with concise short articles (limited to a total of ten journal pages). Such concise articles will be easily accessible by readers working in any aspects of optimization and wish to be informed of recent developments.