Parameterized algorithms and data reduction for safe convoy routing
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We study a problem that models safely routing a convoy through a transportation network, where any vertex adjacent to the travel path of the convoy requires additional precaution: Given a graph G=(V,E), two vertices s,t∈ V, and two integers k,ℓ, we search for a simple s-t-path with at most k vertices and at most ℓ neighbors. We study the problem in two types of transportation networks: graphs with small crossing number, as formed by road networks, and tree-like graphs, as formed by waterways. For graphs with constant crossing number, we provide a subexponential 2^O(√(n))-time algorithm and prove a matching lower bound. We also show a polynomial-time data reduction algorithm that reduces any problem instance to an equivalent instance (a so-called problem kernel) of size polynomial in the vertex cover number of the input graph. In contrast, we show that the problem in general graphs is hard to preprocess. Regarding tree-like graphs, we obtain a 2^O(tw)·ℓ^2· n-time algorithm for graphs of treewidth tw, show that there is no problem kernel with size polynomial in tw, yet show a problem kernel with size polynomial in the feedback edge number of the input graph.
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