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Flexible Graph Connectivity: Approximating Network Design Problems Between 1- and 2-connectivity
arXiv - CS - Data Structures and Algorithms Pub Date : 2019-10-29 , DOI: arxiv-1910.13297
David Adjiashvili, Felix Hommelsheim and Moritz M\"uhlenthaler

Graph connectivity and network design problems are among the most fundamental problems in combinatorial optimization. The minimum spanning tree problem, the two edge-connected spanning subgraph problem (2-ECSS) and the tree augmentation problem (TAP) are all examples of fundamental well-studied network design tasks that postulate different initial states of the network and different assumptions on the reliability of network components. In this paper we motivate and study \emph{Flexible Graph Connectivity} (FGC), a problem that mixes together both the modeling power and the complexities of all aforementioned problems and more. In a nutshell, FGC asks to design a connected network, while allowing to specify different reliability levels for individual edges. While this non-uniform nature of the problem makes it appealing from the modeling perspective, it also renders most existing algorithmic tools for dealing with network design problems unfit for approximating FGC. In this paper we develop a general algorithmic approach for approximating FGC that yields approximation algorithms with ratios that are very close to the best known bounds for many special cases, such as 2-ECSS and TAP. Our algorithm and analysis combine various techniques including a weight-scaling algorithm, a charging argument that uses a variant of exchange bijections between spanning trees and a factor revealing min-max-min optimization problem.

中文翻译:

灵活的图连接:在 1 和 2 连接之间近似网络设计问题

图连通性和网络设计问题是组合优化中最基本的问题。最小生成树问题、两个边连接生成子图问题 (2-ECSS) 和树增强问题 (TAP) 都是经过充分研究的基础网络设计任务的示例,这些任务假设网络的不同初始状态和不同的假设网络组件的可靠性。在本文中,我们激发并研究了\emph{Flexible Graph Connectivity} (FGC),这是一个将建模能力和所有上述问题的复杂性以及更多问题混合在一起的问题。简而言之,FGC 要求设计一个连接网络,同时允许为各个边缘指定不同的可靠性级别。虽然问题的这种非统一性质使其从建模的角度来看很有吸引力,但它也使得大多数现有的用于处理网络设计问题的算法工具不适合近似 FGC。在本文中,我们开发了一种近似 FGC 的通用算法方法,该方法产生的近似算法的比率非常接近许多特殊情况的最佳已知界限,例如 2-ECSS 和 TAP。我们的算法和分析结合了各种技术,包括权重缩放算法、使用生成树之间交换双射变体的充电参数和揭示最小-最大-最小优化问题的因子。在本文中,我们开发了一种近似 FGC 的通用算法方法,该方法产生的近似算法的比率非常接近许多特殊情况的最佳已知界限,例如 2-ECSS 和 TAP。我们的算法和分析结合了各种技术,包括权重缩放算法、使用生成树之间交换双射变体的充电参数和揭示最小-最大-最小优化问题的因子。在本文中,我们开发了一种近似 FGC 的通用算法方法,该方法产生的近似算法的比率非常接近许多特殊情况的最佳已知界限,例如 2-ECSS 和 TAP。我们的算法和分析结合了各种技术,包括权重缩放算法、使用生成树之间交换双射变体的充电参数和揭示最小-最大-最小优化问题的因子。
更新日期:2020-01-14
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