Fixed-Parameter Tractability for Non-Crossing Spanning Trees

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 ^*(1.93^k) 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.