Abstract

In this paper, we study the k edge-connected L-hop-constrained network design problem. Given a weighted graph \(G = (V,E)\) , a set D of pairs of nodes, two integers \(L \ge 2\) and \(k \ge 2\) , the problem consists in finding a minimum weight subgraph of G containing at least k edge-disjoint paths of length at most L between every pair \(\{s,t\}\) of D. The problem has several applications in telecommunications network design. It also has applications in reliable container transportation network design and public transportation. Even if the problem has been studied for several decades, it appears, to the best of our knowledge, that the associated polytope is not well known, even when \(L \in \{2,3\}\) . In this paper, we are particularly interested in the polyhedral analysis of the problem as well as an exact solving of the problem for large-scale instances. We consider the case where \(L \in \{2,3\}\) and present two integer programming formulations introduced in Diarrassouba et al. (2016). Then, we consider the polytope associated with these formulations, and present several new classes of valid inequalities, involving the so-called design variables. We also present separation algorithms for these inequalities and devise Branch-and-Cut algorithms for solving the problem. Finally, we present an extensive computational study for \(L \in \{2,3\}\) and \(k \in \{3,4,5\}\) , in which we test the efficiency of our Branch-and-Cut algorithms. We also compare the algorithm against a resolution using CPLEX Branch-and-Cut and CPLEX Benders Decomposition frameworks. The results show that our Branch-and-Cut algorithm can be competitive against these two frameworks.

中文翻译：

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k边连接L跳约束网络设计问题的优化算法

摘要

在本文中，我们研究了k边连接的L跳约束网络设计问题。给定一个加权图\(G = (V,E)\)、一组节点对D 、两个整数\(L \ge 2\)和\(k \ge 2\)，问题在于找到一个G的最小权重子图，在D的每对\(\{s,t\}\)之间包含至少k 个长度至多为L的边不相交路径。该问题在电信网络设计中有多种应用。它还在可靠的集装箱运输网络设计和公共交通方面有应用。即使这个问题已经研究了几十年，据我们所知，相关的多胞形似乎并不为人所知，即使当\(L \in \{2,3\}\)时也是如此。在本文中，我们对问题的多面体分析以及大规模实例问题的精确解决特别感兴趣。我们考虑\(L \in \{2,3\}\)的情况，并提出 Diarrassouba 等人引入的两个整数规划公式。 （2016）。然后，我们考虑与这些公式相关的多面体，并提出几类新的有效不等式，涉及所谓的设计变量。我们还提出了针对这些不等式的分离算法，并设计了用于解决该问题的分支剪切算法。最后，我们对\(L \in \{2,3\}\)和\(k \in \{3,4,5\}\)进行了广泛的计算研究，其中我们测试了分支的效率-and-Cut 算法。我们还将该算法与使用 CPLEX Branch-and-Cut 和 CPLEX Benders 分解框架的解析进行比较。结果表明，我们的分支剪切算法可以与这两个框架竞争。