This paper deals with the Constrained Forward Shortest Path Tour Problem, an NP-complete variant of the Forward Shortest Path Tour Problem. Given a directed weighted graph G = (V, A), where the set of nodes V is partitioned into clusters T₁, …, T_N, the aim is determining a shortest path between two given nodes, s and d, with the properties that clusters must be visited according to a given order, and each arc can be crossed at most once. We introduce a mathematical formulation of the problem, and a reduction procedure to reduce the number of variables involved in the model. Furthermore, we propose a Greedy Randomized Adaptive Search Procedure (GRASP) algorithm to solve large instances of the problem. Computational tests show that the reduction procedure is very effective and its application significantly speeds up the resolution of the model. Moreover, the computational results certify the effectiveness of GRASP that often finds the optimal solution and, in general, provides quickly high-quality sub-optimal solutions.

中文翻译：

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受约束的前向最短路径游览问题：数学建模和 GRASP 近似解

本文处理的是受约束的前向最短路径巡回问题，这是前向最短路径巡回问题的 NP 完全变体。给定一个有向加权图G = ( V , A )，其中节点集V被划分为集群T ₁ , …, T _N，目的是确定两个给定节点s和d之间的最短路径，具有必须按照给定顺序访问集群的特性，并且每个弧最多可以穿过一次。我们引入了问题的数学公式，以及减少模型中涉及的变量数量的简化程序。此外，我们提出了一种贪婪随机自适应搜索程序 (GRASP) 算法来解决该问题的大量实例。计算测试表明，缩减程序非常有效，其应用显着加快了模型的分辨率。此外，计算结果证明了 GRASP 的有效性，它经常找到最佳解决方案，并且通常可以快速提供高质量的次优解决方案。