We propose two new algorithms called BiLAD and ExactBiLAD for the well-known Single-Constrained Shortest Path (SCSP) problem. It is a fundamental problem in quality-of-service (QoS) routing, where one seeks a source-destination path with the least cost satisfying a delay QoS constraint in a network. As pointed out by Jüttner et al. , there is no widely accepted algorithm with polynomial time to the SCSP problem because the SCSP problem is NP-hard. The remarkable feature of BiLAD is that it ensures that the length of iteratively updated angle interval is shrunk at least at a constant ratio. With the help of this feature, we prove its polynomial time complexity. To the best of our knowledge, this is the first time that the polynomial time complexity is proved in details. The numerical results show that, in most QoS routing test instances, the performance of BiLAD is close to their primal optimal solutions. The proposed modified Dijkstra procedure, whose complexity is the same as that of the Dijkstra algorithm, also accelerates BiLAD. In the second part of the paper, based on the information obtained by BiLAD, we design an exact algorithm–ExactBiLAD, in which an optimal solution to the SCSP problem is finally obtained by scanning the steadily reduced optimal-path-candidate triangle area. The simulation results indicate that ExactBiLAD needs only a dozen times of executing the modified Dijkstra algorithm regardless of the network size or the average node degree. Distinguished from many other exact algorithms, ExactBiLAD has a satisfactory performance in the practical computation.

中文翻译：

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基于拉格朗日对偶的二分法和精确算法的单约束最短路径问题

对于著名的单约束最短路径（SCSP）问题，我们提出了两种称为BiLAD和ExactBiLAD的新算法。这是服务质量（QoS）路由中的一个基本问题，在该路由中，人们寻求成本最低的源-目的地路径来满足网络中的延迟QoS约束。正如尤特纳指出的等。 ，因为SCSP问题是NP-hard，所以没有广泛接受的具有多项式时间的算法用于SCSP问题。BiLAD的显着特征是，它确保迭代更新的角度间隔的长度至少以恒定比率缩小。借助此功能，我们证明了其多项式时间复杂度。据我们所知，这是首次详细证明多项式时间复杂度。数值结果表明，在大多数QoS路由测试实例中，BiLAD的性能接近其最初的最佳解决方案。所提出的改进的Dijkstra程序（其复杂度与Dijkstra算法相同）也加速了BiLAD。在本文的第二部分中，基于BiLAD获得的信息，我们设计了一种精确的算法-ExactBiLAD，其中，通过扫描稳步减小的最佳路径候选三角形区域，最终获得了SCSP问题的最佳解决方案。仿真结果表明，无论网络大小或平均节点度如何，ExactBiLAD只需执行十几次即可执行改进的Dijkstra算法。与许多其他精确算法不同，ExactBiLAD在实际计算中具有令人满意的性能。