Polynomial bounds for the Grid-Minor Theorem

Chandra Chekuri, Julia Chuzhoy

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

One of the key results in Robertson and Seymour's seminal work on graph minors is the Grid-Minor Theorem (also called the Excluded Grid Theorem). The theorem states that for every fixed-size grid H, every graph whose treewidth is large enough, contains H as a minor. This theorem has found many applications in graph theory and algorithms. Let f(κ) denote the largest value, such that every graph of treewidth κ contains a grid minor of size f(κ) × f(κ). The best current quantitative bound, due to recent work of Kawarabayashi and Kobayashi [15], and Leaf and Seymour [18], shows that f(κ) = Ω(√ log κ/ log log κ). In contrast, the best known upper bound implies that f(κ) = O(√ κ/ log κ) [22]. In this paper we obtain the first polynomial relationship between treewidth and grid-minor size by showing that f(κ) = Ω(k δ) for some fixed constant δ > 0, and describe an algorithm, whose running time is polynomial in |V (G)| and κ, that finds a model of such a grid-minor in G.

Original languageEnglish (US)
Title of host publicationSTOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery
Pages60-69
Number of pages10
ISBN (Print)9781450327107
DOIs
StatePublished - Jan 1 2014
Event4th Annual ACM Symposium on Theory of Computing, STOC 2014 - New York, NY, United States
Duration: May 31 2014Jun 3 2014

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Other

Other4th Annual ACM Symposium on Theory of Computing, STOC 2014
CountryUnited States
CityNew York, NY
Period5/31/146/3/14

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Keywords

  • Grid minor theorem
  • Treewidth

ASJC Scopus subject areas

  • Software

Cite this

Chekuri, C., & Chuzhoy, J. (2014). Polynomial bounds for the Grid-Minor Theorem. In STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing (pp. 60-69). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/2591796.2591813