Abstract
In certain shared resource applications, such as file access scheduling for a network of computer users, some level of conflict between the elements of the schedule is tolerable. A scheduling problem of this nature may be modelled as a generalisation of the classical vertex colouring problem for graphs, called the maximum degree graph colouring problem. In this paper we present four algorithmic procedures for solving the maximum degree graph colouring problem. The first two of these algorithms (a simple heuristic and a tabu search metaheuristic) produce approximate solutions, while the other two algorithms are exact. The runtime efficiencies of and the solution qualities produced by these procedures are tested with respect to a number of graph benchmarks from the literature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
This approach of access time discretisation during any particular user group’s access time slot is known as the Time Division Multiple Access (TDMA) channel access protocol (Wikipedia—The free exncyclopedia 2011) and is used in a variety of mobile telephone, landline telephone and local area networks. The TDMA protocol allows multiple users to share the same network channel (such as a radio frequency channel, power line, telephone land line, co-axial cable or fibre-optic cable) by dividing the signal into successive time slots. The users access the network by transmitting in rapid succession, one after the other, each at his/her own access time.
A clique is a complete subgraph, i.e. a subgraph in which any two vertices are adjacent. The number of vertices [edges, resp.] in a (sub)graph G is called the order [size, resp.] of G.
The subgraph of a graph G induced by a vertex subset V is denoted by 〈V〉 G and is that subgraph of G which contains exactly the vertices in V and all the edges of G joining vertices in V. If the context of the larger graph G is clear, then the subscript G in the induced subgraph notation is omitted. A colour-induced subgraph of G is the subgraph of G induced by a colour class, i.e. the set of all vertices to which the same colour has been assigned.
The degree of a vertex v in a (sub)graph G is the number of edges incident to v in G, and is denoted by deg G (v) or merely by deg(v) if the context of G is clear. The maximum degree [minimum degree, resp.] of G is the largest [smallest, resp.] degree of a vertex in G, and is denoted by Δ(G) [δ(G), resp.]. In a colouring of G with colour classes \(\mathcal{C}_{1},\ldots,\mathcal{C}_{x}\), we refer to the largest [smallest, resp.] value of \(\varDelta (\langle \mathcal{C}_{i}\rangle_{G})\) for all i=1,…,x as the largest [smallest, resp.] colour class induced maximum degree.
Given an n×n chessboard, a queen’s graph is a graph on n 2 vertices, each corresponding to a square of the board. Two vertices are adjacent if the corresponding squares are in the same row, column or diagonal (Trick 2003).
Mycielski (1955) established the existence of an x-chromatic, triangle-free graph for every positive integer x.
Given a work of literature, a graph is generated where each vertex represents a character in the book. Two vertices are adjacent if the corresponding characters encounter each other in the book (Trick 2003). Three book graphs were selected, using the original names on the DIMACS website, namely david from Charles Dickens’s David Copperfield, huck from Mark Twain’s Huckleberry Finn, and jean from Victor Hugo’s Les Misérables.
Due to computation limitations we only considered graphs of order less than 100 for which neither exact algorithm could achieve any results within three hours. However, we avoided random graphs of too small orders, since we already had enough results on small graphs (not included in the paper because of their simplicity). We therefore chose five values within the range 0 to 100 to include in the paper. We also did not wish to consider random graphs that are too sparse since some of the Class I graphs are sparse—hence our smallest density value of 0.2. On the other hand, the Class I graphs also include complete graphs, and so we limited the density to no more than 0.8. Finally, we chose one more density value in the middle of the range 0.2 to 0.8.
References
AIMMS (2012). CPLEX solver for linear and mixed integer programming [Online], [Cited: March 4th, 2012]. Available from: http://www.aimms.com/features/solvers/cplex.
Alcatel-Lucent Space Technologies (2011). Alcatel-Lucent: at the speed of ideas [Online], [Cited: December 29th, 2011]. Available from: http://www.alcatel-lucent.com.
Alouf, S., Altman, E., Galtier, J., Lalande, J.-F., & Touati, C. (2005). Quasi-optimal bandwidth allocation for multi-spot MFTDMA satellites. In Proceedings of the 24th annual joint conference of the IEEE computer and communications societies held, 13–17 March 2005, Miami (Florida) (Vol. 1, pp. 560–571).
Andrews, J. A., & Jacobson, M. S. (1985). On a generalization of chromatic number. Congressus Numerantium, 47, 33–48.
Andrews, JA, & Jacobson, M. S. (1987). On a generalization of chromatic number and two kinds of Ramsey numbers. Ars Combinatoria, 23, 97–102.
Benadé, G., Broere, I., Jonck, B., & Frick, M. (1996). Uniquely (m,k)τ-colourable graphs and k-τ-saturated graphs. Discrete Mathematics, 162, 13–22.
Bentz, C., Costa, M. C., de Werra, D., Picouleau, C., & Ries, B. (2008). On a graph coloring problem arising from discrete tomography. Networks, 51(4), 256–267.
Brelaz, D. (1979). New methods to colour the vertices of a graph. Communications of the Association for Computing Machinery, 22, 251–256.
Broere, I., & Frick, M. (1991). A characterization of the sequence of generalized chromatic numbers of a graph. In Graph theory, combinatorics and applications: proceedings of the sixth quadrennial international conference on the theory and applications of graphs (pp. 179–186). New York: Wiley.
Broere, I., & Frick, M. (1985). On the order of colour critical graphs. Congressus Numerantium, 47, 125–130.
Broere, I., & Frick, M. (1990a). On the order of uniquely (k,m)-colourable graphs. Discrete Mathematics, 82, 225–232.
Broere, I., & Frick, M. (1990b). Two results on generalized chromatic numbers. Quaestiones Mathematicae, 13, 183–190.
Brown, J. R. (1972). Chromatic scheduling and the chromatic number problem. Management Science, 19(4), 456–463.
Brown, J. I. (1996). The complexity of generalized graph colourings. Discrete Applied Mathematics, 69, 257–270.
Burger, A. P., & Grobler, P. J. P. (1993). Aspects of (m,k)-colourings (Research Report 146/93(7)). University of South Africa, Pretoria.
Chartrand, G., Geller, D. P., & Hedetniemi, S. T. (1968). A generalization of the chromatic number. Proceedings of the Cambridge Philosophical Society, 64, 265–271.
Costa, M. C., de Werra, D., Picouleau, C., & Ries, B. (2009). Graph coloring with cardinality constraints on the neighbourhoods. Discrete Optimization, 6, 362–369.
Cowen, L., Cowen, R., & Woodall, D. (1986). Defective colourings of graphs in surfaces: Partitions into subgraphs of bounded valence. Journal of Graph Theory, 10, 187–195.
Cowen, L., Goddard, W., & Jesurum, C. E. (1997). Defective colouring revisited. Journal of Graph Theory, 24, 205–219.
Frick, M. (1993). A survey of (m,k)-colourings, quo vadis, graph theory? Annals of Discrete Mathematics, 55, 45–58.
Frick, M. (1986). Generalised colourings of graphs. Ph.D.-dissertation, Rand Afrikaans University, Johannesburg.
Frick, M., & Bullock, F. (2001). Detour chromatic numbers. Discussiones Mathematicae Graph Theory, 21, 283–291.
Frick, M., & Henning, M. A. (1994). Extremal results on defective colourings of graphs. Discrete Mathematics, 126, 151–158.
Goddard, W. (1991). Acyclic colourings of planar graphs. Discrete Mathematics, 91, 91–94.
Harary, F. (1985). Conditional colourability in graphs. In Graphs and applications, proceedings of the 1st Colorado symposium on graph theory (pp. 127–136). New York: Wiley.
Harary, F., & Jones, K. (1985). Conditional colourability II: Bipartite variations. Congressus Numerantium, 50, 205–218.
Herrmann, F., & Hertz, A. (2002). Finding the chromatic number by means of critical graphs. The ACM Journal of Experimental Algorithmics, 7, 1–12.
Hertz, A., & De Werra, D. (1987). Using tabu search techniques for graph colouring. Computing, 39, 345–351.
Johns, G., & Saba, F. (1989). On the path chromatic number of a graph. Annals of the New York Academy of Sciences, 576, 275–280.
Johnson, D. S., Aragon, C. R., McGeoch, L. A., & Schevon, C. (1991). Optimization by simulated annealing: An experimental evaluation—Part II: Graph colouring and number partitioning. Operations Research, 39(3), 378–406.
Johnson, D. S., & Trick, M. A. (2006). DIMACS series in discrete mathematics and theoretical computer science [Online], [Cited November 29th, 2006]. Available from: http://dimacs.rutgers.edu/Volumes/Vol26.html.
Mycielski, J. (1955). Sur le coloriage des graphes. Colloquium Mathematicum, 3, 161–162.
Nieuwoudt, I. (2007). On the maximum degree chromatic number of a graph. Ph.D.-Dissertation, Stellenbosch University, Stellenbosch.
Peemöller, J. (1983). A correction to Brelaz’s modification of Brown’s colouring algorithm. Communications of the Association for Computing Machinery, 26(8), 595–597.
Sachs, H. (1969). Finite graphs. In Recent progress in combinatorics, Proceedings of the third Waterloo conference on combinatorics (pp. 175–184). New York: Academic Press.
Trick, M. A. (2003). Network resources for colouring a graph [Online], [Cited May 28th, 2003]. Available from: http://mat.gsia.cmu.edu/COLOR/color.html.
Wikipedia—The free exncyclopedia (2011). Time division multiple access [Online], [Cited: December 29th, 2011]. Available from: http://en.wikipedia.org/wiki/Timedivisionmultipleaccess.
Wolfram Research Inc. (2009). Wolfram Mathematica 7 [Online], [Cited June 7th, 2009]. Available from: http://www.wolfram.com/products/mathematica/index.html.
Woodall, D. (1990). Improper colourings of graphs. In Graph colourings. Pitman Research Notes in Mathematics Series. Harlow: Longman Scientific and Technical.
Acknowledgements
The authors are indebted to Dr. Werner Gründlingh who produced the graphics in this paper. The South African National Research Foundation funded work towards this paper in the form of a research grant (GUN 2072999).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nieuwoudt, I., van Vuuren, J.H. Algorithms for a shared resource scheduling problem in which some level of conflict is tolerable. J Sched 15, 681–702 (2012). https://doi.org/10.1007/s10951-012-0291-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10951-012-0291-z