<?xml version="1.0" encoding="utf-8"?>
<TEI xmlns="http://www.tei-c.org/ns/1.0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:hal="http://hal.archives-ouvertes.fr/" xmlns:gml="http://www.opengis.net/gml/3.3/" xmlns:gmlce="http://www.opengis.net/gml/3.3/ce" version="1.1" xsi:schemaLocation="http://www.tei-c.org/ns/1.0 http://api.archives-ouvertes.fr/documents/aofr-sword.xsd">
  <teiHeader>
    <fileDesc>
      <titleStmt>
        <title>HAL TEI export of inria-00635882</title>
      </titleStmt>
      <publicationStmt>
        <distributor>CCSD</distributor>
        <availability status="restricted">
          <licence target="https://creativecommons.org/publicdomain/zero/1.0/">CC0 1.0 - Universal</licence>
        </availability>
        <date when="2026-05-03T09:58:40+02:00"/>
      </publicationStmt>
      <sourceDesc>
        <p part="N">HAL API Platform</p>
      </sourceDesc>
    </fileDesc>
  </teiHeader>
  <text>
    <body>
      <listBibl>
        <biblFull>
          <titleStmt>
            <title xml:lang="en">Weighted Improper Colouring</title>
            <author role="aut">
              <persName>
                <forename type="first">Julio</forename>
                <surname>Araujo</surname>
              </persName>
              <email type="md5">6a3486cf2270c59b35dc824a28edf949</email>
              <email type="domain">inria.fr</email>
              <idno type="idhal" notation="numeric">912809</idno>
              <idno type="halauthorid" notation="string">449889-912809</idno>
              <affiliation ref="#struct-2466"/>
              <affiliation ref="#struct-131562"/>
            </author>
            <author role="aut">
              <persName>
                <forename type="first">Jean-Claude</forename>
                <surname>Bermond</surname>
              </persName>
              <email type="md5">6152b43c5dee4ff74e213a261cfe369d</email>
              <email type="domain">inria.fr</email>
              <idno type="idhal" notation="string">jean-claude-bermond</idno>
              <idno type="idhal" notation="numeric">739520</idno>
              <idno type="halauthorid" notation="string">13047-739520</idno>
              <idno type="ORCID">https://orcid.org/0009-0001-0454-0050</idno>
              <idno type="IDREF">https://www.idref.fr/032696256</idno>
              <affiliation ref="#struct-2466"/>
            </author>
            <author role="aut">
              <persName>
                <forename type="first">Frédéric</forename>
                <surname>Giroire</surname>
              </persName>
              <email type="md5">6b9d377baf3ca6a6eb902ee1ecda6bee</email>
              <email type="domain">cnrs.fr</email>
              <idno type="idhal" notation="string">frederic-giroire</idno>
              <idno type="idhal" notation="numeric">5597</idno>
              <idno type="halauthorid" notation="string">19508-5597</idno>
              <idno type="IDREF">https://www.idref.fr/11611598X</idno>
              <idno type="ORCID">https://orcid.org/0000-0002-3727-051X</idno>
              <orgName ref="#struct-441569"/>
              <affiliation ref="#struct-2466"/>
            </author>
            <author role="aut">
              <persName>
                <forename type="first">Frédéric</forename>
                <surname>Havet</surname>
              </persName>
              <email type="md5">7ad9297a403d0a4dfa0bf25d740b57aa</email>
              <email type="domain">cnrs.fr</email>
              <idno type="idhal" notation="string">frederic-havet</idno>
              <idno type="idhal" notation="numeric">9383</idno>
              <idno type="halauthorid" notation="string">20978-9383</idno>
              <idno type="IDREF">https://www.idref.fr/110354842</idno>
              <idno type="ORCID">https://orcid.org/0000-0002-3447-8112</idno>
              <affiliation ref="#struct-2466"/>
            </author>
            <author role="aut">
              <persName>
                <forename type="first">Dorian</forename>
                <surname>Mazauric</surname>
              </persName>
              <email type="md5">5d34b01b31db3910b8f71c236f14d425</email>
              <email type="domain">inria.fr</email>
              <idno type="idhal" notation="string">dorian-mazauric</idno>
              <idno type="idhal" notation="numeric">6470</idno>
              <idno type="halauthorid" notation="string">23522-6470</idno>
              <idno type="IDREF">https://www.idref.fr/157983455</idno>
              <affiliation ref="#struct-2466"/>
            </author>
            <author role="aut">
              <persName>
                <forename type="first">Remigiusz</forename>
                <surname>Modrzejewski</surname>
              </persName>
              <email type="md5">f3000954729ca7cccd7d4c9503c95392</email>
              <email type="domain">inria.fr</email>
              <idno type="idhal" notation="numeric">891945</idno>
              <idno type="halauthorid" notation="string">533601-891945</idno>
              <affiliation ref="#struct-2466"/>
            </author>
            <editor role="depositor">
              <persName>
                <forename>Julio</forename>
                <surname>Araujo</surname>
              </persName>
              <email type="md5">e2ebf8b4eb6887a4d504c9ea08c10236</email>
              <email type="domain">mat.ufc.br</email>
            </editor>
            <funder ref="#projanr-31126"/>
            <funder>This work was partially supported by r'egion PACA and ANR International Taiwan GRATEL (ANR-09-BLAN-0373).</funder>
          </titleStmt>
          <editionStmt>
            <edition n="v1" type="current">
              <date type="whenSubmitted">2011-10-26 11:20:26</date>
              <date type="whenModified">2025-10-07 15:32:02</date>
              <date type="whenReleased">2011-10-26 11:26:53</date>
              <date type="whenProduced">2011-07-20</date>
              <date type="whenEndEmbargoed">2011-10-26</date>
              <ref type="file" target="https://inria.hal.science/inria-00635882v1/document">
                <date notBefore="2011-10-26"/>
              </ref>
              <ref type="file" subtype="author" n="1" target="https://inria.hal.science/inria-00635882v1/file/jcb-coloring-lncs-submitted.pdf" id="file-635882-335348">
                <date notBefore="2011-10-26"/>
              </ref>
            </edition>
            <respStmt>
              <resp>contributor</resp>
              <name key="149554">
                <persName>
                  <forename>Julio</forename>
                  <surname>Araujo</surname>
                </persName>
                <email type="md5">e2ebf8b4eb6887a4d504c9ea08c10236</email>
                <email type="domain">mat.ufc.br</email>
              </name>
            </respStmt>
          </editionStmt>
          <publicationStmt>
            <distributor>CCSD</distributor>
            <idno type="halId">inria-00635882</idno>
            <idno type="halUri">https://inria.hal.science/inria-00635882</idno>
            <idno type="halBibtex">araujo:inria-00635882</idno>
            <idno type="halRefHtml">&lt;i&gt;22th International Workshop, IWOCA 2011&lt;/i&gt;, Jul 2011, Victoria, Canada</idno>
            <idno type="halRef">22th International Workshop, IWOCA 2011, Jul 2011, Victoria, Canada</idno>
            <availability status="restricted">
              <licence target="https://about.hal.science/hal-authorisation-v1/">HAL Authorization<ref corresp="#file-635882-335348"/></licence>
            </availability>
          </publicationStmt>
          <seriesStmt>
            <idno type="stamp" n="UNICE">Université Nice Sophia Antipolis</idno>
            <idno type="stamp" n="CNRS">CNRS - Centre national de la recherche scientifique</idno>
            <idno type="stamp" n="INRIA">INRIA - Institut National de Recherche en Informatique et en Automatique</idno>
            <idno type="stamp" n="INRIA-SOPHIA">INRIA Sophia Antipolis - Méditerranée</idno>
            <idno type="stamp" n="I3S">Laboratoire d'Informatique, Signaux et Systèmes de Sophia-Antipolis</idno>
            <idno type="stamp" n="INRIASO">INRIA-SOPHIA</idno>
            <idno type="stamp" n="INRIA_TEST">INRIA - Institut National de Recherche en Informatique et en Automatique</idno>
            <idno type="stamp" n="TESTALAIN1">TESTALAIN1</idno>
            <idno type="stamp" n="INRIA2">INRIA 2</idno>
            <idno type="stamp" n="UNIV-COTEDAZUR">Université Côte d'Azur</idno>
            <idno type="stamp" n="ANR">ANR</idno>
            <idno type="stamp" n="TEST-NICE">test nice</idno>
          </seriesStmt>
          <notesStmt>
            <note type="audience" n="2">International</note>
            <note type="invited" n="0">No</note>
            <note type="popular" n="0">No</note>
            <note type="peer" n="1">Yes</note>
            <note type="proceedings" n="1">Yes</note>
          </notesStmt>
          <sourceDesc>
            <biblStruct>
              <analytic>
                <title xml:lang="en">Weighted Improper Colouring</title>
                <author role="aut">
                  <persName>
                    <forename type="first">Julio</forename>
                    <surname>Araujo</surname>
                  </persName>
                  <email type="md5">6a3486cf2270c59b35dc824a28edf949</email>
                  <email type="domain">inria.fr</email>
                  <idno type="idhal" notation="numeric">912809</idno>
                  <idno type="halauthorid" notation="string">449889-912809</idno>
                  <affiliation ref="#struct-2466"/>
                  <affiliation ref="#struct-131562"/>
                </author>
                <author role="aut">
                  <persName>
                    <forename type="first">Jean-Claude</forename>
                    <surname>Bermond</surname>
                  </persName>
                  <email type="md5">6152b43c5dee4ff74e213a261cfe369d</email>
                  <email type="domain">inria.fr</email>
                  <idno type="idhal" notation="string">jean-claude-bermond</idno>
                  <idno type="idhal" notation="numeric">739520</idno>
                  <idno type="halauthorid" notation="string">13047-739520</idno>
                  <idno type="ORCID">https://orcid.org/0009-0001-0454-0050</idno>
                  <idno type="IDREF">https://www.idref.fr/032696256</idno>
                  <affiliation ref="#struct-2466"/>
                </author>
                <author role="aut">
                  <persName>
                    <forename type="first">Frédéric</forename>
                    <surname>Giroire</surname>
                  </persName>
                  <email type="md5">6b9d377baf3ca6a6eb902ee1ecda6bee</email>
                  <email type="domain">cnrs.fr</email>
                  <idno type="idhal" notation="string">frederic-giroire</idno>
                  <idno type="idhal" notation="numeric">5597</idno>
                  <idno type="halauthorid" notation="string">19508-5597</idno>
                  <idno type="IDREF">https://www.idref.fr/11611598X</idno>
                  <idno type="ORCID">https://orcid.org/0000-0002-3727-051X</idno>
                  <orgName ref="#struct-441569"/>
                  <affiliation ref="#struct-2466"/>
                </author>
                <author role="aut">
                  <persName>
                    <forename type="first">Frédéric</forename>
                    <surname>Havet</surname>
                  </persName>
                  <email type="md5">7ad9297a403d0a4dfa0bf25d740b57aa</email>
                  <email type="domain">cnrs.fr</email>
                  <idno type="idhal" notation="string">frederic-havet</idno>
                  <idno type="idhal" notation="numeric">9383</idno>
                  <idno type="halauthorid" notation="string">20978-9383</idno>
                  <idno type="IDREF">https://www.idref.fr/110354842</idno>
                  <idno type="ORCID">https://orcid.org/0000-0002-3447-8112</idno>
                  <affiliation ref="#struct-2466"/>
                </author>
                <author role="aut">
                  <persName>
                    <forename type="first">Dorian</forename>
                    <surname>Mazauric</surname>
                  </persName>
                  <email type="md5">5d34b01b31db3910b8f71c236f14d425</email>
                  <email type="domain">inria.fr</email>
                  <idno type="idhal" notation="string">dorian-mazauric</idno>
                  <idno type="idhal" notation="numeric">6470</idno>
                  <idno type="halauthorid" notation="string">23522-6470</idno>
                  <idno type="IDREF">https://www.idref.fr/157983455</idno>
                  <affiliation ref="#struct-2466"/>
                </author>
                <author role="aut">
                  <persName>
                    <forename type="first">Remigiusz</forename>
                    <surname>Modrzejewski</surname>
                  </persName>
                  <email type="md5">f3000954729ca7cccd7d4c9503c95392</email>
                  <email type="domain">inria.fr</email>
                  <idno type="idhal" notation="numeric">891945</idno>
                  <idno type="halauthorid" notation="string">533601-891945</idno>
                  <affiliation ref="#struct-2466"/>
                </author>
              </analytic>
              <monogr>
                <meeting>
                  <title>22th International Workshop, IWOCA 2011</title>
                  <date type="start">2011-07-20</date>
                  <date type="end">2011-07-22</date>
                  <settlement>Victoria</settlement>
                  <country key="CA">Canada</country>
                </meeting>
                <imprint>
                  <date type="datePub">2011-11-15</date>
                </imprint>
              </monogr>
            </biblStruct>
          </sourceDesc>
          <profileDesc>
            <langUsage>
              <language ident="en">English</language>
            </langUsage>
            <textClass>
              <classCode scheme="halDomain" n="info.info-ni">Computer Science [cs]/Networking and Internet Architecture [cs.NI]</classCode>
              <classCode scheme="halTypology" n="COMM">Conference papers</classCode>
              <classCode scheme="halOldTypology" n="COMM">Conference papers</classCode>
              <classCode scheme="halTreeTypology" n="COMM">Conference papers</classCode>
            </textClass>
            <abstract xml:lang="en">
              <p>In this paper, we study a colouring problem motivated by a practical frequency assignment problem and up to our best knowledge new. In wireless networks, a node interferes with the other nodes the level of interference depending on numerous parameters: distance between the nodes, geographical topography, obstacles, etc.We model this with a weighted graph G where the weights on the edges represent the noise (interference) between the two end-nodes. The total interference in a node is then the sum of all the noises of the nodes emitting on the same frequency. A weighted t-improper k-colouring of G is a k-colouring of the nodes of G (assignment of k frequencies) such that the interference at each node does not exceed some threshold t. The Weighted Improper Colouring problem, that we consider here consists in determining the weighted t-improper chromatic number defined as the minimum integer k such that G admits a weighted t-improper k-colouring. We also consider the dual problem, denoted the Threshold Improper Colouring problem, where given a number k of colours (frequencies) we want to determine the minimum real t such that G admits a weighted t-improper k-colouring. We show that both problems are NP-hard and first present general upper bounds; in particular we show a generalisation of Lov'asz's Theorem for the weighted t-improper chromatic number. We then show how to transform an instance of the Threshold Improper Colouring problem into another equivalent one where the weights are either 1 or M, for a sufficient big value M. Motivated by the original application, we study a special interference model on various grids (square, triangular, hexagonal) where a node produces a noise of intensity 1 for its neighbours and a noise of intensity 1/2 for the nodes that are at distance 2. Consequently, the problem consists of determining the weighted t-improper chromatic number when G is the square of a grid and the weights of the edges are 1, if their end nodes are adjacent in the grid, and 1/2 otherwise. Finally, we model the problem using linear integer programming, propose and test heuristic and exact Branch-and-Bound algorithms on random cell-like graphs, namely the Poisson-Voronoi tessellations.</p>
            </abstract>
          </profileDesc>
        </biblFull>
      </listBibl>
    </body>
    <back>
      <listOrg type="structures">
        <org type="researchteam" xml:id="struct-2466" status="OLD">
          <idno type="RNSR">200018393X</idno>
          <orgName>Algorithms, simulation, combinatorics and optimization for telecommunications</orgName>
          <orgName type="acronym">MASCOTTE</orgName>
          <desc>
            <address>
              <country key="FR"/>
            </address>
          </desc>
          <listRelation>
            <relation active="#struct-34586" type="direct"/>
            <relation active="#struct-300009" type="indirect"/>
            <relation active="#struct-451999" type="direct"/>
            <relation active="#struct-13009" type="indirect"/>
            <relation active="#struct-117617" type="indirect"/>
            <relation name="UMR7271" active="#struct-441569" type="indirect"/>
            <relation active="#struct-1039632" type="indirect"/>
          </listRelation>
        </org>
        <org type="laboratory" xml:id="struct-131562" status="VALID">
          <orgName>Parallelism, Graphs and Optimization Research Group</orgName>
          <orgName type="acronym">ParGO</orgName>
          <desc>
            <address>
              <addrLine>Universidade Federal do Ceará Departamento de Computação Campus do Pici, Bloco 910 60440-554 Fortaleza, CE Brazil</addrLine>
              <country key="BR"/>
            </address>
            <ref type="url">http://www.lia.ufc.br/~pargo</ref>
          </desc>
          <listRelation>
            <relation active="#struct-117182" type="direct"/>
          </listRelation>
        </org>
        <org type="laboratory" xml:id="struct-34586" status="VALID">
          <idno type="RNSR">198318250R</idno>
          <idno type="ROR">https://ror.org/01nzkaw91</idno>
          <orgName>Centre Inria d'Université Côte d'Azur</orgName>
          <desc>
            <address>
              <addrLine>2004 route des Lucioles BP 93 06902 Sophia Antipolis</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.inria.fr/centre/sophia/</ref>
          </desc>
          <listRelation>
            <relation active="#struct-300009" type="direct"/>
          </listRelation>
        </org>
        <org type="institution" xml:id="struct-300009" status="VALID">
          <idno type="ROR">https://ror.org/02kvxyf05</idno>
          <orgName>Institut National de Recherche en Informatique et en Automatique</orgName>
          <orgName type="acronym">Inria</orgName>
          <desc>
            <address>
              <addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.inria.fr/en/</ref>
          </desc>
        </org>
        <org type="department" xml:id="struct-451999" status="VALID">
          <orgName>COMmunications, Réseaux, systèmes Embarqués et Distribués</orgName>
          <orgName type="acronym">Laboratoire I3S - COMRED</orgName>
          <date type="start">2016-03-02</date>
          <desc>
            <address>
              <addrLine>Laboratoire I3SCS 4012106903 Sophia Antipolis Cedex</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.i3s.unice.fr/comred</ref>
          </desc>
          <listRelation>
            <relation active="#struct-13009" type="direct"/>
            <relation active="#struct-117617" type="indirect"/>
            <relation name="UMR7271" active="#struct-441569" type="indirect"/>
            <relation active="#struct-1039632" type="indirect"/>
          </listRelation>
        </org>
        <org type="laboratory" xml:id="struct-13009" status="VALID">
          <idno type="IdRef">050320491</idno>
          <idno type="ISNI">0000000105613840</idno>
          <idno type="RNSR">201220432L</idno>
          <idno type="ROR">https://ror.org/01215r597</idno>
          <orgName>Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis</orgName>
          <orgName type="acronym">I3S</orgName>
          <desc>
            <address>
              <addrLine>2000, route des Lucioles - Les Algorithmes - bât. Euclide B 06900 Sophia Antipolis</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.i3s.unice.fr/</ref>
          </desc>
          <listRelation>
            <relation active="#struct-117617" type="direct"/>
            <relation name="UMR7271" active="#struct-441569" type="direct"/>
            <relation active="#struct-1039632" type="direct"/>
          </listRelation>
        </org>
        <org type="institution" xml:id="struct-117617" status="VALID">
          <idno type="IdRef">026403498</idno>
          <idno type="ISNI">0000000123372892</idno>
          <idno type="ROR">https://ror.org/02k9vew78</idno>
          <orgName>Université Nice Sophia Antipolis (1965 - 2019)</orgName>
          <orgName type="acronym">UNS</orgName>
          <date type="start">1965-10-23</date>
          <date type="end">2019-12-31</date>
          <desc>
            <address>
              <addrLine>Parc Valrose, 06100 Nice</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://unice.fr/</ref>
          </desc>
        </org>
        <org type="regroupinstitution" xml:id="struct-441569" status="VALID">
          <idno type="IdRef">02636817X</idno>
          <idno type="ISNI">0000000122597504</idno>
          <idno type="ROR">https://ror.org/02feahw73</idno>
          <orgName>Centre National de la Recherche Scientifique</orgName>
          <orgName type="acronym">CNRS</orgName>
          <date type="start">1939-10-19</date>
          <desc>
            <address>
              <country key="FR"/>
            </address>
            <ref type="url">https://www.cnrs.fr/</ref>
          </desc>
        </org>
        <org type="regroupinstitution" xml:id="struct-1039632" status="VALID">
          <idno type="IdRef">241035694</idno>
          <idno type="ROR">https://ror.org/019tgvf94</idno>
          <orgName>Université Côte d'Azur</orgName>
          <orgName type="acronym">UniCA</orgName>
          <date type="start">2020-01-01</date>
          <desc>
            <address>
              <addrLine>Parc Valrose, 28, avenue Valrose 06108 Nice Cedex 2</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">https://univ-cotedazur.fr</ref>
          </desc>
        </org>
        <org type="institution" xml:id="struct-117182" status="VALID">
          <idno type="ISNI">0000 0001 2160 0329</idno>
          <idno type="ROR">https://ror.org/03srtnf24</idno>
          <orgName>Universidade Federal do Ceará = Federal University of Ceará</orgName>
          <orgName type="acronym">UFC</orgName>
          <desc>
            <address>
              <addrLine>Av. da Universidade, 2853 - Benfica, Fortaleza - CE, CEP 60020-181</addrLine>
              <country key="BR"/>
            </address>
            <ref type="url">http://www.ufc.br/</ref>
          </desc>
        </org>
      </listOrg>
      <listOrg type="projects">
        <org type="anrProject" xml:id="projanr-31126" status="VALID">
          <idno type="anr">ANR-09-BLAN-0159</idno>
          <idno type="program">Blanc</idno>
          <orgName>AGAPE</orgName>
          <desc>Algorithmes de graphes parametres et exacts</desc>
          <date type="start">2009</date>
        </org>
      </listOrg>
    </back>
  </text>
</TEI>