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 -