Fixed-parameter tractability for non-crossing spanning trees

Authors:
Magnús M. Halldórsson;Christian Knauer;Andreas Spillner;Takeshi Tokuyama
Affiliations:
Dept. of Computer Science, Reykjavik University;Institute of Computer Science, Freie Universität Berlin;Institute of Computer Science, Friedrich-Schiller-Universität Jena;Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Venue:
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
Year:
2007

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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 l 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*(2O(√k)), O*(µO(µ2/3)), and O*(2O(√l)). By giving a reduction from 3-SAT, we show that the O*(2√k) complexity is hard to improve under a hypothesis of the complexity of 3-SAT.