How to navigate through obstacles?
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Given a set of obstacles and two points, is there a path between the two points that does not cross more than k different obstacles? This is a fundamental problem that has undergone a tremendous amount of work. It is known to be NP-hard, even when the obstacles are very simple geometric shapes (e.g., unit-length line segments). The problem can be 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, 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 that without the color-connectivity property, the problem is W[SAT]-hard parameterized by k. A corollary of this result is that, unless W[2] = FPT, the problem cannot be approximated in FPT time to within a factor that is a function of k. By describing a generic plane embedding of the graph instances, we show that our hardness results translate to the geometric instances of the problem. We then focus on graphs satisfying the color-connectivity property. By exploiting the planarity of the graph and the connectivity of the colors, we develop topological results to "represent" the valid s-t paths containing subsets of colors from any vertex v. We employ these results to design an FPT algorithm for the problem parameterized by both k and the treewidth of the graph, and extend this result to obtain an FPT algorithm for the parameterization by both k and the length of the path. The latter result directly implies previous FPT results for various obstacle shapes, such as unit disks and fat regions.
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