{"title":"Partial inverse min–max spanning tree problem under the weighted bottleneck hamming distance","authors":"Qingzhen Dong, Xianyue Li, Yu Yang","doi":"10.1007/s10878-023-01093-8","DOIUrl":null,"url":null,"abstract":"<p>Min–max spanning tree problem is a classical problem in combinatorial optimization. Its purpose is to find a spanning tree to minimize its maximum edge in a given edge weighted graph. Given a connected graph <i>G</i>, an edge weight vector <i>w</i> and a forest <i>F</i>, the partial inverse min–max spanning tree problem (PIMMST) is to find a new weighted vector <span>\\(w^*\\)</span>, so that <i>F</i> can be extended into a min–max spanning tree with respect to <span>\\(w^*\\)</span> and the gap between <i>w</i> and <span>\\(w^*\\)</span> is minimized. In this paper, we research PIMMST under the weighted bottleneck Hamming distance. Firstly, we study PIMMST with value of optimal tree restriction, a variant of PIMMST, and show that this problem can be solved in strongly polynomial time. Then, by characterizing the properties of the value of optimal tree, we present first algorithm for PIMMST under the weighted bottleneck Hamming distance with running time <span>\\(O(|E|^2\\log |E|)\\)</span>, where <i>E</i> is the edge set of <i>G</i>. Finally, by giving a necessary and sufficient condition to determine the feasible solution of this problem, we present a better algorithm for this problem with running time <span>\\(O(|E|\\log |E|)\\)</span>. Moreover, we show that these algorithms can be generalized to solve these problems with capacitated constraint.\n</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-023-01093-8","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}
引用次数: 1
Abstract
Min–max spanning tree problem is a classical problem in combinatorial optimization. Its purpose is to find a spanning tree to minimize its maximum edge in a given edge weighted graph. Given a connected graph G, an edge weight vector w and a forest F, the partial inverse min–max spanning tree problem (PIMMST) is to find a new weighted vector \(w^*\), so that F can be extended into a min–max spanning tree with respect to \(w^*\) and the gap between w and \(w^*\) is minimized. In this paper, we research PIMMST under the weighted bottleneck Hamming distance. Firstly, we study PIMMST with value of optimal tree restriction, a variant of PIMMST, and show that this problem can be solved in strongly polynomial time. Then, by characterizing the properties of the value of optimal tree, we present first algorithm for PIMMST under the weighted bottleneck Hamming distance with running time \(O(|E|^2\log |E|)\), where E is the edge set of G. Finally, by giving a necessary and sufficient condition to determine the feasible solution of this problem, we present a better algorithm for this problem with running time \(O(|E|\log |E|)\). Moreover, we show that these algorithms can be generalized to solve these problems with capacitated constraint.
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.