<p>`<br>UNIVERSITA DEGLI STUDI DI BOLOGNA </p><p>Dottorato di Ricerca in <br>Automatica e Ricerca Operativa </p><p>XIX Ciclo </p><p>The Vertex Coloring Problem and its Generalizations </p><p>Enrico Malaguti </p><p>A.A. 2003–2006 </p><p>Contents </p><p></p><ul style="display: flex;"><li style="flex:1">I</li><li style="flex:1">Vertex Coloring Problems </li><li style="flex:1">9</li></ul><p></p><p>CONTENTS </p><p></p><ul style="display: flex;"><li style="flex:1">II Fair&nbsp;Routing </li><li style="flex:1">93 </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">CONTENTS </li><li style="flex:1">CONTENTS </li></ul><p></p><p>Acknowledgments </p><p>ACKNOWLEDGMENTS </p><p>Keywords </p><p>Keyworks </p><p>List of Figures </p><p>LIST OF FIGURES </p><p>List of Tables </p><p>LIST OF TABLES </p><p>Chapter 1 </p><p>Introduction </p><p>1.1 The&nbsp;Vertex Coloring Problem and its Generalizations </p><p>1</p><p><sup style="top: -0.3176em;">1</sup>Four are enough for any map, see Appel, Haken and Koch [10], the Four Color Conjecture was proposed by Francis Guthrie in 1852 </p><p>Introduction </p><p>∈</p><p>The Vertex Coloring Problem and its Generalizations </p><p>=1 </p><p>∀ ∈ </p><p>=1 </p><p></p><ul style="display: flex;"><li style="flex:1">≤</li><li style="flex:1">∀</li><li style="flex:1">∈</li></ul><p></p><ul style="display: flex;"><li style="flex:1">∈ { </li><li style="flex:1">}</li></ul><p>}<br>∀ ∈ <br>∈ { </p><p></p><ul style="display: flex;"><li style="flex:1">≤</li><li style="flex:1">∀</li></ul><p></p><p>=1 </p><p>Introduction </p><p>∈</p><ul style="display: flex;"><li style="flex:1">|</li><li style="flex:1">−</li><li style="flex:1">| ≥ </li></ul><p></p><ul style="display: flex;"><li style="flex:1">≥</li><li style="flex:1">∈</li></ul><p>∈</p><p>∈</p><p>≤<br>≤<br>∈ { ∈ { </p><ul style="display: flex;"><li style="flex:1">∈</li><li style="flex:1">∈</li><li style="flex:1">∈ {&nbsp;− </li><li style="flex:1">− } </li></ul><p>∈∈<br>∈<br>∈</p><ul style="display: flex;"><li style="flex:1">∈</li><li style="flex:1">}</li></ul><p>}</p><p>∈<br>∈</p><p>=1 </p><p>The Vertex Coloring Problem and its Generalizations </p><p></p><ul style="display: flex;"><li style="flex:1">≥</li><li style="flex:1">∈</li></ul><p>∈</p><p>=1 </p><p></p><ul style="display: flex;"><li style="flex:1">≤</li><li style="flex:1">∈</li></ul><p></p><ul style="display: flex;"><li style="flex:1">∈ { </li><li style="flex:1">}</li><li style="flex:1">∈</li></ul><p></p><p>Introduction </p><p>1.2 Fair&nbsp;Routing </p><p>••</p><p>Fair Routing </p><p></p><ul style="display: flex;"><li style="flex:1">∈</li><li style="flex:1">∈</li></ul><p></p><p>Introduction </p><p>Part I </p><p>Vertex Coloring Problems </p><p>Chapter 2 </p><p>A Metaheuristic Approach for the Vertex Coloring Problem </p><p>1</p><p>2.1 Introduction </p><p><sup style="top: -0.3176em;">1</sup>The results of this chapter appear in [84]. </p><p>A Metaheuristic Approach for the Vertex Coloring Problem </p><p>2.1.1 The&nbsp;Heuristic Algorithm MMT </p><p>Introduction </p><p>S<sup style="top: -0.3301em;">0 </sup><br>S<sup style="top: -0.3301em;">0 </sup></p><p>S<sup style="top: -0.3301em;">0 </sup></p><p>∅</p><p>S<sup style="top: -0.3301em;">0 </sup><br>S<sup style="top: -0.3301em;">0 </sup></p><p>2.1.2 Initialization&nbsp;Step </p><p>A Metaheuristic Approach for the Vertex Coloring Problem </p><p>∅<br>| | </p><p>:</p><p></p><ul style="display: flex;"><li style="flex:1">∈</li><li style="flex:1">\</li></ul><p></p><p>:</p><p></p><ul style="display: flex;"><li style="flex:1">∈</li><li style="flex:1">\</li></ul><p></p><p>and </p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li></ul><p></p><p>∪ {&nbsp;} </p><p></p><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">1</li></ul><p></p><p>0<br>0</p><p>••<br>∈</p><p>0</p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p>∈</p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p>∈</p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p>∈<br>••</p><p>2<br>2<br>2</p><p>PHASE 1: Evolutionary Algorithm </p><p>2<br>2</p><p>•</p><p>2<br>2</p><p>•</p><p>0</p><p>2<br>2<br>2</p><p>2.2 PHASE&nbsp;1: Evolutionary Algorithm </p><p>2.2.1 Tabu&nbsp;Search Algorithm </p><p>•••</p><p>0</p><p>∈</p><p>0<br>0</p><p>A Metaheuristic Approach for the Vertex Coloring Problem </p><p></p><ul style="display: flex;"><li style="flex:1">{</li><li style="flex:1">}</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">+1 </li></ul><p></p><p>0</p><p>∈<br>|<br>∈</p><p>+1 </p><p>0</p><p>|</p><p>+1 </p><p>∈</p><p>+1 </p><p>0</p><p>∈</p><p>0</p><p></p><ul style="display: flex;"><li style="flex:1">|</li><li style="flex:1">|</li></ul><p></p><p>+1 </p><p>•••••</p><p>∗</p>