Weighted Improper Colouring

  • Julio Araujo
  • Jean-Claude Bermond
  • Frédéric Giroire
  • Frédéric Havet
  • Dorian Mazauric
  • Remigiusz Modrzejewski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7056)

Abstract

In this paper, we study a colouring problem motivated by a practical frequency assignment problem and up to our best knowledge new. In wireless networks, a node interferes with the other nodes the level of interference depending on numerous parameters: distance between the nodes, geographical topography, obstacles, etc. We model this with a weighted graph G where the weights on the edges represent the noise (interference) between the two end-nodes. The total interference in a node is then the sum of all the noises of the nodes emitting on the same frequency. A weighted t-improper k-colouring of G is a k-colouring of the nodes of G (assignment of k frequencies) such that the interference at each node does not exceed some threshold t. The Weighted Improper Colouring problem, that we consider here consists in determining the weighted t-improper chromatic number defined as the minimum integer k such that G admits a weighted t-improper k-colouring. We also consider the dual problem, denoted the Threshold Improper Colouring problem, where given a number k of colours (frequencies) we want to determine the minimum real t such that G admits a weighted t-improper k-colouring. We show that both problems are NP-hard and first present general upper bounds; in particular we show a generalisation of Lovász’s Theorem for the weighted t-improper chromatic number. We then show how to transform an instance of the Threshold Improper Colouring problem into another equivalent one where the weights are either 1 or M, for a sufficient big value M. Motivated by the original application, we study a special interference model on various grids (square, triangular, hexagonal) where a node produces a noise of intensity 1 for its neighbours and a noise of intensity 1/2 for the nodes that are at distance 2. Consequently, the problem consists of determining the weighted t-improper chromatic number when G is the square of a grid and the weights of the edges are 1, if their end nodes are adjacent in the grid, and 1/2 otherwise. Finally, we model the problem using linear integer programming, propose and test heuristic and exact Branch-and-Bound algorithms on random cell-like graphs, namely the Poisson-Voronoi tessellations.

Keywords

Voronoi Diagram Chromatic Number Partition Problem Colouring Problem Triangular Grid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aardal, K.I., van Hoesel, S.P.M., Koster, A.M.C.A., Mannino, C., Sassano, A.: Models and solution techniques for frequency assignment problems. Annals of Operations Research 153(1), 79–129 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alouf, S., Altman, E., Galtier, J., Lalande, J.F., Touati, C.: Quasi-optimal bandwidth allocation for multi-spot MFTDMA satellites. In: INFOCOM 2005. 24th Annual Joint Conference of the IEEE Computer and Communications Societies. Proceedings IEEE, vol. 1, pp. 560–571. IEEE (2005)Google Scholar
  3. 3.
    Araujo, J., Bermond, J.-C., Giroire, F., Havet, F., Mazauric, D., Modrzejewski, R.: Weighted Improper Colouring. Research Report RR-7590, INRIA (April 2011)Google Scholar
  4. 4.
    Baccelli, F., Klein, M., Lebourges, M., Zuyev, S.: Stochastic geometry and architecture of communication networks. Telecom. Systems 7(1), 209–227 (1997)CrossRefGoogle Scholar
  5. 5.
    Brooks, R.L.: On colouring the nodes of a network. Mathematical Proceedings of the Cambridge Philosophical Society 37(02), 194–197 (1941)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Correa, R., Havet, F., Sereni, J.-S.: About a Brooks-type theorem for improper colouring. Australasian Journal of Combinatorics 43, 219–230 (2009)MathSciNetMATHGoogle Scholar
  7. 7.
    Fischetti, M., Lepschy, C., Minerva, G., Romanin-Jacur, G., Toto, E.: Frequency assignment in mobile radio systems using branch-and-cut techniques. European Journal of Operational Research 123(2), 241–255 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gupta, P., Kumar, P.R.: The capacity of wireless networks. IEEE Transactions on Information Theory 46(2), 388–404 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Haenggi, M., Andrews, J.G., Baccelli, F., Dousse, O., Franceschetti, M.: Stochastic geometry and random graphs for the analysis and design of wireless networks. IEEE Journal on Selected Areas in Communications 27(7), 1029–1046 (2009)CrossRefGoogle Scholar
  10. 10.
    Havet, F., Reed, B., Sereni, J.-S.: L(2,1)-labelling of graphs. In: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, pp. 621–630. Society for Industrial and Applied Mathematics, Philadelphia (2008)Google Scholar
  11. 11.
    Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press (1972)Google Scholar
  12. 12.
    Lovász, L.: On decompositions of graphs. Studia Sci. Math. Hungar. 1, 238–278 (1966)Google Scholar
  13. 13.
    Mannino, C., Sassano, A.: An enumerative algorithm for the frequency assignment problem. Discrete Applied Mathematics 129(1), 155–169 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Woodall, D.R.: Improper colorings of graphs. In: Nelson, R., Wilson, R.J. (eds.) Pitman Res. Notes Math. Ser., vol. 218, pp. 45–63. Longman Scientific and Technical (1990)Google Scholar
  15. 15.
    Yeh, R.K.: A survey on labeling graphs with a condition at distance two. Discrete Mathematics 306(12), 1217–1231 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Julio Araujo
    • 1
    • 2
  • Jean-Claude Bermond
    • 1
  • Frédéric Giroire
    • 1
  • Frédéric Havet
    • 1
  • Dorian Mazauric
    • 1
  • Remigiusz Modrzejewski
    • 1
  1. 1.Mascotte, joint project I3S(CNRS/Univ. de Nice)/INRIAFrance
  2. 2.ParGO Research GroupUniversidade Federal do Ceará - UFCBrazil

Personalised recommendations