IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v16y2008i4d10.1007_s10878-008-9152-2.html
   My bibliography  Save this article

Minimum entropy coloring

Author

Listed:
  • Jean Cardinal

    (Université Libre de Bruxelles (U.L.B.))

  • Samuel Fiorini

    (Université Libre de Bruxelles (U.L.B.))

  • Gwenaël Joret

    (Université Libre de Bruxelles (U.L.B.))

Abstract

We study an information-theoretic variant of the graph coloring problem in which the objective function to minimize is the entropy of the coloring. The minimum entropy of a coloring is called the chromatic entropy and was shown by Alon and Orlitsky (IEEE Trans. Inform. Theory 42(5):1329–1339, 1996) to play a fundamental role in the problem of coding with side information. In this paper, we consider the minimum entropy coloring problem from a computational point of view. We first prove that this problem is NP-hard on interval graphs. We then show that, for every constant ε>0, it is NP-hard to find a coloring whose entropy is within (1−ε)log n of the chromatic entropy, where n is the number of vertices of the graph. A simple polynomial case is also identified. It is known that graph entropy is a lower bound for the chromatic entropy. We prove that this bound can be arbitrarily bad, even for chordal graphs. Finally, we consider the minimum number of colors required to achieve minimum entropy and prove a Brooks-type theorem.

Suggested Citation

  • Jean Cardinal & Samuel Fiorini & Gwenaël Joret, 2008. "Minimum entropy coloring," Journal of Combinatorial Optimization, Springer, vol. 16(4), pages 361-377, November.
    • Handle: RePEc:spr:jcomop:v:16:y:2008:i:4:d:10.1007_s10878-008-9152-2
    DOI: 10.1007/s10878-008-9152-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-008-9152-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10878-008-9152-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jcomop:v:16:y:2008:i:4:d:10.1007_s10878-008-9152-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.