Minimum Spanning Trees in Infinite Graphs: Theory and Algorithms

IF 2.6 1区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Optimization Pub Date : 2024-09-11 DOI:10.1137/23m157627x

Christopher T. Ryan, Robert L. Smith, Marina A. Epelman

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Abstract

SIAM Journal on Optimization, Volume 34, Issue 3, Page 3112-3135, September 2024.
Abstract. We discuss finding minimum spanning trees (MSTs) on connected graphs with countably many nodes of finite degree. When edge costs are summable and an MST exists (which is not guaranteed in general), we show that an algorithm that finds MSTs on finite subgraphs (called layers) converges in objective value to the cost of an MST of the whole graph as the sizes of the layers grow to infinity. We call this the layered greedy algorithm since a greedy algorithm is used to find MSTs on each finite layer. We stress that the overall algorithm is not greedy since edges can enter and leave iterate spanning trees as larger layers are considered. However, in the setting where the underlying graph has the finite cycle (FC) property (meaning that every edge is contained in at most finitely many cycles) and distinct edge costs, we show that a unique MST [math] exists and the layered greedy algorithm produces iterates that converge to [math] by eventually “locking in" edges after finitely many iterations. Applications to network deployment are discussed.

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无限图中的最小生成树：理论与算法

SIAM 优化期刊》，第 34 卷第 3 期，第 3112-3135 页，2024 年 9 月。摘要。我们讨论在具有有限度的可数节点的连通图上寻找最小生成树（MST）。当边缘成本可求和且存在 MST 时（一般情况下无法保证），我们证明了一种在有限子图（称为层）上寻找 MST 的算法，当层的大小增长到无穷大时，其目标值收敛于整个图的 MST 成本。我们称其为分层贪婪算法，因为在每个有限层上都使用了贪婪算法来寻找 MST。我们强调整体算法并不贪婪，因为在考虑更大的层时，边可以进入和离开迭代生成树。然而，在底层图具有有限循环 (FC) 属性（即每条边最多包含在有限多个循环中）和不同边成本的情况下，我们证明存在唯一的 MST [math]，而且分层贪婪算法产生的迭代在有限次迭代后通过最终 "锁定 "边收敛到 [math]。我们还讨论了网络部署中的应用。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Optimization 数学-应用数学

CiteScore

5.30

自引率

9.70%

发文量

101

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Optimization contains research articles on the theory and practice of optimization. The areas addressed include linear and quadratic programming, convex programming, nonlinear programming, complementarity problems, stochastic optimization, combinatorial optimization, integer programming, and convex, nonsmooth and variational analysis. Contributions may emphasize optimization theory, algorithms, software, computational practice, applications, or the links between these subjects.