Skip to main content

Advertisement

Springer Nature Link
Log in

Circular coloring of graphs via linear programming and tabu search

  • Original Paper
  • Published:
Central European Journal of Operations Research Aims and scope Submit manuscript

Abstract

Circular coloring is a popular branch of graph theory which has been exhaustively studied for two decades mainly from a theoretical perspective. Since it is a refinement of the traditional proper coloring, it provides a more accurate model for cyclic scheduling problems which often arise in industrial applications. The present paper briefly surveys a special class of open shop scheduling that can be solved via circular coloring, and then proposes a new mathematical programming model and tabu search algorithm to compute the circular chromatic number of a graph effectively.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. This part of the argument is the same as in Zhu (2001). The existence of the claimed finite \(G'\) follows by the ‘compactness principle’; see e.g. Truszczyński and Zs Tuza (1992) for its various formulations and many applications.

  2. Depending on the vertex order under consideration, the number of essential constraints may be smaller: if \(v_i^k\) and \(v_j^{k+1}\) are adjacent then the corresponding line of (2) automatically follows from that of (5); and if \(j,k,l,m\) are four distinct indices such that \(v_jv_k,v_lv_m\in E(G)\) and both of \(v_j,v_k\) precede both of \(v_l,v_m\) then only (5) is necessary for them. The same simplification applies in the situation where \(v_k\) and \(v_l\) coincide. As a consequence, each vertex has to be involved in at most two constraints of type (5)–(6).

References

  • Asanovic K et al (2009) A view of the parallel computing landscape. Commun ACM 52:56–67

    Article  Google Scholar 

  • Brandes U, Eiglsperger M, Herman I, Himsolt M, Marshall MS (2002) GraphML progress report structural layer proposal. In: Mutzel P, Junger M, Leipert S (eds) Graph drawing. Springer, Berlin, Heidelberg, pp 501–512

  • Coin-OR Cbc. https://projects.coin-or.org/Cbc

  • Webpage of Brendan McKay. http://cs.anu.edu.au/bdm/data/graphs.html

  • Dagum L, Menon R (1998) OpenMP: an industry standard API for shared-memory programming. IEEE Comput Sci Eng 5:46–55

    Article  Google Scholar 

  • Deuber W, Zhu X (1996) Circular coloring of weighted graphs. J Graph Theor 23:365–376

    Article  MATH  MathSciNet  Google Scholar 

  • Ernst AT, Jiang H, Krishnamoorthy M, Sier D (2004) Staff scheduling and rostering: a review of applications, methods and models. Eur J Oper Res 153:3–27

    Article  MATH  MathSciNet  Google Scholar 

  • Ghebleh M (2007) Theorems and computations in circular colourings of graphs. PhD Thesis, Simon Fraser University

  • Glover F (1989) Tabu search—part I. ORSA J Comput 1:190–206

    Article  MATH  MathSciNet  Google Scholar 

  • Glover F (1990) Tabu search—part II. ORSA J Comput 2:4–32

    Article  MATH  Google Scholar 

  • Guichard DR (1993) Acyclic graph coloring and the complexity of the star chromatic number. J Graph Theor 17:129–134

    Article  MATH  MathSciNet  Google Scholar 

  • Hatami H, Tusserkani R (2004) On the complexity of the circular chromatic number. J Graph Theor 47: 226–230

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Kemnitz A, Wellmann P (2004) Circular chromatic numbers of certain planar graphs and small graphs. Congr Numer 169:199–209

    MATH  MathSciNet  Google Scholar 

  • Kubale M (2004) Graph colorings. American Mathematical Society, Providence, Rhode Island

  • Levner E, Kats V, de Pablo D, Cheng TCE (2010) Complexity of cyclic scheduling problems: a state-of-the-art survey. Comput Ind Eng 59:352–361

    Article  Google Scholar 

  • Maros I (2003) Computational techniques of the simplex method. Kluwer Academic Publishers, Boston

  • Marx D (2004) Graph colouring problems and their applications in scheduling. Period Polytech Ser El Eng 48:11–16

    Google Scholar 

  • Modarres M, Ghandehari M (2008) Applying circular coloring to open shop scheduling. Sci Iran 15:652–660

    MATH  MathSciNet  Google Scholar 

  • Pardalos PM, Mavridou T, Xue J (1999) The graph coloring problem: a bibliographic survey. In: Du DZ, Pardalos PM (eds) Handbook of combinatorial optimization. Springer, US, pp 1077–1141

  • Pinedo M (2005) Planning and scheduling in manufacturing and services, vol 24. Springer, New York

  • Randall, M., Abramson, D.: (1999) A general parallel tabu search algorithm for combinatorial optimisation problems. In: Proceedings of 1999 parallel and real time conference, Melbourne, pp 68–79

  • Stadtler H (2005) Supply chain management and advanced planning—basics, overview and challenges. Eur J Oper Res 163:575–588

    Article  MATH  Google Scholar 

  • Truszczyński M, Zs Tuza (1992) Rado’s selection principle: applications to binary relations, graph and hypergraph colorings and partially ordered sets. Discret Math 103:301–312

    Article  MATH  Google Scholar 

  • Vince A (1988) Star chromatic number. J Graph Theor 12:551–559

    Article  MATH  MathSciNet  Google Scholar 

  • Yeh HG, Zhu X (2005) Resource-sharing system scheduling and circular chromatic number. Theor Comput Sci 332:447–460

    Article  MATH  MathSciNet  Google Scholar 

  • Zhu X (1992) Star chromatic numbers and products of graphs. J Graph Theor 16:557–569

    Article  MATH  Google Scholar 

  • Zhu X (2001) Circular chromatic number: a survey. Discret Math 229:371–410

    Article  MATH  Google Scholar 

  • Zhu X, Thomas R (2006) Recent developments in circular colouring of graphs. In: Klazar M, Kratochvil J, Loebl M, Thomas R, Valtr P (eds) Topics in discrete mathematics. Springer, Berlin, Heidelberg, pp 497–550

Download references

Acknowledgments

Research of the second author was supported in part by the Hungarian Scientific Research Fund, OTKA grant 81493.

Author information

Authors and Affiliations

  1. Department of Computer Science and Systems Technology, University of Pannonia, Egyetem u. 10, Veszprém, 8200, Hungary

    Mate Barany & Zsolt Tuza

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, Budapest, 1053, Hungary

    Zsolt Tuza

Authors
  1. Mate Barany
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Zsolt Tuza
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Mate Barany.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barany, M., Tuza, Z. Circular coloring of graphs via linear programming and tabu search. Cent Eur J Oper Res 23, 833–848 (2015). https://doi.org/10.1007/s10100-014-0345-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10100-014-0345-8

Keywords