Contemporary Methods for Graph Coloring as an Example of Discrete Optimization
Abstract
This paper provides an insight into graph coloring
application of the contemporary heuristic methods. It discusses a
variety of algorithmic solutions for The Graph Coloring Problem
(GCP) and makes recommendations for implementation. The
GCP is the NP-hard problem, which aims at finding the minimum
number of colors for vertices in such a way, that none of two
adjacent vertices are marked with the same color.With the advent
of multicore processing technology, the metaheuristic approach
to solving GCP reemerged as means of discrete optimization. To
explain the phenomenon of these methods, the author makes a
thorough survey of AI-based algorithms for GCP, while pointing
out the main differences between all these techniques.
References
Zuckerman D.: Linear degree extractors and the inapproximability of Max Clique and Chromatic Number. Proc. STOC06, Seattle, 2006, p. 681-690.
Jensen T.R., Tof t B.: Graph Coloring Problems. New York, Wiley, 1995.
Appel K., Haken W.: Every planar graph is four colorable. Pt. I:
Discharging. Illinois J. Math., vol. 21, 1977, p. 429-490.
Holyer, Ian (1981), ”The NP-completeness of edge-coloring”, SIAM
Journal on Computing 10 (4): 718720
Chartrand G., Zhang P., Chromatic Graph Theory, Chapman and Hall,
Vizing V.G.: On an estimate of the chromatic class of a p-graph (po
rosyjsku). Diskret. Analiz, vol.. 3, 1964, p. 25-30.
Tait P.G.: On the coloring of maps. Proc. Royal Soc. Edinburgh, Sec. A,
vol. 10, 1880, p. 501-503.
Mycielski J., On the coloring of graphs, Coll. Math. 1955, 3, s. 161-162.
Halldorsson M.M.: A still better performance guarantee for approximate graph coloring, Inf. Process. Lett., vol. 45, 1993, p. 19-23.
Tait P.G.: On the coloring of maps. Proc. Royal Soc. Edinburgh, Sec.
A, vol. 10, 1880, p. 501-503.
Brlaz D.: New methods to color the vertices of a graph. Comm. ACM,
vol. 22,1979, p. 251-256.
Arbaugh W., Banerjee S., Mishra A., Weighted Coloring Based Channel Assignment for WLAN’s”, ACM SIGMOBILE Mobile Computing and
Communications Review, July 2005, col. 9, no. 3, pp. 19-31.
Kubale M., Optymalizacja dyskretna. Modele i metody kolorwania
grafow, (in polish), WNT, 2002, Warszawa.
Furmanowicz P., Tanas K., A survey of graph coloring - its types, methods and applications, Foundation of Computing and Decision Sciences, vol. 37, no. 3, 2012, pp. 223-238.
Yegnanarayanan V., Graph colourings and partitions, Theoretical Computer Science, vol. 263, 2001, pp. 59-79.
K. Schnabel, Representation of graphs by integers, in: B. Rainer, H.
Ruddf (Eds.), Topics in Combinatorics and Graph Theory, Physica-Verlag,
Heidelberg.
R.C. Entringer, D.E. Jackson, D.A. Smyder, Distance in graphs,
Czechoslovak Math. J. 26 (101) (1976) 283296.
I. Tomescu, A.M. Robert, On distances in chromatic graphs, Quart. J.
Math. Oxford 40 (2) (1989) 475480
Dorne R., Hao J.K.: Tabu search for graph coloring, T-coloring and set Tcoloring, in: Metaheuristics: Theory and Applications. Kluwer Academic Publishers 1997.
Szyfelbein D.: Genetic algorithms for graph coloring. Proc. 4th Conference Neural Networks and Their Applications. 1999, s. 605-610.
Zhu X., Star Chromatic number and products of graphs, Journal of Graph Theory, 16:557–569, 1992.
Xuding Zhu, Circular chromatic number and graph minors, TAIWANESE JOURNAL OF MATHEMATICS Vol. 4, No. 4, pp. 643-660,
December 2000
Zhu X., On the chromatic number of the products of hypergraphs. Ars Combinatoria, 34:25–31, 1992.
Hermann F., Hertz A., Finding the chromatic number by means of
critical graphs, ACM Journal of Experimental Algorithmics, vol. 7, s.
-9, 2002.
Mendez-Diaz I., Zabala P., A branch-and-cut algorithm for graph coloring, Discrete Applied Mathematics, vol. 154, s. 826-847, 2006.
de Werra D., Kobler D., Graph coloring problems, Paradigms of Combinatorial Optimization, vol. 2, Paschos V.Th.(ed.), ISTE-Wiley, s. 265-310, 2010.
Welsh D.J., Powell M.B.: An upper bound for the chromatic number of a graph and its application to timetabling problem. Comp. J., 1967, 10,
s. 85-86.
Matula D.W., Marble G., Isaacson J.D., Graph coloring algorithms, in:
Graph Theory of and Computing, Read R.C(ed.), Academic Press, New
York, 1972, s. 109-122.
Brelaz D., New methods to color the vertices of a graph, Communications of the ACM, 1979, vol. 22, s. 251-256.
Johnson D.S., Worst case behavior of graph coloring algorithms, Proc. 5th Conf. on Combinatorics, Graph Theory and Computing, Utilitas
Mathematica Publishing, Winnipeg, 1974, s. 513-527.
Lewandowski G., Condon A., Experiments with Parallel Graph Coloring Heuristics and Applications of Graph Coloring, Technical Report, 1994
Leighton F.T.: A graph coloring algorithm for large scheduling problems. J. RBNS. 1979, 84, s. 489-506.
Kirkpatrick S., Gelatt Jr. C. D. and Vecchi M.P., Optimization by
Simulated Annealing, Science, vol. 220, 1983, pp. 671-680.
Glover F. (1990). ”Tabu Search: A Tutorial”. Interfaces.
Holland, J. H., Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI, 1975.
Hopfield, J.J., and D.W.Tank, Neural Computation of Decisions in
Optimization Problems, Biological Cybernetics, vol. 52, pp. 141-152.
Press W. H. , Teukolsky S. A. , Vetterling W. T., Flannery B. P.,
Numerical Recipes in C, Numerical Recipes 3rd Edition: The Art of
Scientific Computing, Cambridge University Press, 2007
Costa D., Hertz A.: Ants can color graphs. J. Oper. Soc. 1996, 47, s.
-11.
Glover F. (1990). ”Tabu Search - Part 2”. ORSA Journal on Computing 2 (1): 432.
Goldberg D.E: Algorytmy genetyczne i ich zastosowania. Warszawa,
WNT 1995.
DiBlas A., Jagota A., Hughey R., Energy function-based approaches to graph coloring, Transactions on Neural Networks, vol 13, no. 1, 2002,
pp. 81-91.
Gassen D.W., Carothers J.D., Graph color minimization using neural
networks, in: Proceedings of the IEEE International Conference on Neural
Networks, 1993, pp. 1541-1544.
Berger M.O., k-coloring vertices using a neural network with convergence to valid solutions, in: Proceedings of the IEEE International
Conference on Neural Networks, vol. 7, 1994, pp. 4515-4517.
DiBlas A., Jagota A., Hughey R., Optimization neural networks on
SIMD parallel computers, Parallel Computing, vol. 31, no 1, 2005, 97-
Jagota A., An adaptive, multiple restarts neural network algorithm for graph coloring, European Journal of Operational Research, vol. 91, 1996, pp. 257-270.
Jagota A, Scheduling problems in radio networks using Hopfield networks”, In: Alspector J., Godman R., and Brown T., eds., Proceedings
of the International Workshop on Applications of Neural Networks to
Telecommunications, Lawrence Erlbaun Associates, Hillsdale, NY, 1993,
pp. 67-76.
Fiat A., Woeginger G.J. (eds.): Online Algorithms The State of the Art, LNCS 1442, Berlin, Springer, 1998.
Hall, M. M., Frugal Methods for the Independent Set and Graph
Coloring Problems, PhD. thesis, The State University of New Jersey,
New Brunswick, New Jersey, October 1991.
Gyarfas A., Lehel J.: First-Fit and on-line chromatic number of families of graphs. Ars Combinatoria. 1990, 29C, s. 168-176.
M. Duque-Anton, D. Kunz, and B. Ruber. Channel assignment for cellular radio using simulated annealing, IEEE Trans. Vehicular Technology, vol. 42, pp. 1421, 1993.
Wang and N. Ansari. Optimal broadcast scheduling in packet radio
networks using mean field annealing. IEEE Journal on Selected Areas
in Communications, 15(2):250260, 1997.
Dongming Zhaoa, Liang Luo , Kai Zhang, An improved ant colony optimization for the communication network routing problem, Mathematical and Computer Modelling 52 (2010) 19761981
C.H. Chen, C.J. Ting, An improved ant system algorithm for routing
problem, Journal of the Chinese Institute of Industrial Engineers 23 (2)
(2006) 115126.
Xu J., Chiu S.Y., Glover F., Tabu search for dynamic routing communications network design, Telecommunication Systems, vol. 8, Issue 1, pp. 55-77, 1997.
Xu J., Chiu S.Y., Glover F., Probabilistic tabu search for telecommunications network design, Combinatorial Optimization: Theory and Practice, vol. 1, no. 1, pp. 69-94, 1996
Edwards T, Tansley D.S.W, Frank R.J., Davey N. (1997) Traffic Trends
Analysis Using Neural Networks. Proceedings. InternationalWorkshop on
Applications of Neural Networks to Telecommunications 3 (IWANNT’3),
-164, 1997
Yeo, H. Lee, and S. Kim. An efficient broadcast scheduling algorithm
for TDMA ad hoc networks. Computers and Operations Research,
:17931806, 2002.
Javedankherad, Mostafa and Zeinalpour-Yazdi, Zolfa. (2017). Content Placement in Cache Networks Using Graph-Coloring.
Ashok Singh Sairam, Sangita Roy, Rishikesh Sahay, Coloring networks for attacker identification and response, Security Comm. Networks, 2015;8: 751-768.
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