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Safe Lower Bounds for Graph Coloring

  • Conference paper
Integer Programming and Combinatoral Optimization (IPCO 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6655))

Abstract

The best known method for determining lower bounds on the vertex coloring number of a graph is the linear-programming column-generation technique first employed by Mehrotra and Trick in 1996. We present an implementation of the method that provides numerically safe results, independent of the floating-point accuracy of linear-programming software. Our work includes an improved branch-and-bound algorithm for maximum-weight stable sets and a parallel branch-and-price framework for graph coloring. Computational results are presented on a collection of standard test instances, including the unsolved challenge problems created by David S. Johnson in 1989.

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References

  1. Andrade, D.V., Resende, M.G.C., Werneck, R.F.: Fast local search for the maximum independent set problem. In: Proceedings of Workshop on Experimental Algorithms, pp. 220–234 (2008)

    Google Scholar 

  2. Applegate, D.L., Cook, W., Dash, S., Espinoza, D.G.: Exact solutions to linear programming problems. Operations Research Letters 35(6), 693–699 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babel, L.: A fast algorithm for the maximum weight clique problem. Computing 52, 31–38 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Balas, E., Xue, J.: Minimum weighted coloring of triangulated graphs with application to maximum weight vertex packing and clique finding in arbitrary graphs. SIAM Journal of Computing 20(2), 209–221 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Balas, E., Yu, C.S.: Finding a maximum clique in an arbitrary graph. SIAM Journal of Computing 15(4), 1054–1068 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brélaz, D.: New methods to color the vertices of a graph. Communications of the ACM 22(4), 251–256 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caramia, M., Dell’Olmo, P.: Bounding vertex coloring by truncated multistage branch and bound. Networks 44(4), 231–242 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gualandi, S., Malucelli, F.: Exact solution of graph coloring problems via constraint programming and column generation. Optimization Online in press for INFORMS Journal on Computing (2010)

    Google Scholar 

  9. Held, S., Sewell, E.C., Cook, W.: Exact colors project webpage (2010), http://code.google.com/p/exactcolors/

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Malaguti, E., Monaci, M., Toth, P.: An exact approach for the vertex coloring problem. Discrete Optimization (2010) (in press)

    Google Scholar 

  12. Malaguti, E., Toth, P.: A survey on vertex coloring problems. International Transactions in Operational Research 17, 1–34 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mehrotra, A., Trick, M.A.: A Column Generation Approach for Graph Coloring. INFORMS Journal on Computing 8(4), 344–354 (1996)

    Article  MATH  Google Scholar 

  14. Mycielski, J.: Sur le coloriage des graphes. Colloq. Math. 3, 161–162 (1955)

    MathSciNet  MATH  Google Scholar 

  15. Méndez-Díaz, I., Zabala, P.: A branch-and-cut algorithm for graph coloring. Discrete Applied Mathematics 154(5), 826–847 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Méndez Díaz, I., Zabala, P.: A cutting plane algorithm for graph coloring. Discrete Applied Mathematics 156(2), 159–179 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Porumbel, D.C., Hao, J.-K., Kuntz, P.: An evolutionary approach with diversity guarantee and well-informed grouping recombination for graph coloring. Computers and Operations Research 37(10), 1822–1832 (2010)

    Article  MATH  Google Scholar 

  18. Sewell, E.C.: A branch and bound algorithm for the stability number of a sparse graph. INFORMS Journal on Computing 10(4), 438–447 (1998)

    Article  MathSciNet  Google Scholar 

  19. Titiloye, O., Crispin, A.: Quantum annealing of the graph coloring problem. Discrete Optimization (2011) (in Press)

    Google Scholar 

  20. Trick, M.A.: DIMACS Graph Coloring Instances (2002), http://mat.gsia.cmu.edu/COLOR02/

  21. Warren, J.S., Hicks, I.V.: Combinatorial branch-and-bound for the maximum weight independent set problem. Technical report, Texas A&M University (2006)

    Google Scholar 

  22. Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Computing 3(1), 103–128 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Östergård, P.R.J.: A new algorithm for the maximum-weight clique problem. Electronic Notes in Discrete Mathematics 3, 153–156 (1999)

    Article  MathSciNet  Google Scholar 

  24. Östergård, P.R.J., Niskanen, S.: Cliquer home page (2010), http://users.tkk.fi/pat/cliquer.html

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Author information

Authors and Affiliations

  1. University of Bonn, Germany

    Stephan Held

  2. Georgia Institute of Technology, USA

    William Cook

  3. Southern Illinois University Edwardsville, United States

    Edward C. Sewell

Authors
  1. Stephan Held
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  2. William Cook
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  3. Edward C. Sewell
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Editor information

Editors and Affiliations

  1. Mathematical Sciences Department, IBM T. J. Watson Research Center, 10598, Yorktown Heights, NY, USA

    Oktay Günlük

  2. Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600, Eindhoven, MB, The Netherlands

    Gerhard J. Woeginger

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© 2011 Springer-Verlag Berlin Heidelberg

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Held, S., Cook, W., Sewell, E.C. (2011). Safe Lower Bounds for Graph Coloring. In: Günlük, O., Woeginger, G.J. (eds) Integer Programming and Combinatoral Optimization. IPCO 2011. Lecture Notes in Computer Science, vol 6655. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20807-2_21

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  • DOI: https://doi.org/10.1007/978-3-642-20807-2_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-20806-5

  • Online ISBN: 978-3-642-20807-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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