The precedence constrained traveling salesman (TSP-PC), also known as sequential ordering problem (SOP), consists of finding an optimal tour that satisfies the namesake constraints. Mixed integer-linear programming works well with the ‘lightly constrained’ TSP-PCs, close to asymmetric TSP, as well as the with the ‘heavily constrained’ (Gouveia, Ruthmair, 2015). Dynamic programming (DP) works well with the heavily constrained (Salii, 2019). However, judging by the open TSPLIB SOP instances, the worst for any method are the ‘medium’.

We implement a parallel Morin–Marsten branch-and-bound scheme for DP (DPBB). We show how the lower bound heuristic parameterizes DPBB's worst-case complexity and DPBB ‘inherits’ the abstract travel cost aggregation feature of the DP, permitting its direct use with both the conventional and bottleneck TSP-PC.

The scheme was tested on TSPLIB instances, with best known upper bounds (TSP-PC), or those found by restricted DP (Bottleneck TSP-PC), and lower bounds from a greedy-type heuristic. Our OPENMP-based parallel implementation achieved 20-fold speedup for larger instances. We close the long-standing kro124p.4.sop (conventional TSP-PC) and both kro124p.4.sop and ry48p.2.sop (Bottleneck TSP-PC).

优先约束旅行商 (TSP-PC)，也称为顺序排序问题 (SOP)，包括找到满足同名约束的最优旅行。混合整数线性规划适用于接近非对称 TSP 的“轻度约束”TSP-PC，以及“高度约束”（Gouveia，Ruthmair，2015）。动态规划 (DP) 与严重受限的情况下效果很好(Salii, 2019)。然而，从开放的 TSPLIB SOP 实例来看，任何方法的最差都是“中等”。

我们为 DP (DPBB) 实现了一个并行的Morin-Marsten分支定界方案。我们展示了下界启发式如何参数化 DPBB 的最坏情况复杂性，并且 DPBB“继承”了 DP 的抽象旅行成本聚合特征，允许其直接用于传统和瓶颈TSP-PC。

该方案在 TSPLIB 实例上进行了测试，具有最广为人知的上限 (TSP-PC)，或由受限DP (Bottleneck TSP-PC) 发现的上限，以及来自贪婪型启发式的下限。我们基于 O PEN MP 的并行实现在较大的实例中实现了 20 倍的加速。我们关闭了长期存在的kro124p.4.sop（传统 TSP-PC）以及kro124p.4.sop和ry48p.2.sop（瓶颈 TSP-PC）。