Abstract
We consider the problem of computing non-crossing spanning trees in topological graphs. It is known that it is NP-hard to decide whether a topological graph has a non-crossing spanning tree, and that it is hard to approximate the minimum number of crossings in a spanning tree. We consider the parametric complexities of the problem for the following natural input parameters: the number k of crossing edge pairs, the number μ of crossing edges in the given graph, and the number ι of vertices in the interior of the convex hull of the vertex set. We start with an improved strategy of the simple search-tree method to obtain an O *(1.93k) time algorithm. We then give more sophisticated algorithms based on graph separators, with a novel technique to ensure connectivity. The time complexities of our algorithms are \(O^*(2^{O(\sqrt{k})})\), \(O^*(\mu^{O(\mu^{2/3})})\), and \(O^*(2^{O(\sqrt{\iota})})\). By giving a reduction from 3-SAT, we show that the \(O^*(2^{\sqrt{k}})\) complexity is hard to improve under a hypothesis of the complexity of 3-SAT.
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Halldórsson, M.M., Knauer, C., Spillner, A., Tokuyama, T. (2007). Fixed-Parameter Tractability for Non-Crossing Spanning Trees. In: Dehne, F., Sack, JR., Zeh, N. (eds) Algorithms and Data Structures. WADS 2007. Lecture Notes in Computer Science, vol 4619. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73951-7_36
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DOI: https://doi.org/10.1007/978-3-540-73951-7_36
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