Fixed-parameter tractability for non-crossing spanning trees

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

Research output: Chapter in Book/Report/Conference proceedingConference 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* (1.93k) 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.

Original languageEnglish
Title of host publicationAlgorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings
PublisherSpringer Verlag
Pages410-421
Number of pages12
ISBN (Print)3540739483, 9783540739487
DOIs
Publication statusPublished - 2007
Event10th International Workshop on Algorithms and Data Structures, WADS 2007 - Halifax, Canada
Duration: 2007 Aug 152007 Aug 17

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4619 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other10th International Workshop on Algorithms and Data Structures, WADS 2007
CountryCanada
CityHalifax
Period07/8/1507/8/17

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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