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12 - Chromatic scheduling

Published online by Cambridge University Press:  05 May 2015

By
Dominique de Werra  and
Alain Hertz
Edited by
Lowell W. Beineke  and
Robin J. Wilson
Dominique de Werra
Affiliation:
Ecole Polytechnique Fédérale de Lausanne in Switzerland
Alain Hertz
Affiliation:
École Polytechnique Fédérale de Lausanne in Switzerland
Lowell W. Beineke
Affiliation:
Purdue University, Indiana
Robin J. Wilson
Affiliation:
The Open University, Milton Keynes
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Summary

Variations and extensions of the basic vertex-colouring and edge-colouring models have been developed to deal with increasingly complex scheduling problems. We present and illustrate them in specific situations where additional requirements are imposed. We include list-colouring, mixed graph colouring, co-colouring, colouring with preferences and bandwidth colouring, and we present applications of edge-colourings to open shop, school timetabling and sports scheduling problems. We also discuss balancing and compactness constraints which often appear in practical situations.

Introduction

We show here how graph colouring models may provide a natural tool for dealing with a variety of scheduling problems. Starting from the basic vertex-colouring model, we will introduce some variations and extensions that are motivated by their applications to some scheduling issues. In each case we give references for further results and for extensions of the various models presented. For algorithms, see Chapter 13.

In chromatic scheduling problems we have a collection V of items, such as operations of jobs to be performed. In V there are some pairs v, w that are subject to an incompatibility condition and we call E the set of such incompatibility pairs. These data are represented by the graph G = (V, E) in which the items are associated with the vertices and the incompatible pairs v,w with the edges vw between the corresponding vertices.

We also have a set C = {1, 2, …, k} of time periods (of unit duration). Assuming that each item (considered as an operation) has unit completion time, we may ask whether we can find a schedule taking the incompatibilities into account and using at most k periods of time. This is precisely the vertex k-colouring problem: there exists a feasible schedule if and only if the set V of vertices can be partitioned into subsets S1, S2, …, Sk, where each Si contains no two incompatible items.

In some instances, we may try to find the smallest set C of periods (that is, the smallest k) for which a schedule in time k = |C| exists.

Type
Chapter
Information
Topics in Chromatic Graph Theory , pp. 255 - 276
Publisher: Cambridge University Press
Print publication year: 2015

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References

1. S., Altinakar, G., Caporossi and A., Hertz, On compact k-edge colorings: a polynomial time reduction from linear to cyclic, Discrete Optimization 8 (2011), 502–512.Google Scholar
2. C., Archetti, N., Bianchessi and A., Hertz, A branch-and-price algorithm for the robust graph coloring problem, Les Cahiers du Gerad, G-2011-75, Montréal, 2011.Google Scholar
3. M., Bouchard, M., Čangalovič and A., Hertz, About equivalent interval colorings of weighted graphs, Discrete Appl. Math. 157 (2009), 3615–3624.Google Scholar
4. M., Bouchard, A., Hertz and G., Desaulniers, Lower bounds and a tabu search algorithm for the minimum deficiency problem, J. Combin. Optimization 17 (2009), 168–191.Google Scholar
5. D., Brélaz, Y., Nicolier and D., de Werra, Compactness and balancing in scheduling, Zeitschrift für Operations Research 21 (1977), 65–73.Google Scholar
6. E. K., Burke, D., de Werra and J., Kingston, Applications to timetabling, Handbook of Graph Theory (eds. J. L., Gross and J., Yellen), CRC Press (2004), 445–174.Google Scholar
7. D., Cartwright and F., Harary, Structural balance: a generalization of Heider's theory, Psychological Review 63 (1956), 277–293.Google Scholar
8. D., de Werra, A note on graph coloring, RAIRO, Revue Française d'Automatique, d'Informatique et de Recherche Opérationnelle R-1 (1974), 49–53.Google Scholar
9. D., de Werra, A few remarks on chromatic scheduling, Combinatorial programming, Methods and Applications (ed. B., Roy), D. Reidel (1975), 337–342.Google Scholar
10. D., de Werra, Some models of graphs for scheduling sports competitions, Discrete Appl. Math. 21 (1988), 47–65.Google Scholar
11. D., de Werra, Graph theoretical models for preemptive scheduling, Advances in Project Scheduling (eds. R., Slowinski and J., Weglarz), Elsevier (1989), 171–185.Google Scholar
12. D., de Werra and Y., Gay, Chromatic scheduling and frequency assignment, Discrete Appl. Math. 49 (1994), 165–174.Google Scholar
13. D., de Werra, A., Hertz, D., Kobler and N. V. R., Mahadev, Feasible edge colorings of trees with cardinality constraints, Discrete Math. 222 (2000), 61–72.Google Scholar
14. D., de Werra and Ph., Solot, Compact cylindrical chromatic scheduling, SIAM J. Discrete Math. 4 (1991), 528–534.Google Scholar
15. M., Demange, D., de Werra, J., Monnot and V., Paschos, Time slot scheduling of compatible jobs, J. Scheduling 10 (2007), 111–127.Google Scholar
16. M., Demange, T., Ekim and D., de Werra, On the approximation of min split coloring and min cocoloring, J. Graph Algorithms and Appl. 10 (2006), 297–315.Google Scholar
17. M., Demange, T., Ekim and D., de Werra, A tutorial on the use of graph coloring for some problems in robotics, Europ. J. Oper. Res. 192 (2009), 41–55.Google Scholar
18. T., Ekim and D., de Werra, On split-coloring problems, J. Combin. Optimization 10 (2005), 211–225.Google Scholar
19. P., Erdős, A. L., Rubin and H., Taylor, Choosability in graphs, Congr. Numer. 26 (1979), 125–157.Google Scholar
20. S., Even, A., Itai and A., Shamir, On the complexity of timetable and multicommodity flow problems, SIAM J. Comput. 5 (1976), 691–703.Google Scholar
21. M. R., Garey and D. S., Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, 1979.Google Scholar
22. K., Giaro, The complexity of consecutive Δ-coloring of bipartite graphs: 4 is easy, 5 is hard, Ars Combin. 47 (1997), 287–298.Google Scholar
23. K., Giaro, M., Kubale and M., Malafiejski, Consecutive colorings of the edges of general graphs, Discrete Math. 236 (2001), 131–143.Google Scholar
24. J., Gimbel, D., Kratsch and L., Stewart, On cocolourings and cochromatic numbers of graphs, Discrete Appl. Math. 48 (1994), 111–127.Google Scholar
25. T., Gonzales and S., Sahni, Open shop scheduling to minimize finish time, J. ACM 23 (1976), 665–679.Google Scholar
26. S., Gutner, The complexity of planar graph choosability, Discrete Math. 159 (1996), 119–130.Google Scholar
27. A., Hajnal and E., Szemerédi, Proof of a conjecture of P. Erdős, Combinatorial Theory and its Applications, Balatonfured (eds. P., Erdős, A., Rényi and V. T., Sős), North-Holland (1970), 601–623.Google Scholar
28. M. M., Halldórsson and G., Kortsarz, Multicoloring: problems and techniques, Math. Foundations of Computer Science,LNCS 3153 (2004), 25–41.Google Scholar
29. P., Hansen, J., Kuplinski and D., de Werra, Mixed graph coloring, Math. Methods of Operations Research 45 (1997), 145–160.Google Scholar
30. A., Hertz and B., Ries, On r-equitable colorings of trees and forests, Les Cahiers du Gerad, G-2011-40, Montreal, 2011.Google Scholar
31. G., Kendall, S., Knust, C. C., Ribeiro and S., Urrutia, Scheduling in sports: an annotated bibliography, Computers Oper. Res. 37 (2010), 1–19.Google Scholar
32. H. A., Kierstead and A. V., Kostochka, A short proof of the Hajnal-Szemerédi theorem on equitable coloring, Combin. Probab. Comput. 17 (2008), 265–270.Google Scholar
33. S., Prestwich, Constrained bandwidth multicoloration neighborhoods, Proc. Computational Symp. on Graph Coloring and its Generalizations, Ithaca(eds. D. S., Johnson, A., Mehrotra and M., Trick), (2002), 126–133.Google Scholar
34. R. V., Rassmussen and M. A., Trick, The timetable constrained distance minimization problem, Annals of Operations Research 171 (2006), 167–181.Google Scholar
35. B., Ries, Coloring some classes of mixed graph, Discrete Appl. Math. 155 (2007), 1–6.Google Scholar
36. B., Ries and D., de Werra, On two coloring problems in mixed graphs, Europ. J. Combin. 29 (2008), 712–725.Google Scholar
37. B., Robillard, Vérification Formelle et Optimisation de l'Allocation de Registres, Ph.D. Thesis, CNAM, Paris, 2010.Google Scholar
38. B., Roy, Nombre chromatique etplus longs chemins d'un graphe, RAIRO, Revue Française d'Automatique, d'Informatique et de Recherche Opérationnelle 1 (1967), 129–132.Google Scholar
39. A., Schwartz, The deficiency of a regular graph, Discrete Math. 306 (2006), 1947–1954.Google Scholar
40. Y. N., Sotskov, A., Dolgui and F., Werner, Mixed graph coloring for unit-time job-shop scheduling, Internat. J. Math. Algorithms 4 (2001), 289–323.Google Scholar
41. M., ČangaloviČ and J. A. M., Schreuder, Exact colouring algorithm for weighted graphs applied to timetabling problems with lectures of different lengths, Europ. J. Operational Research 51 (1991), 248–258.Google Scholar
42. M., Voigt, List colourings of planar graphs, Discrete Math. 120 (1993), 215–219.Google Scholar
43. J., Yánez and J., Ramírez, The robust coloring problem, Europ. J. Operational Research 148 (2003), 546–558.Google Scholar

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