The landscape of applications and subroutines relying on shortest path computations continues to grow steadily. This growth is driven by the undeniable success of shortest path algorithms in theory and practice. It also introduces new challenges as the models and assessing the optimality of paths become more complicated. Hence, multiple recent publications in the field adapt existing labeling methods in an ad-hoc fashion to their specific problem variant without considering the underlying general structure: they always deal with multi-criteria scenarios and those criteria define different partial orders on the paths. In this paper, we introduce the partial order shortest path problem (POSP), a generalization of the multi-objective shortest path problem (MOSP) and in turn also of the classical shortest path problem. POSP captures the particular structure of many shortest path applications as special cases. In this generality, we study optimality conditions or the lack of them, depending on the objective functions' properties. Our final contribution is a big lookup table summarizing our findings and providing the reader an easy way to choose among the most recent multicriteria shortest path algorithms depending on their problem's weight structure. Examples range from time-dependent shortest path and bottleneck path problems to the fuzzy shortest path problem and complex financial weight functions studied in the public transportation community. Our results hold for general digraphs and therefore surpass previous generalizations that were limited to acyclic graphs.

翻译：依赖最短路径计算的应用和子程序的数量持续稳定增长。这种增长源于最短路径算法在理论与实践中的显著成功，同时也带来了新的挑战，因为路径模型的复杂性及最优性评估标准变得更加复杂。因此，该领域近年来的多篇论文采用即席方式将现有标记方法适配到其特定问题变体中，却未考虑底层通用结构：这些场景始终涉及多准则问题，而这些准则在路径上定义了不同的偏序关系。本文提出偏序最短路径问题（POSP），这是多目标最短路径问题（MOSP）的泛化，同时也是经典最短路径问题的推广。POSP将许多最短路径应用的特殊结构作为特例进行统一描述。在此通用框架下，我们根据目标函数的性质研究了最优性条件的存在性。最终贡献在于构建了一个大型查找表，汇总了我们的发现，使读者能够根据其问题的权重结构，便捷地从最新多准则最短路径算法中作出选择。应用实例涵盖从时间依赖最短路径、瓶颈路径问题，到模糊最短路径问题以及公共交通领域研究的复杂金融权重函数。我们的结论适用于一般有向图，因此超越了此前仅限于无环图的泛化结果。