University Library
  • Login
A gateway to Melbourne's research publications
Minerva Access is the University's Institutional Repository. It aims to collect, preserve, and showcase the intellectual output of staff and students of the University of Melbourne for a global audience.
View Item 
  • Minerva Access
  • Science
  • School of Mathematics and Statistics
  • School of Mathematics and Statistics - Research Publications
  • View Item
  • Minerva Access
  • Science
  • School of Mathematics and Statistics
  • School of Mathematics and Statistics - Research Publications
  • View Item
JavaScript is disabled for your browser. Some features of this site may not work without it.

    A Linear-Time Algorithm to Find a Separator in a Graph Excluding a Minor

    Thumbnail
    Citations
    Scopus
    Web of Science
    Altmetric
    17
    14
    Author
    Reed, B; Wood, DR
    Date
    2009-10-01
    Source Title
    ACM Transactions on Algorithms
    Publisher
    ASSOC COMPUTING MACHINERY
    University of Melbourne Author/s
    WOOD, DAVID
    Affiliation
    Mathematics and Statistics
    Metadata
    Show full item record
    Document Type
    Journal Article
    Citations
    Reed, B. & Wood, D. R. (2009). A Linear-Time Algorithm to Find a Separator in a Graph Excluding a Minor. ACM TRANSACTIONS ON ALGORITHMS, 5 (4), https://doi.org/10.1145/1597036.1597043.
    Access Status
    This item is currently not available from this repository
    URI
    http://hdl.handle.net/11343/29373
    DOI
    10.1145/1597036.1597043
    Abstract
    <jats:p> Let <jats:italic>G</jats:italic> be an <jats:italic>n</jats:italic> -vertex <jats:italic>m</jats:italic> -edge graph with weighted vertices. A pair of vertex sets <jats:italic>A</jats:italic> , <jats:italic>B</jats:italic> ⊆ <jats:italic>V</jats:italic> ( <jats:italic>G</jats:italic> ) is a 2/3 <jats:italic>-separation</jats:italic> of <jats:italic>order</jats:italic> | <jats:italic>A</jats:italic> ∩ <jats:italic>B</jats:italic> | if <jats:italic>A</jats:italic> ∪ <jats:italic>B</jats:italic> = <jats:italic>V</jats:italic> ( <jats:italic>G</jats:italic> ), there is no edge between <jats:italic>A</jats:italic> − <jats:italic>B</jats:italic> and <jats:italic>B</jats:italic> − <jats:italic>A</jats:italic> , and both <jats:italic>A</jats:italic> − <jats:italic>B</jats:italic> and <jats:italic>B</jats:italic> − <jats:italic>A</jats:italic> have weight at most 2/3 the total weight of <jats:italic>G</jats:italic> . Let ℓ ∈ Z <jats:sup>+</jats:sup> be fixed. Alon et al. [1990] presented an algorithm that in <jats:italic>O</jats:italic> ( <jats:italic>n</jats:italic> <jats:sup>1/2</jats:sup> <jats:italic>m</jats:italic> ) time, outputs either a <jats:italic>K</jats:italic> <jats:sub>ℓ</jats:sub> -minor of <jats:italic>G</jats:italic> , or a separation of <jats:italic>G</jats:italic> of order <jats:italic>O</jats:italic> ( <jats:italic>n</jats:italic> <jats:sup>1/2</jats:sup> ). Whether there is a <jats:italic>O</jats:italic> ( <jats:italic>n</jats:italic> + <jats:italic>m</jats:italic> )-time algorithm for this theorem was left as an open problem. In this article, we obtain a <jats:italic>O</jats:italic> ( <jats:italic>n</jats:italic> + <jats:italic>m</jats:italic> )-time algorithm at the expense of a <jats:italic>O</jats:italic> ( <jats:italic>n</jats:italic> <jats:sup>2/3</jats:sup> ) separator. Moreover, our algorithm exhibits a trade-off between time complexity and the order of the separator. In particular, for any given ϵ ∈ [0,1/2], our algorithm outputs either a <jats:italic>K</jats:italic> <jats:sub>ℓ</jats:sub> -minor of <jats:italic>G</jats:italic> , or a separation of <jats:italic>G</jats:italic> with order <jats:italic>O</jats:italic> ( <jats:italic>n</jats:italic> <jats:sup>(2−ϵ)/3</jats:sup> in <jats:italic>O</jats:italic> ( <jats:italic>n</jats:italic> <jats:sup>1 + ϵ</jats:sup> + <jats:italic>m</jats:italic> ) time. As an application we give a fast approximation algorithm for finding an independent set in a graph with no <jats:italic>K</jats:italic> ℓ-minor. </jats:p>
    Keywords
    Computation Theory and Mathematics

    Export Reference in RIS Format     

    Endnote

    • Click on "Export Reference in RIS Format" and choose "open with... Endnote".

    Refworks

    • Click on "Export Reference in RIS Format". Login to Refworks, go to References => Import References


    Collections
    • Minerva Elements Records [54083]
    • School of Mathematics and Statistics - Research Publications [864]
    Minerva AccessDepositing Your Work (for University of Melbourne Staff and Students)NewsFAQs

    BrowseCommunities & CollectionsBy Issue DateAuthorsTitlesSubjectsThis CollectionBy Issue DateAuthorsTitlesSubjects
    My AccountLoginRegister
    StatisticsMost Popular ItemsStatistics by CountryMost Popular Authors