TY - GEN

T1 - Fixed-parameter tractability for non-crossing spanning trees

AU - Halldórsson, Magnús M.

AU - Knauer, Christian

AU - Spillner, Andreas

AU - Tokuyama, Takeshi

PY - 2007

Y1 - 2007

N2 - 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 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(√)). 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.

AB - 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 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(√)). 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.

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U2 - 10.1007/978-3-540-73951-7_36

DO - 10.1007/978-3-540-73951-7_36

M3 - Conference contribution

AN - SCOPUS:38149055034

SN - 3540739483

SN - 9783540739487

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 410

EP - 421

BT - Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings

PB - Springer Verlag

T2 - 10th International Workshop on Algorithms and Data Structures, WADS 2007

Y2 - 15 August 2007 through 17 August 2007

ER -