research-article

Bidimensionality and Kernels

Authors:

Fedor V. Fomin,

Daniel Lokshtanov,

Saket Saurabh, and

Dimitrios M. ThilikosAuthors Info & Claims

SIAM Journal on Computing, Volume 49, Issue 6

Pages 1397 - 1422

https://doi.org/10.1137/16M1080264

Published: 01 January 2020 Publication History

Abstract

Bidimensionality theory was introduced by [E. D. Demaine et al., J. ACM, 52 (2005), pp. 866--893] as a tool to obtain subexponential time parameterized algorithms on H-minor-free graphs. In [E. D. Demaine and M. Hajiaghayi, Bidimensionality: New connections between FPT algorithms and PTASs, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadelphia, 2005, pp. 590--601] this theory was extended in order to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this work, we establish a third meta-algorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In particular, we prove that every minor (resp., contraction) bidimensional problem that satisfies a separation property and is expressible in Countable Monadic Second Order Logic (CMSO) admits a linear kernel for classes of graphs that exclude a fixed graph (resp., an apex graph) H as a minor. Our results imply that a multitude of bidimensional problems admit linear kernels on the corresponding graph classes. For most of these problems no polynomial kernels on H-minor-free graphs were known prior to our work.

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Cited By

Golovach PStamoulis GThilikos D(2023)Hitting Topological Minor Models in Planar Graphs is Fixed Parameter TractableACM Transactions on Algorithms10.1145/358368819:3(1-29)Online publication date: 10-Feb-2023
https://dl.acm.org/doi/10.1145/3583688
Bonnet ÉKim EReinald AThomassé SWatrigant R(2022)Twin-width and Polynomial KernelsAlgorithmica10.1007/s00453-022-00965-584:11(3300-3337)Online publication date: 1-Nov-2022
https://dl.acm.org/doi/10.1007/s00453-022-00965-5

Index Terms

Bidimensionality and Kernels
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
  2. Mathematical analysis
    1. Functional analysis
      1. Approximation
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis

Index terms have been assigned to the content through auto-classification.

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cover image SIAM Journal on Computing

SIAM Journal on Computing Volume 49, Issue 6

ISSN:0097-5397

DOI:10.1137/smjcat.49.6

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© 2020, Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

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Published: 01 January 2020

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Cited By

Golovach PStamoulis GThilikos D(2023)Hitting Topological Minor Models in Planar Graphs is Fixed Parameter TractableACM Transactions on Algorithms10.1145/358368819:3(1-29)Online publication date: 10-Feb-2023
https://dl.acm.org/doi/10.1145/3583688
Bonnet ÉKim EReinald AThomassé SWatrigant R(2022)Twin-width and Polynomial KernelsAlgorithmica10.1007/s00453-022-00965-584:11(3300-3337)Online publication date: 1-Nov-2022
https://dl.acm.org/doi/10.1007/s00453-022-00965-5

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