{"title":"On the k-maximally-disjoint weighted spanning trees problem: variants, complexity and algorithms","authors":"Walid Astaoui, Youcef Magnouche, Sébastien Martin","doi":"10.1007/s10479-025-06592-x","DOIUrl":null,"url":null,"abstract":"<div><p>Let us consider a connected undirected graph <span>\\(G = (V, E,d,w)\\)</span> with a set of nodes <i>V</i>, a set of edges <i>E</i>, an edge distance vector <i>d</i>, and an edge weight vector <i>w</i>. For a given integer <span>\\(k \\ge 2\\)</span>, we investigate the problem of finding <i>k</i>-maximally weighted edge-disjoint spanning trees <span>\\(S_1,S_2\\ldots S_k\\)</span>, where <span>\\(S_i\\subseteq E\\)</span>, <span>\\(i\\in \\{1,\\dots , k\\}\\)</span>. Given <i>k</i> root nodes <span>\\(r_1,\\ldots r_k \\in V\\)</span>, we also impose additional constraints, leading to two new variants: (1) <span>\\(S_1\\)</span> must be a shortest-path tree, with respect to <i>d</i>, rooted on <span>\\(r_1\\)</span> and 2) all trees must be shortest-path trees, with respect to <i>d</i>, rooted on <span>\\(r_1,\\ldots r_k\\)</span>, respectively. We consider two different objective functions: (1) the weight of <span>\\(S_1\\)</span> is minimum, and (2) the total weight of <span>\\(S_1,\\dots , S_k\\)</span> is minimum. We show that each variant belongs to <span>\\(\\mathcal {P}\\)</span> class for some values of <i>k</i>. This leads to exact polynomial matroid-based algorithms. We present and discuss the numerical results for every variant, and analyze the properties of the trees returned by the algorithms.</p></div>","PeriodicalId":8215,"journal":{"name":"Annals of Operations Research","volume":"351 1","pages":"849 - 881"},"PeriodicalIF":4.5000,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Operations Research","FirstCategoryId":"91","ListUrlMain":"https://link.springer.com/article/10.1007/s10479-025-06592-x","RegionNum":3,"RegionCategory":"管理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"OPERATIONS RESEARCH & MANAGEMENT SCIENCE","Score":null,"Total":0}
引用次数: 0
Abstract
Let us consider a connected undirected graph \(G = (V, E,d,w)\) with a set of nodes V, a set of edges E, an edge distance vector d, and an edge weight vector w. For a given integer \(k \ge 2\), we investigate the problem of finding k-maximally weighted edge-disjoint spanning trees \(S_1,S_2\ldots S_k\), where \(S_i\subseteq E\), \(i\in \{1,\dots , k\}\). Given k root nodes \(r_1,\ldots r_k \in V\), we also impose additional constraints, leading to two new variants: (1) \(S_1\) must be a shortest-path tree, with respect to d, rooted on \(r_1\) and 2) all trees must be shortest-path trees, with respect to d, rooted on \(r_1,\ldots r_k\), respectively. We consider two different objective functions: (1) the weight of \(S_1\) is minimum, and (2) the total weight of \(S_1,\dots , S_k\) is minimum. We show that each variant belongs to \(\mathcal {P}\) class for some values of k. This leads to exact polynomial matroid-based algorithms. We present and discuss the numerical results for every variant, and analyze the properties of the trees returned by the algorithms.
期刊介绍:
The Annals of Operations Research publishes peer-reviewed original articles dealing with key aspects of operations research, including theory, practice, and computation. The journal publishes full-length research articles, short notes, expositions and surveys, reports on computational studies, and case studies that present new and innovative practical applications.
In addition to regular issues, the journal publishes periodic special volumes that focus on defined fields of operations research, ranging from the highly theoretical to the algorithmic and the applied. These volumes have one or more Guest Editors who are responsible for collecting the papers and overseeing the refereeing process.