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与子例程场景持续稳步增长，这一趋势源于最短路径算法在理论与实践中的显著成功。然而，随着路径模型及其最优性评估日趋复杂，新的挑战也随之涌现。近年来，本领域多项研究以临时性方式将现有标记方法适配至具体问题变体，却未能考虑其底层通用结构：这些研究始终处理多标准场景，而各标准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），该问题既泛化了多目标最短路径问题（MOSP），亦泛化了经典最短路径问题。POSP将众多最短路径应用的特殊结构作为特例进行统一建模。在此通用框架下，我们依据目标函数的性质，深入研究了最优性条件的存在性及其缺失机制。最终贡献为一张综合查找表，系统总结研究成果，帮助读者根据问题权重结构，便捷选择最新多标准最短路径算法。应用实例涵盖时间依赖最短路径、瓶颈路径问题、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。本文结论适用于一般有向图，因此超越了先前仅适用于无环图的泛化研究。