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

4 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 et al.

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

Halldórsson, Magnús M. ; Knauer, Christian ; Spillner, Andreas et al. / 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)).

@inproceedings{949e9dd90d0143b79d79b4a9093e05f0,

title = "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 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.",

author = "Halld{\'o}rsson, {Magn{\'u}s M.} and Christian Knauer and Andreas Spillner and Takeshi Tokuyama",

year = "2007",

doi = "10.1007/978-3-540-73951-7_36",

language = "English",

isbn = "3540739483",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer Verlag",

pages = "410--421",

booktitle = "Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings",

note = "10th International Workshop on Algorithms and Data Structures, WADS 2007 ; Conference date: 15-08-2007 Through 17-08-2007",

}

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.

UR - http://www.scopus.com/inward/record.url?scp=38149055034&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38149055034&partnerID=8YFLogxK

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 -

Fixed-parameter tractability for non-crossing spanning trees

Abstract

Publication series

Other

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this