Finding paths is a fundamental problem in graph theory and algorithm design due to its many applications. Recently, this problem has been considered on temporal graphs, where edges may change over a discrete time domain. The analysis of graphs has also taken into account the relevance of vertex properties, modeled by assigning to vertices labels or colors. In this work, we deal with a problem that, given a static or temporal graph, whose vertices are colored graph looks for a path such that (1) the vertices of the path have distinct colors and (2) that path includes the maximum number of colors. We analyze the approximation complexity of the problem on static and temporal graphs, and we prove an inapproximability bound. Then, we consider the problem on temporal graphs, and we design a heuristic for it. We present an experimental evaluation of our heuristic, both on synthetic and real-world graphs. The experimental results show that for many instances of the problem, our method is able to return near-optimal solutions.

由于其广泛的应用，寻找路径是图论和算法设计中的一个基本问题。最近，这个问题已经在时间图上得到考虑，其中边缘可能在离散时域上发生变化。图的分析还考虑了顶点属性的相关性，通过分配给顶点标签或颜色来建模。在这项工作中，我们处理一个问题，给定一个静态或时间图，其顶点是彩色的，图寻找一条路径，使得（1）路径的顶点具有不同的颜色，并且（2）该路径包含最大数量颜色。我们分析了静态和时间图上问题的近似复杂性，并证明了不可近似性界限。然后，我们考虑时间图上的问题，并为其设计启发式。我们在合成图和真实世界图上对我们的启发式进行了实验评估。实验结果表明，对于问题的许多实例，我们的方法能够返回接近最优的解决方案。