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.
Downloads
Published
Issue
Section
License
Copyright (c) 2019 International Journal of Electronics and Telecommunications
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
1. License
The non-commercial use of the article will be governed by the Creative Commons Attribution license as currently displayed on https://creativecommons.org/licenses/by/4.0/.
2. Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. The undersigned also warrants that the manuscript (or its essential substance) has not been published other than as an abstract or doctorate thesis and has not been submitted for consideration elsewhere, for print, electronic or digital publication.
3. User Rights
Under the Creative Commons Attribution license, the author(s) and users are free to share (copy, distribute and transmit the contribution) under the following conditions: 1. they must attribute the contribution in the manner specified by the author or licensor, 2. they may alter, transform, or build upon this work, 3. they may use this contribution for commercial purposes.
4. Rights of Authors
Authors retain the following rights:
- copyright, and other proprietary rights relating to the article, such as patent rights,
- the right to use the substance of the article in own future works, including lectures and books,
- the right to reproduce the article for own purposes, provided the copies are not offered for sale,
- the right to self-archive the article
- the right to supervision over the integrity of the content of the work and its fair use.
5. Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
6. Termination
This agreement can be terminated by the author or the Journal Owner upon two months’ notice where the other party has materially breached this agreement and failed to remedy such breach within a month of being given the terminating party’s notice requesting such breach to be remedied. No breach or violation of this agreement will cause this agreement or any license granted in it to terminate automatically or affect the definition of the Journal Owner. The author and the Journal Owner may agree to terminate this agreement at any time. This agreement or any license granted in it cannot be terminated otherwise than in accordance with this section 6. This License shall remain in effect throughout the term of copyright in the Work and may not be revoked without the express written consent of both parties.
7. Royalties
This agreement entitles the author to no royalties or other fees. To such extent as legally permissible, the author waives his or her right to collect royalties relative to the article in respect of any use of the article by the Journal Owner or its sublicensee.
8. Miscellaneous
The Journal Owner will publish the article (or have it published) in the Journal if the article’s editorial process is successfully completed and the Journal Owner or its sublicensee has become obligated to have the article published. Where such obligation depends on the payment of a fee, it shall not be deemed to exist until such time as that fee is paid. The Journal Owner may conform the article to a style of punctuation, spelling, capitalization and usage that it deems appropriate. The Journal Owner will be allowed to sublicense the rights that are licensed to it under this agreement. This agreement will be governed by the laws of Poland.
By signing this License, Author(s) warrant(s) that they have the full power to enter into this agreement. This License shall remain in effect throughout the term of copyright in the Work and may not be revoked without the express written consent of both parties.
