Recent Advances in Graph Vertex Coloring

Graph vertex coloring is one of the most studied NP-hard combinatorial optimization problems. Given the hardness of the problem, various heuristic algorithms have been proposed for practical graph coloring, based on local search, population-based approaches and hybrid methods. The research in graph coloring heuristics is very active and improved results have been obtained recently, notably for coloring large and very large graphs. This chapter surveys and analyzes graph coloring heuristics with a focus on the most recent advances.

[1]  Jin-Kao Hao,et al.  A New Genetic Local Search Algorithm for Graph Coloring , 1998, PPSN.

[2]  Thomas Stützle,et al.  An Analysis of Heuristics for Vertex Colouring , 2010, SEA.

[3]  Jin-Kao Hao,et al.  A Study of Evaluation Functions for the Graph K-Coloring Problem , 2007, Artificial Evolution.

[4]  D. Werra,et al.  Some experiments with simulated annealing for coloring graphs , 1987 .

[5]  Alain Hertz,et al.  A survey of local search methods for graph coloring , 2004, Comput. Oper. Res..

[6]  Gregory J. Chaitin,et al.  Register allocation & spilling via graph coloring , 1982, SIGPLAN '82.

[7]  Laurence A. Wolsey,et al.  Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, 4th International Conference, CPAIOR 2007, Brussels, Belgium, May 23-26, 2007, Proceedings , 2007, CPAIOR.

[8]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[9]  Alain Hertz,et al.  Using tabu search techniques for graph coloring , 1987, Computing.

[10]  Jin-Kao Hao,et al.  An adaptive multistart tabu search approach to solve the maximum clique problem , 2013, J. Comb. Optim..

[11]  Yoshiyasu Takefuji,et al.  Artificial neural networks for four-coloring map problems and K-colorability problems , 1991 .

[12]  Craig A. Morgenstern,et al.  Coloration neighborhood structures for general graph coloring , 1990, SODA '90.

[13]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1994, JACM.

[14]  Nicolas Zufferey,et al.  A graph coloring heuristic using partial solutions and a reactive tabu scheme , 2008, Comput. Oper. Res..

[15]  Jan van Leeuwen,et al.  lambda-Coloring of Graphs , 2000, STACS.

[16]  Christian Blum,et al.  Metaheuristics in combinatorial optimization: Overview and conceptual comparison , 2003, CSUR.

[17]  Thomas Stützle,et al.  An application of Iterated Local Search to Graph Coloring , 2002 .

[18]  Paolo Toth,et al.  An evolutionary approach for bandwidth multicoloring problems , 2008, Eur. J. Oper. Res..

[19]  Carlo Mannino,et al.  Models and solution techniques for frequency assignment problems , 2003, 4OR.

[20]  Jano I. van Hemert,et al.  Graph Coloring with Adaptive Evolutionary Algorithms , 1998, J. Heuristics.

[21]  John Shawe-Taylor,et al.  Ants and Graph Coloring , 2001 .

[22]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[23]  Nicolas Zufferey,et al.  Ant Local Search and its efficient adaptation to graph colouring , 2010, J. Oper. Res. Soc..

[24]  Alain Hertz,et al.  A New Ant Algorithm for Graph Coloring , 2006 .

[25]  Jin-Kao Hao,et al.  Scatter Search for Graph Coloring , 2001, Artificial Evolution.

[26]  Alain Hertz,et al.  On edge orienting methods for graph coloring , 2007, J. Comb. Optim..

[27]  William J. Cook,et al.  Safe Lower Bounds for Graph Coloring , 2011, IPCO.

[28]  Barry A. Tesman List T-Colorings of Graphs , 1993, Discret. Appl. Math..

[29]  Daniel Brélaz,et al.  New methods to color the vertices of a graph , 1979, CACM.

[30]  Christine L. Mumford New Order-Based Crossovers for the Graph Coloring Problem , 2006, PPSN.

[31]  Frank Thomson Leighton,et al.  A Graph Coloring Algorithm for Large Scheduling Problems. , 1979, Journal of research of the National Bureau of Standards.

[32]  Katsumi Inoue,et al.  Distance based hybrid genetic algorithm: an application for the graph coloring problem , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[33]  Jin-Kao Hao,et al.  Hybrid Evolutionary Algorithms for Graph Coloring , 1999, J. Comb. Optim..

[34]  Fred W. Glover,et al.  A Study of Tabu Search for Coloring Random 3-Colorable Graphs Around the Phase Transition , 2010, Int. J. Appl. Metaheuristic Comput..

[35]  Jin-Kao Hao,et al.  A memetic algorithm for graph coloring , 2010, Eur. J. Oper. Res..

[36]  Lawrence. Davis,et al.  Handbook Of Genetic Algorithms , 1990 .

[37]  Hakan Yildiz,et al.  A Large Neighborhood Search Heuristic for Graph Coloring , 2007, CPAIOR.

[38]  Jin-Kao Hao,et al.  An evolutionary approach with diversity guarantee and well-informed grouping recombination for graph coloring , 2010, Comput. Oper. Res..

[39]  Samuel Pierre,et al.  A tabu search heuristic for the routing and wavelength assignment problem in optical networks , 2005, IEEE Communications Letters.

[40]  Pablo Moscato,et al.  Memetic algorithms: a short introduction , 1999 .

[41]  Jin-Kao Hao,et al.  A search space "cartography" for guiding graph coloring heuristics , 2010, Comput. Oper. Res..

[42]  David S. Johnson,et al.  Cliques, Coloring, and Satisfiability , 1996 .

[43]  Alain Hertz,et al.  An adaptive memory algorithm for the k-coloring problem , 2003, Discret. Appl. Math..

[44]  Emanuel Falkenauer,et al.  Genetic Algorithms and Grouping Problems , 1998 .

[45]  Edmund K. Burke,et al.  Parallel Problem Solving from Nature - PPSN IX: 9th International Conference, Reykjavik, Iceland, September 9-13, 2006, Proceedings , 2006, PPSN.

[46]  Tommy R. Jensen,et al.  Graph Coloring Problems: Jensen/Graph , 1994 .

[47]  Nicholas J. Radcliffe,et al.  The algebra of genetic algorithms , 1994, Annals of Mathematics and Artificial Intelligence.

[48]  Emile H. L. Aarts,et al.  Combinatorial Optimization on a Boltzmann Machine , 1989, J. Parallel Distributed Comput..

[49]  Andrea Schaerf,et al.  A Survey of Automated Timetabling , 1999, Artificial Intelligence Review.

[50]  Jin-Kao Hao,et al.  An Extraction and Expansion Approach for Graph Coloring , 2013, Asia Pac. J. Oper. Res..

[51]  Celia A. Glass,et al.  Genetic Algorithm for Graph Coloring: Exploration of Galinier and Hao's Algorithm , 2003, J. Comb. Optim..

[52]  Daniel Br New Methods to Color the Vertices of a Graph , 1979 .

[53]  Bernard Gendron,et al.  On a generalization of the Gallai-Roy-Vitaver theorem to the bandwidth coloring problem , 2008, Oper. Res. Lett..

[54]  Gerhard J. Woeginger,et al.  Integer Programming and Combinatoral Optimization - 15th International Conference, IPCO 2011, New York, NY, USA, June 15-17, 2011. Proceedings , 2011, IPCO.

[55]  Alain Hertz,et al.  Embedding a sequential procedure within an evolutionary algorithm for coloring problems in graphs , 1995, J. Heuristics.

[56]  Riccardo Poli,et al.  New ideas in optimization , 1999 .

[57]  Charles Fleurent,et al.  Genetic and hybrid algorithms for graph coloring , 1996, Ann. Oper. Res..

[58]  Chirag M. Patel,et al.  An ant-based algorithm for coloring graphs , 2008, Discret. Appl. Math..

[59]  Paolo Toth,et al.  A Metaheuristic Approach for the Vertex Coloring Problem , 2008, INFORMS J. Comput..

[60]  Paolo Toth,et al.  An exact approach for the Vertex Coloring Problem , 2011, Discret. Optim..

[61]  Alain Hertz,et al.  Variable space search for graph coloring , 2006, Discret. Appl. Math..

[62]  Alan Crispin,et al.  Quantum annealing of the graph coloring problem , 2011, Discret. Optim..

[63]  Alain Hertz,et al.  A variable neighborhood search for graph coloring , 2003, Eur. J. Oper. Res..

[64]  A. Jagota An adaptive, multiple restarts neural network algorithm for graph coloring , 1996 .

[65]  Craig A. Morgenstern Distributed coloration neighborhood search , 1993, Cliques, Coloring, and Satisfiability.

[66]  Michael A. Trick,et al.  Appendix: Second DIMACS Challenge test problems , 1993, Cliques, Coloring, and Satisfiability.

[67]  Jin-Kao Hao,et al.  Spacing memetic algorithms , 2011, GECCO '11.

[68]  David Chalupa Population-based and learning-based metaheuristic algorithms for the graph coloring problem , 2011, GECCO '11.

[69]  Jin-Kao Hao,et al.  Coloring large graphs based on independent set extraction , 2012, Comput. Oper. Res..

[70]  Tommy R. Jensen,et al.  Graph Coloring Problems , 1994 .

[71]  Cecilia R. Aragon,et al.  Optimization by Simulated Annealing: An Experimental Evaluation; Part II, Graph Coloring and Number Partitioning , 1991, Oper. Res..

[72]  Marc Schoenauer,et al.  Artificial Evolution , 2000, Lecture Notes in Computer Science.

[73]  David Zuckerman Linear Degree Extractors and the Inapproximability of Max Clique and Chromatic Number , 2007, Theory Comput..

[74]  W. K. Hale Frequency assignment: Theory and applications , 1980, Proceedings of the IEEE.

[75]  Thomas Bäck,et al.  Parallel Problem Solving from Nature — PPSN V , 1998, Lecture Notes in Computer Science.

[76]  Dan Gusfield,et al.  Partition-distance: A problem and class of perfect graphs arising in clustering , 2002, Inf. Process. Lett..

[77]  Alan Crispin,et al.  Graph Coloring with a Distributed Hybrid Quantum Annealing Algorithm , 2011, KES-AMSTA.

[78]  Jin-Kao Hao,et al.  An efficient algorithm for computing the distance between close partitions , 2011, Discret. Appl. Math..