On the k-maximally-disjoint weighted spanning trees problem: variants, complexity and algorithms

IF 4.5 3区 管理学 Q1 OPERATIONS RESEARCH & MANAGEMENT SCIENCE
Walid Astaoui, Youcef Magnouche, Sébastien Martin
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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.

Abstract Image

关于k-最大不相交加权生成树问题:变量、复杂度和算法
让我们考虑一个连通无向图\(G = (V, E,d,w)\),它有一组节点V,一组边E,一个边距离向量d和一个边权向量w。对于给定的整数\(k \ge 2\),我们研究找到k个最大加权边不相交生成树\(S_1,S_2\ldots S_k\)的问题,其中\(S_i\subseteq E\), \(i\in \{1,\dots , k\}\)。给定k个根节点\(r_1,\ldots r_k \in V\),我们还施加了额外的约束,导致两个新的变体:(1)\(S_1\)必须是相对于d的最短路径树,扎根于\(r_1\);(2)所有的树必须是相对于d的最短路径树,分别扎根于\(r_1,\ldots r_k\)。我们考虑两个不同的目标函数:(1)\(S_1\)的权值最小,(2)\(S_1,\dots , S_k\)的总权值最小。我们表明,对于某些k值,每个变体都属于\(\mathcal {P}\)类。这导致了基于精确多项式矩阵的算法。我们给出并讨论了每个变量的数值结果,并分析了算法返回的树的性质。
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来源期刊
Annals of Operations Research
Annals of Operations Research 管理科学-运筹学与管理科学
CiteScore
7.90
自引率
16.70%
发文量
596
审稿时长
8.4 months
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.
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