Given a set of obstacles and two points in the plane, is there a path between the two points that does not cross more than k different obstacles? Equivalently, can we remove k obstacles so that there is an obstacle-free path between the two designated points? This is a fundamental NP-hard problem that has undergone a tremendous amount of research work. The problem can be formulated and generalized into the following graph problem: Given a planar graph G whose vertices are colored by color sets, two designated vertices s , t ∈ V ( G ), and k ∈ N, is there an s - t path in G that uses at most k colors? If each obstacle is connected, then the resulting graph satisfies the color-connectivity property, namely that each color induces a connected subgraph. We study the complexity and design algorithms for the above graph problem with an eye on its geometric applications. We prove a set of hardness results, including a result showing that the color-connectivity property is crucial for any hope for fixed-parameter tractable (FPT) algorithms. We also show that our hardness results translate to the geometric instances of the problem. We then focus on graphs satisfying the color-connectivity property. We design an FPT algorithm for this problem parameterized by both k and the treewidth of the graph and extend this result further to obtain an FPT algorithm for the parameterization by both k and the length of the path. The latter result implies and explains previous FPT results for various obstacle shapes.

中文翻译：

---

彩色路径问题及其应用

给定平面上的一组障碍物和两个点，两点之间是否存在一条不超过ķ不同的障碍？等效地，我们可以删除ķ障碍物，以便在两个指定点之间存在无障碍路径？这是一个基本的 NP-hard 问题，已经进行了大量的研究工作。该问题可以表述并推广为以下图问题： 给定一个平面图G其顶点由颜色集着色，两个指定顶点s,吨∈五(G）， 和ķ∈ N，是否存在s-吨路径G最多使用ķ颜色？如果每个障碍物都是连通的，那么得到的图满足颜色连通性的性质，即每种颜色都导出一个连通子图。我们研究了上述图问题的复杂性和设计算法，着眼于它的几何应用。我们证明了一组硬度结果，包括一个结果表明颜色连接属性对于固定参数易处理 (FPT) 算法的任何希望都是至关重要的。我们还表明，我们的硬度结果可以转化为问题的几何实例。然后，我们专注于满足颜色连通性属性的图。我们为这个问题设计了一个 FPT 算法，由两者参数化ķ和图的树宽，并进一步扩展此结果以获得 FPT 算法，用于通过两者进行参数化ķ和路径的长度。后一个结果暗示并解释了先前针对各种障碍物形状的 FPT 结果。