article

Tree decompositions with small cost

Authors:

Hans L. Bodlaender,

Fedor V. FominAuthors Info & Claims

Discrete Applied Mathematics, Volume 145, Issue 2

Pages 143 - 154

Published: 15 January 2005 Publication History

Abstract

The f-cost of a tree decomposition ({X"i|i@?I},T=(I,F)) for a function f:N->R^+ is defined as @?"i"@?"If(|X"i|). This measure associates with the running time or memory use of some algorithms that use the tree decomposition. In this paper, we investigate the problem to find tree decompositions of minimum f-cost. A function f:N->R^+ is fast, if for every i@?N: f(i+1)>=2f(i). We show that for fast functions f, every graph G has a tree decomposition of minimum f-cost that corresponds to a minimal triangulation of G; if f is not fast, this does not hold. We give polynomial time algorithms for the problem, assuming f is a fast function, for graphs that have a polynomial number of minimal separators, for graphs of treewidth at most two, and for cographs, and show that the problem is NP-hard for bipartite graphs and for cobipartite graphs. We also discuss results for a weighted variant of the problem derived of an application from probabilistic networks.

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Cited By

Korhonen TBerg JJärvisalo M(2019)Solving Graph Problems via Potential Maximal CliquesACM Journal of Experimental Algorithmics10.1145/330129724(1-19)Online publication date: 18-Feb-2019
https://dl.acm.org/doi/10.1145/3301297
Kangas KKoivisto MSalonen S(2019)A Faster Tree-Decomposition Based Algorithm for Counting Linear ExtensionsAlgorithmica10.1007/s00453-019-00633-182:8(2156-2173)Online publication date: 3-Oct-2019
https://dl.acm.org/doi/10.1007/s00453-019-00633-1
Dereniowski DStański A(2019)On Tradeoffs Between Width- and Fill-like Graph ParametersTheory of Computing Systems10.1007/s00224-018-9882-163:3(450-465)Online publication date: 1-Apr-2019
https://dl.acm.org/doi/10.1007/s00224-018-9882-1
Show More Cited By

Index Terms

Tree decompositions with small cost
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Network flows
      2. Trees
  2. Probability and statistics
    1. Probabilistic algorithms
    2. Probabilistic reasoning algorithms
      1. Markov-chain Monte Carlo methods
      2. Sequential Monte Carlo methods
2. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis
      1. Network flows

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Published In

cover image Discrete Applied Mathematics

Discrete Applied Mathematics Volume 145, Issue 2

Structural decompositions, width parameters, and graph labelings (DAS 5)

15 January 2005

263 pages

ISSN:0166-218X

Issue’s Table of Contents

Copyright © Elsevier B.V. © 2004.

Publisher

Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 15 January 2005

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Citations

Cited By

Korhonen TBerg JJärvisalo M(2019)Solving Graph Problems via Potential Maximal CliquesACM Journal of Experimental Algorithmics10.1145/330129724(1-19)Online publication date: 18-Feb-2019
https://dl.acm.org/doi/10.1145/3301297
Kangas KKoivisto MSalonen S(2019)A Faster Tree-Decomposition Based Algorithm for Counting Linear ExtensionsAlgorithmica10.1007/s00453-019-00633-182:8(2156-2173)Online publication date: 3-Oct-2019
https://dl.acm.org/doi/10.1007/s00453-019-00633-1
Dereniowski DStański A(2019)On Tradeoffs Between Width- and Fill-like Graph ParametersTheory of Computing Systems10.1007/s00224-018-9882-163:3(450-465)Online publication date: 1-Apr-2019
https://dl.acm.org/doi/10.1007/s00224-018-9882-1
Fomin FLiedloff MMontealegre PTodinca I(2018)Algorithms Parameterized by Vertex Cover and Modular Width, Through Potential Maximal CliquesAlgorithmica10.1007/s00453-017-0297-180:4(1146-1169)Online publication date: 1-Apr-2018
https://dl.acm.org/doi/10.1007/s00453-017-0297-1
Abseher MMusliu NWoltran S(2017)Improving the efficiency of dynamic programming on tree decompositions via machine learningJournal of Artificial Intelligence Research10.5555/3176764.317678558:1(829-858)Online publication date: 1-Jan-2017
https://dl.acm.org/doi/10.5555/3176764.3176785
Goldberg LGysel RLapinskas J(2016)Approximately counting locally-optimal structuresJournal of Computer and System Sciences10.1016/j.jcss.2016.04.00182:6(1144-1160)Online publication date: 1-Sep-2016
https://dl.acm.org/doi/10.1016/j.jcss.2016.04.001
Scarcello FGottlob GGreco G(2008)Uniform Constraint Satisfaction Problems and Database TheoryComplexity of Constraints10.1007/978-3-540-92800-3_7(156-195)Online publication date: 23-Dec-2008
https://dl.acm.org/doi/10.1007/978-3-540-92800-3_7

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