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Heuristic Algorithms for Graph Coloring Problems Wen Sun

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Heuristic Algorithms for Graph Coloring Problems Wen Sun Heuristic Algorithms for Graph Coloring Problems Wen Sun To cite this version: Wen Sun. Heuristic Algorithms for Graph Coloring Problems. Data Structures and Algorithms [cs.DS]. Université d’Angers, 2018. English. NNT : 2018ANGE0027. tel-02136810 HAL Id: tel-02136810 https://tel.archives-ouvertes.fr/tel-02136810 Submitted on 22 May 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THESE DE DOCTORAT DE L'UNIVERSITE D'ANGERS COMUE UNIVERSITE BRETAGNE LOIRE ECOLE DOCTORALE N° 601 Mathématiques et Sciences et Technologies de l'Information et de la Communication Spécialité : Informatique, section CNU 27 Par Wen SUN Heuristic Algorithms for Graph Coloring Problems Thèse présentée et soutenue à Angers, le 29/11/2018 Unité de recherche : Laboratoire d'Étude et de Recherche en Informatique d'Angers (LERIA) Rapporteurs avant soutenance : Djamal HABET MC HDR, Université d'Aix-Marseille Olivier BAILLEUX MC HDR, Université de Bourgogne Composition du Jury : Examinateurs : Béatrice DUVAL Professeur, Université d'Angers Djamal HABET MC HDR, Université d'Aix-Marseille Marc SCHOENAUER Directeur de Recherche, INRIA Olivier BAILLEUX MC HDR, Université de Bourgogne Dir. de thèse : Jin-Kao HAO Professeur, Université d'Angers Co-dir. de thèse : Alexandre CAMINADA Professeur, Université Côte d’Azur Contents General Introduction7 1 Introduction 11 1.1 Graph coloring problem.................................. 12 1.1.1 Problem Introduction............................... 12 1.1.2 Exact algorithm................................... 12 1.1.3 Heuristic algorithm................................. 13 1.1.4 Summary...................................... 14 1.2 Equitable coloring problem................................ 15 1.2.1 Theoretical Studies................................. 15 1.2.2 Exact approaches.................................. 15 1.2.3 Heuristic approaches................................ 16 1.2.4 Summary...................................... 16 1.3 Weighted vertex coloring problem............................ 17 1.3.1 Exact approaches.................................. 18 1.3.2 Heuristic approaches................................ 18 1.3.3 Summary...................................... 18 1.4 k-vertex critical subgraphs problem............................ 19 1.4.1 Problem Introduction............................... 19 1.4.2 Exact approaches.................................. 20 1.4.3 Heuristic approaches................................ 20 1.4.4 Summary...................................... 20 2 A reduction-based memetic algorithm for GCP 21 2.1 Introduction......................................... 23 2.2 Memetic algorithm for the GCP.............................. 23 2.2.1 General approach.................................. 23 2.2.2 Population initialization.............................. 24 2.2.3 The adaptive multi-parent crossover procedure................ 27 2.2.4 Backbone-based group matching......................... 27 2.2.5 Weight tabu search improvement......................... 30 2.2.6 Uncoarsening phase................................ 31 2.2.7 Pool updating strategy............................... 32 2.3 Experimental results and comparisons.......................... 32 2.3.1 Benchmark instances................................ 33 2.3.2 Experiment settings................................ 33 2.3.3 Comparison with state-of-the-art algorithms.................. 35 2.3.4 Comparative results on easy instances...................... 35 2.3.5 Comparative results on difficult instances.................... 36 2.4 Analysis........................................... 37 3 4 CONTENTS 2.4.1 Effectiveness of the number of parents for RMA................ 37 2.4.2 Effectiveness of the different matching strategies for RMA.......... 37 2.4.3 Effectiveness of the perturbation operation................... 38 2.5 Conclusions......................................... 38 3 On feasible and infeasible search for ECP 41 3.1 Introduction......................................... 42 3.2 Basic definitions....................................... 42 3.3 Feasible and infeasible search algorithm for ECP.................... 43 3.3.1 General approach.................................. 43 3.3.2 Searching equity-feasible solutions........................ 44 3.3.3 Searching equity-infeasible solutions....................... 45 3.3.4 Perturbation of infeasible search......................... 47 3.4 Experimental results and comparisons.......................... 48 3.4.1 Experiment settings................................ 48 3.4.2 Comparison with state-of-the-art algorithms.................. 48 3.5 Analysis........................................... 49 3.5.1 Analysis of the penalty coefficient j ....................... 49 3.5.2 Impact of the perturbation operation....................... 49 3.6 Conclusions......................................... 52 4 Adaptive feasible and infeasible search for WVCP 53 4.1 Introduction......................................... 54 4.2 Adaptive feasible and infeasible search for the WVCP................. 54 4.2.1 General approach.................................. 54 4.2.2 Initial solution.................................... 55 4.2.3 Search space and penalty-based evaluation function.............. 55 4.2.4 Searching feasible and infeasible solutions with tabu search......... 57 4.2.5 Perturbation strategy................................ 59 4.2.6 Connections with existing studies........................ 60 4.3 Experimental results and comparisons.......................... 60 4.3.1 Benchmark instances................................ 60 4.3.2 Experimental settings............................... 61 4.3.3 Computational results and comparisons with state-of-the-art algorithms.. 62 4.4 Analysis........................................... 65 4.4.1 Impact of the penalty coefficient......................... 66 4.4.2 Benefit of searching both feasible and infeasible solutions.......... 67 4.4.3 Impact of the perturbation operation....................... 68 4.5 Conclusions......................................... 69 5 Iterated backtrack removal search for finding k-VCS 71 5.1 Introduction......................................... 72 5.2 Notations........................................... 72 5.3 The Iterated Backtrack-based Removal Algorithm................... 73 5.3.1 General structure of the IBR algorithm...................... 73 5.3.2 The removal algorithm............................... 76 5.3.3 Heuristic coloring algorithm........................... 76 5.3.4 Backtrack-based removal approach....................... 77 5.3.5 Perturbation operator of backtrack-based removal algorithm......... 78 5.3.6 Update procedure................................. 79 5.4 Experimental results and analysis............................. 81 CONTENTS 5 5.4.1 Experiment settings................................ 81 5.4.2 Instances and experimental settings....................... 83 5.4.3 Comparison with state of the art algorithm................... 83 5.5 Analysis and discussions.................................. 85 5.5.1 Effectiveness of different backtrack strategies for k-VCS detection...... 86 5.5.2 Effectiveness of perturbation for k-VCS detection............... 86 5.5.3 Effectiveness of backtrack for k-VCS detection................. 87 5.6 Conclusion.......................................... 87 General Conclusion 89 Appendix 93 List of Publications 95 List of Figures 97 List of Tables 100 References 109 General Introduction Context Graph vertex coloring problems are known to be very general and useful models to formulate numerous practical problems [Lewis, 2015]. Given an undirected graph G = (V, E) with the vertex set V = f1, 2, . , ng, the edge set E 2 V × V, graph vertex coloring problems typically involve assigning a color to each vertex of V such that two vertices linked by an edge must receive different colors while optimizing a given optimization objective. This thesis focuses on four generalized graph coloring problems, namely the graph color- ing problem (GCP), the equitable coloring problem (ECP), the weighted vertex coloring problem (WVCP) and the k-vertex critical subgraphs (k-VCS). The first one is the most basic graph coloring problem. The ECP requires that the sizes of two arbitrary color classes differ in at most one unit while the WVCP aims to minimize the sum of the largest weights of the vertices of each color class. The k-VCS is to find a subgraph H of G with minimum vertices such that removing any vertex from the subgraph H decreases its chromatic number. These four problems are practically relevant since they have wide applications in real world such as garbage collection, load balancing, timetabling, scheduling, operating system, manufac- turing, etc. From the perspective of computational theory, graph coloring problems belong to the class of the NP-hard problems, meaning that optimal solutions cannot be found in polynomial time in the general case. For solving large and challenging problem instances, heuristic and metaheuristic approaches are commonly used with the purpose of finding sub-optimal
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