局部彩虹路径
Locally Rainbow Paths
AI总结：
本文研究顶点着色有向图中连接两指定顶点的局部彩虹路径问题，证明其在多种受限情形下计算难解，并给出当绕行较短且参数r较小时接近最优的高效算法。
摘要：
我们引入一个算法问题：在顶点着色的有向图中，寻找一条长度为 $\ell$ 的局部彩虹路径，连接两个指定顶点 $s$ 和 $t$。这里，一条路径被称为局部彩虹路径，如果在任意两次访问相同颜色顶点之间，该路径连续经过至少 $r$ 个不同颜色的顶点。该问题推广了著名的寻找彩虹路径问题。它在需要保护不同类型资源免于过度使用的场景中有自然应用，例如作物轮作序列优化或生产过程调度。我们证明该问题在计算上是难解的，即使 $r=2$ 或者是在最短路径中寻找局部彩虹路径。从积极的一面看，如果寻找的路径只走一个较短的绕行（即仅比最短路径略长）且 $r$ 较小，则该问题可以高效求解。实际上，除非指数时间假设（ETH）不成立，否则相应算法的运行时间接近最优。
英文摘要：
We introduce the algorithmic problem of finding a locally rainbow path of length $\ell$ connecting two distinguished vertices $s$ and $t$ in a vertex-colored directed graph. Herein, a path is locally rainbow if between any two visits of equally colored vertices, the path traverses consecutively at least $r$ differently colored vertices. This problem generalizes the well-known problem of finding a rainbow path. It finds natural applications whenever there are different types of resources that must be protected from overuse, such as crop sequence optimization or production process scheduling. We show that the problem is computationally intractable even if $r=2$ or if one looks for a locally rainbow among the shortest paths. On the positive side, if one looks for a path that takes only a short detour (i.e., it is slightly longer than the shortest path) and if $r$ is small, the problem can be solved efficiently. Indeed, the running time of the respective algorithm is near-optimal unless the ETH fails.
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