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This paper addresses the Forward Shortest Path Tour Problem (FSPTP). Given a weighted directed graph, whose nodes are partitioned into clusters, the FSPTP consists of finding a shortest path from a source node to a destination node and which crosses all the clusters in a fixed order. We propose a polynomial time algorithm to solve the problem and show that our algorithm can be easily adapted to solve the shortest path tour problem, a slightly different variant of the FSPTP. Moreover, we carried out some preliminary computational tests to verify how the performance of the algorithm is affected by parameters of the instances

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, …, TN, 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.

This article addresses the 2‐edge‐connected minimum branch vertices problem, a variant of the minimum branch vertices problem in which the spanning subgraph is required to be 2‐edge‐connected for survivability reasons. The problem has been recently introduced and finds application in optical networks design scenarios, where branch vertices are associated to switch devices that allow to split the entering light signals and send them to several adjacent vertices. An exact approach to the problem has been proposed in the literature. In this paper, we formally prove its NP‐completeness and propose a genetic algorithm, which exploits some literature‐provided procedures for efficiently checking and restoring solutions feasibility, and makes use of novel ad‐hoc designed operators aiming to improve their values, reducing the number of branch vertices. The computational tests show that, on the benchmark instances, the genetic algorithm very often finds the optimal solution. Moreover, in order to further investigate the effectiveness and the performance of our algorithm, we generated a new set of random instances where the optimal solution is known a priori.