Fixed-parameter tractability for non-crossing spanning trees

Magnús M. Halldórsson; Christian Knauer; Andreas Spillner; Takeshi Tokuyama

doi:10.1007/978-3-540-73951-7_36

Fixed-parameter tractability for non-crossing spanning trees

Magnús M. Halldórsson, Christian Knauer, Andreas Spillner, Takeshi Tokuyama

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

2 Citations (Scopus)

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 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(√k)), O* (μ_O(μ2/3)), and O*(2^O(√)). 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.

Original language	English
Title of host publication	Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings
Publisher	Springer Verlag
Pages	410-421
Number of pages	12
ISBN (Print)	3540739483, 9783540739487
DOIs	https://doi.org/10.1007/978-3-540-73951-7_36
Publication status	Published - 2007
Externally published	Yes
Event	10th International Workshop on Algorithms and Data Structures, WADS 2007 - Halifax, Canada Duration: 2007 Aug 15 → 2007 Aug 17

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	4619 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Other

Other	10th International Workshop on Algorithms and Data Structures, WADS 2007
Country/Territory	Canada
City	Halifax
Period	07/8/15 → 07/8/17

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science(all)

Access to Document

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

Cite this

Halldórsson, M. M., Knauer, C., Spillner, A., & Tokuyama, T. (2007). Fixed-parameter tractability for non-crossing spanning trees. In Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings (pp. 410-421). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4619 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-540-73951-7_36

Fixed-parameter tractability for non-crossing spanning trees. / Halldórsson, Magnús M.; Knauer, Christian; Spillner, Andreas; Tokuyama, Takeshi.

Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings. Springer Verlag, 2007. p. 410-421 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4619 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Halldórsson, MM, Knauer, C, Spillner, A & Tokuyama, T 2007, Fixed-parameter tractability for non-crossing spanning trees. in Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4619 LNCS, Springer Verlag, pp. 410-421, 10th International Workshop on Algorithms and Data Structures, WADS 2007, Halifax, Canada, 07/8/15. https://doi.org/10.1007/978-3-540-73951-7_36

Halldórsson MM, Knauer C, Spillner A, Tokuyama T. Fixed-parameter tractability for non-crossing spanning trees. In Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings. Springer Verlag. 2007. p. 410-421. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-540-73951-7_36

Halldórsson, Magnús M. ; Knauer, Christian ; Spillner, Andreas ; Tokuyama, Takeshi. / Fixed-parameter tractability for non-crossing spanning trees. Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings. Springer Verlag, 2007. pp. 410-421 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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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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Fixed-parameter tractability for non-crossing spanning trees

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