Skip to main content
SpringerLink
Log in

Simple decentralized graph coloring

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

Graph coloring is a classical NP-hard combinatorial optimization problem with many practical applications. A broad range of heuristic methods exist for tackling the graph coloring problem: from fast greedy algorithms to more time-consuming metaheuristics. Although the latter produce better results in terms of minimizing the number of colors, the former are widely employed due to their simplicity. These heuristic methods are centralized since they operate by using complete knowledge of the graph. However, in real-world environmets where each component only interacts with a limited number of other components, the only option is to apply decentralized methods. This paper explores a novel and simple algorithm for decentralized graph coloring that uses a fixed number of colors and iteratively reduces the edge conflicts in the graph. We experimentally demonstrate that, for most of the tested instances, the new algorithm outperforms a recent and very competitive algorithm for decentralized graph coloring in terms of coloring quality. In our experiments, the fixed number of colors used by the new algorithm is controlled in a centralized manner.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Al-Omari, H., Sabri, K.E.: New graph coloring algorithms. Am. J. Math. Stat. 2(4), 739–741 (2006)

    MathSciNet  MATH  Google Scholar 

  2. Avanthay, C., Hertz, A., Zufferey, N.: A variable neighborhood search for graph coloring. Eur. J. Oper. Res. 151, 379–388 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barnier, N., Brisset, P.: Graph coloring for air traffic flow management. Ann. Oper. Res. 130(1), 163–178 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blöchliger, I., Zufferey, N.: A graph coloring heuristic using partial solutions and a reactive tabu scheme. Comp. Oper. Res. 35(3), 960–975 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blum, C., Calvo, B., Blesa, M.J.: FrogCOL and FrogMIS: new decentralized algorithms for finding large independent sets in graphs. Swarm Intell. 9, 205–227 (2015)

    Article  Google Scholar 

  6. Brélaz, D.: New methods to color the vertices of a graph. Commun. ACM 22, 251–256 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brown, J.R.: Chromatic scheduling and the chromatic number problem. Manag. Sci. 19, 456–463 (1972)

    Article  MATH  Google Scholar 

  8. Burke, E.K., Elliman, D.G., Weare, R.F.: A university timetabling system based on graph colouring and constraint manipulation. J. Res. Comput. Educ. 27(1), 1–18 (1994)

    Article  Google Scholar 

  9. Caprara, A., Kroon, L., Monaci, M., Peeters, M., Tooth, P.: Passenger railway optimization. In: Barnhart, C., Laporte, G. (eds.) Handbook in Operations Research and Management Science, pp. 129–187. Elsevier, Amsterdam (2007)

    Google Scholar 

  10. Chaitin, G.J.: Register allocation & spilling via graph coloring. In: Proceedings of the 1982 SIGPLAN Symposium on Compiler Construction, pp. 98–105. ACM (1982)

  11. Chams, M., Hertz, A., de Werra, D.: Some experiments with simulated annealing for coloring graphs. Eur. J. Oper. Res. 32(2), 260–266 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Coleman, T.F., Moré, J.J.: Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal. 20(1), 187–209 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  13. Culberson, J.C., Luo, F.: Exploring the k-colorable landscape with iterated greedy. In: DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pp. 245–284. American Mathematical Society (1995)

  14. Davis, L.: Order-based genetic algorithms and the graph coloring problem. In: Davis, L. (ed.) Handbook of Genetic Algorithms, pp. 72–90. Van Nostrand Reinhold, New York (1991)

    Google Scholar 

  15. Fitzpatrick, S., Meertens, L.: Soft, real-time, distributed graph coloring using decentralized, synchronous, stochastic, iterative-repair, anytime algorithms. Technical Report KES.U.01.05, Kestrel Institute, Palo Alto, CA (2001)

  16. Fleurent, C., Ferland, J.: Genetic and hybrid algorithms for graph coloring. Ann. Oper. Res. 63(3), 437–461 (1996)

    Article  MATH  Google Scholar 

  17. Galinier, P., Hamiez, J.P., Hao, J.K., Porumbel, D.: Recent advances in graph vertex coloring. In: Zelinka, I., Snášel, V., Abraham, A. (eds.) Handbook of Optimization: From Classical to Modern Approach, pp. 505–528. Springer, Berlin (2013)

    Chapter  Google Scholar 

  18. Galinier, P., Hao, J.K.: Hybrid evolutionary algorithms for graph coloring. J. Comb. Optim. 3, 379–397 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1974)

    MATH  Google Scholar 

  20. Gebremedhin, A.H., Manne, F., Pothen, A.: What color is your Jacobian? Graph coloring for computing derivatives. SIAM Rev. 47(4), 629–705 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Glass, C.A.: Bag rationalisation for a food manufacturer. J. Oper. Res. Soc. 53, 544–551 (2002)

    Article  MATH  Google Scholar 

  22. Gualandi, S., Malucelli, F.: Exact solution of graph coloring problems via constraint programming and column generation. INFORMS J. Comput. 24(1), 81–100 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Guo, C., Zhong, L.C., Rabaey, J.M.: Low power distributed mac for ad hoc sensor radio networks. In: Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM 2001), pp. 2944–2948. IEEE Press (2001)

  24. Hale, W.K.: Frequency assignment theory and applications. Proc. IEEE 68(12), 1497–1514 (1980)

    Article  Google Scholar 

  25. Hernández, H., Blum, C.: Distributed graph coloring: an approach based on the calling behavior of Japanese tree frogs. Swarm Intell. 6, 117–150 (2012)

    Article  Google Scholar 

  26. Hernández, H., Blum, C.: FrogSim: distributed graph coloring in wireless ad hoc networks. Telecommun. Syst. 55, 211–223 (2014)

    Article  Google Scholar 

  27. Hertz, A., Plumettaz, M., Zufferey, N.: Variable space search for graph coloring. Discret. Appl. Math. 156(13), 2551–2560 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hertz, A., de Werra, D.: Using tabu search techniques for graph coloring. Computing 39(4), 345–351 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  29. Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9(3), 256–278 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  30. Johnson, D.S., Aragon, C.R., McGeoch, L.A., Schevon, C.: Optimization by simulated annealing: an experimental evaluation. Part II: graph coloring and number partitioning. Oper. Res. 39(3), 378–406 (1991)

    Article  MATH  Google Scholar 

  31. Johnson, D.S., Trick, M.A.: Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge. American Mathematical Society, Providence (1996)

    MATH  Google Scholar 

  32. Keshavarzian, A., Lee, H., Venkatraman, L.: Wakeup scheduling in wireless sensor networks. In: Proceedings of the 7th ACM International Symposium on Mobile Ad-Hoc Networking and Computing (MobiHoc 06), pp. 322–333. ACM Press (2006)

  33. Kosowski, A., Manuszewski, K.: Classical coloring of graphs. In: Kubale, M. (ed.) Graph Colorings, pp. 1–20. American Mathematical Society, Providence (2004)

    Chapter  Google Scholar 

  34. Laurent, B., Hao, J.K.: List-graph colouring for multiple depot vehicle scheduling. Int. J. Math. Oper. Res. 1(1), 228–245 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Leighton, F.T.: A graph coloring algorithm for large scheduling problems. J. Res. Natl. Bur. Stand. 84(6), 489–503 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  36. Leith, D.J., Clifford, P.: Convergence of distributed learning algorithms for optimal wireless channel allocation. In: Proceedings of the 45th IEEE Conference on Decision and Control, pp. 2980–2985. IEEE (2006)

  37. Lewandowski, G., Condon, A.: Experiments with parallel graph coloring heuristics and applications of graph coloring. In: Johnson, D.J., Trick, M.A. (eds.) Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, pp. 309–334. American Mathematical Society, Providence (1996)

    Google Scholar 

  38. Lim, A., Wang, F.: Robust graph coloring for uncertain supply chain management. In: Proceedings of the 38th Annual Hawaii International Conference on System Sciences (HICSS’05), pp. 263–277. IEEE (2005)

  39. Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. ACM 41(5), 960–981 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  40. Malaguti, E., Monaci, M., Toth, P.: A metaheuristic approach for the vertex coloring problem. INFORMS J. Comput. 20(2), 302–316 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Malaguti, E., Monaci, M., Toth, P.: An exact approach for the vertex coloring problem. Discret. Optim. 8(2), 174–190 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Matula, D.W., Marble, G., Isaacson, J.: Graph coloring algorithms. In: Read, R. (ed.) Graph Theory and Computing, pp. 109–122. Academic Press, New York (1972)

    Chapter  Google Scholar 

  43. Meertens, L., Fitzpatrick, S.: Asynchronous execution and communication latency in distributed constraint optimization. In: Proceedings of the 3rd International Workshop on Distributed Constraint Reasoning, International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS’2002), pp. 80–85 (2002)

  44. Mehrotra, A., Trick, M.A.: A column generation approach for graph coloring. INFORMS J. Comput. 8(4), 344–354 (1996)

    Article  MATH  Google Scholar 

  45. Méndez-Díaz, I., Zabala, P.: A cutting plane algorithm for graph coloring. Discret. Appl. Math. 156, 159–179 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  46. Minton, S., Johnston, M.D., Philips, A.B., Laird, P.: Minimizing conflicts: a heuristic repair method for constraint satisfaction and scheduling problems. Artif. Intell. 58, 161–205 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  47. Park, T., Lee, C.Y.: Application of the graph coloring algorithm to the frequency assignment problem. J. Oper. Res. Soc. Jpn. 39(2), 258–265 (1996)

    MathSciNet  MATH  Google Scholar 

  48. Peemöller, J.: A correction to Brélaz’s modification of Brown’s coloring algorithm. Commun. ACM 26, 593–597 (1983)

    Article  Google Scholar 

  49. Thompson, J., Dowsland, K.: An improved ant colony optimisation heuristic for graph colouring. Discret. Appl. Math. 156, 313–324 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  50. Turner, J.S.: Almost all k-colorable graphs are easy to color. J. Algorithms 9(1), 63–82 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  51. Welsh, D., Powell, M.: An upper bound for the chromatic number of a graph and its application to timetabling problems. Comput. J. 12, 317–322 (1967)

    MATH  Google Scholar 

  52. de Werra, D.: An introduction to timetabling. Eur. J. Oper. Res. 19(2), 151–162 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  53. Wilensky, U.: Center for Connected Learning and Computer Science, Northwestern University, Evanston, IL. NetLogo. http://ccl.northwestern.edu/netlogo/ (1999)

  54. Woo, T.K., Su, S.Y.W., Newman-Wolfe, R.: Resource allocation in a dynamically partitionable bus network using a graph coloring algorithm. IEEE Trans. Commun. 39(12), 1794–1801 (1991)

    Article  Google Scholar 

  55. Zhang, W., Wang, G., Xing, Z., Wittenburg, L.: Distributed stochastic search and distributed breakout: properties, comparison and applications to constraint optimization problems in sensor networks. Artif. Intell. 161, 55–87 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Artificial Intelligence Department, National Distance University of Spain (UNED), C/ Juan del Rosal, 16, 28040, Madrid, Spain

    Severino F. Galán

Authors
  1. Severino F. Galán
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Severino F. Galán.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Galán, S.F. Simple decentralized graph coloring. Comput Optim Appl 66, 163–185 (2017). https://doi.org/10.1007/s10589-016-9862-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-016-9862-9

Keywords

Search

Navigation