The computation of short paths in graphs with arc lengths is a pillar of graph algorithmics and network science. In a more diverse world, however, not every short path is equally valuable. For the setting where each vertex is assigned to a group (color), we provide a framework to model multiple natural fairness aspects. We seek to find short paths in which the number of occurrences of each color is within some given lower and upper bounds. Among other results, we prove the introduced problems to be computationally intractable (NP-hard and parameterized hard with respect to the number of colors) even in very restricted settings (such as each color should appear with exactly the same frequency), while also presenting an encouraging algorithmic result ("fixed-parameter tractability") related to the length of the sought solution path for the general problem.

翻译：在带有弧段长度的图中计算短路径是图算法和网络科学的基石。然而，在一个更加多样化的世界中，并非每条短路径都具有同等价值。针对每个顶点被赋予一个组别（颜色）的场景，我们提供了一个框架来建模多种自然的公平性方面。我们旨在寻找满足每种颜色出现次数在给定的下限和上限范围内的短路径。除其他结果外，我们证明所引入的问题即使在非常受限的设置中（例如每种颜色应恰好出现相同频率）在计算上也是难解的（NP难且关于颜色数量为参数化难），同时针对一般问题中与所求解路径长度相关的方面，我们也展示了一个令人鼓舞的算法结果（“固定参数可解性”）。