Abstract
Graph coloring problem is a famous NP-complete problem and there exist several methods which have been projected to resolve this issue. For a graph colouring algorithm to be efficient, it ought to paint the input graph by minimum colours and must also find the solution in the minimum possible time. Here, we have proposed a different method to solve the graph coloring problem using maximal independent set. In our method, we used the concept of maximal independent sets using trees. In the first part, it converts a massive graph into a sequence of step by step smaller graphs by eliminating big independent sets from the initial graph. The second part starts by assigning a proper colour to each maximal independent set within the sequence. The proposed method is estimated on the DIMACS standards and presented reasonable outcomes concerning to other latest methods.
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References
Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. San Francisco: W.H. Freeman and Company.
de Werra, D., Eisenbeis, C., Lelait, S., & Marmol, B. (1999). On a graph-theoretical model for cyclic register allocation. Discrete Applied Mathematics,93(2–3), 191–203.
Burke, E. K., McCollum, B., Meisels, A., Petrovic, S., & Qu, R. (2007). A graph-based hyper heuristic for timetabling problems. European Journal of Operational Research,176, 177–192.
Smith, D. H., Hurley, S., & Thiel, S. U. (1998). Improving heuristics for the frequency assignment problem. European Journal of Operational Research,107(1), 76–86.
Zufferey, N., Amstutz, P., & Giaccari, P. (2008). Graph colouring approaches for a satellite range scheduling problem. Journal of Scheduling,11(4), 263–277.
Karp, R. M. (1972). Reducibility among combinatorial problems (pp. 85–103). New York: Plenum Press.
Brelez, D. (1979). New methods to color the vertices of a graph. Communications of the ACM,22(4), 251–256.
Leighton, F. T. (1979). A graph coloring algorithm for large scheduling problems. Journal of Research of the National Bureau of Standards,84(6), 489–506.
Blochligerand, I., & Zufferey, N. (2008). A graph coloring heuristic using partial solutions and a reactive tabu scheme. Computers & Operations Research,35(3), 960–975.
Dorne, R., & Hao, J. K. (1998). Tabu search for graph coloring, T-colorings and set T-colorings. In S. Voß, S. Martello, I. H. Osman, & C. Roucairol (Eds.), Meta-Heuristics: Advances and Trends in Local Search Paradigms for Optimization (pp. 77–92). New York: Springer.
Hertz, A., & de Werra, D. (1987). Using tabu search techniques for graph coloring. Computing,39, 345–351.
Porumbel, D. C., Hao, J. K., & Kuntz, P. (2010). An evolutionary approach with diversity guarantee and well-informed grouping recombination for graph coloring. Computers & Operations Research,37(10), 1822–1832.
Chams, M., Hertz, A., & de Werra, D. (1987). Some experiments with simulated annealing for coloring graphs. European Journal of Operational Research,32, 260–266.
Johnson, D. S., Aragon, C. R., McGeoch, L. A., & Schevon, C. (1991). Optimization by simulated annealing: An experimental evaluation; Part II, graph coloring and number partitioning. Operations Research,39(3), 378–406.
Dorne, R., & Hao, J. K. (1998). A new genetic local search algorithm for graph coloring. Lecture Notes in Computer Science,1498, 745–754.
Galinier, P., Hertz, A., & Zufferey, N. (2008). An adaptive memory algorithm for the K-colouring problem. Discrete Applied Mathematics,156(2), 267–279.
Lu, Z., & Hao, J. K. (2010). A memetic algorithm for graph coloring. European Journal of Operational Research,200(1), 235–244.
Malaguti, E., Monaci, M., & Toth, P. (2008). A metaheuristic approach for the vertex coloring problem. INFORMS Journal on Computing,20(2), 302–316.
Galinier, P., & Hertz, A. (2006). A survey of local search methods for graph coloring. Computers & Operations Research,33(9), 2547–2562.
Fleurent, C., & Ferland, J. A. (1996). Genetic and hybrid algorithms for graph coloring. Annals of Operations Research,63, 437–461.
Wu, Q., & Hao, J. K. (2012). Coloring large graphs based on independent set extraction. Computers & Operations Research,39(2), 283–290.
Hao, J. K., & Wu, Q. (2012). Improving the extraction and expansion method for large graph coloring. Discrete Applied Mathematics,160(16–17), 2397–2407.
Campelo, M. B., Campos, V. A., & Correa, R. C. (2008). On the asymmetric representatives formulation for the vertex coloring problem. Discrete Applied Mathematics,156(7), 1097–1111.
Hansen, P., Labbe, M., & Schindl, D. (2009). Set covering and packing formulation of graph coloring: Algorithms and first polyhedral results. Discrete Optimization,6(2), 135–147.
Malaguti, E., Monaci, M., & Toth, P. (2011). An exact approach for the vertex coloring problem. Discrete Optimization,8(2), 174–190.
Gualandi, S., & Malucelli, F. (2012). Exact solution of graph coloring problems via constraint programming and column generation. INFORMS Journal on Computing,24(1), 81–100.
Seidman, S. (1983). Network structure and minimum degree. Social Networks,5(3), 269–287.
Rossi, R. A., & Ahmed, N. K. (2014). Coloring large complex networks. Social Network Analysis: Mining,4(1), 228–239.
Verma, A., Buchanan, A., & Butenko, S. (2015). Solving the maximum clique and vertex coloring problems on very large sparse networks. INFORMS Journal on Computing,27(1), 164–177.
Wu, Q., & Hao, J. K. (2013). An extraction and expansion approach for graph coloring. Asia-Pacific Journal of Operational Research,30(5), 132–147.
Peng, Y., Choi, B., He, B., Zhou, S., Xu, R., & Yu, X. (2016). Vcolor: A practical vertex-cut based approach for coloring large graphs. In 32nd IEEE international conference on data engineering, Helsinki, Finland, May 2016 (pp. 97–108).
Lin, J., Cai, S., Luo, C., & Su, K. (2017). A reduction based method for coloring very large graphs. In Proceedings of the twenty-sixth international joint conference on artificial intelligence (IJCAI-17) (pp. 517–523).
DIMACS Implementation Challenges. Retrieved from 10 Sept 2016 http://dimacs.rutgers.edu/Challenges/.
Graph coloring instances. http://mat.gsia.cmu.edu/COLOR/instances.html.
Mendez-Diaz, I., & Zabala, P. (2006). A branch and cut algorithm for graph coloring. Discrete Applied Mathematics,154(5), 826–847.
Malaguti, E., Monaci, M., & Toth, A. (2008). A metaheuristic approach for the vertex coloring problem. INFORMS Journal on Computing,20(2), 302–316.
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Sharma, P.C., Chaudhari, N.S. A Tree Based Novel Approach for Graph Coloring Problem Using Maximal Independent Set. Wireless Pers Commun 110, 1143–1155 (2020). https://doi.org/10.1007/s11277-019-06778-0
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DOI: https://doi.org/10.1007/s11277-019-06778-0