Abstract
We consider the relationship between the graph coloring problem (GCP) and the vector assignment problem (VAP). Given an undirected graph, VAP asks to assign a vector to each vertex so as to maximize the minimum angle between the vectors corresponding to adjacent vertices. We show that any solution to the VAP in the 2-dimensional space, which we call the 2-dimensional VAP (2VAP), gives a feasible coloring, and that such transformation can be computed efficiently. We also show that any optimal solution to 2VAP gives an optimal coloring for GCP. Based on this fact, we propose a heuristic algorithm for GCP, whose search space is the set of solutions for 2VAP. The algorithm is quite simple and can be considered as a variant of the threshold accepting. The experiments show that our algorithm works well for graphs with relatively low degree.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Avanthay, C., Hertz, A., Zufferey, N.: A variable neighborhood search for graph coloring. European Journal of Operational Research 151, 379–388 (2003)
Brélaz, D.: New methods to color vertices of a graph. Communications of the ACM 22, 251–256 (1979)
Burer, S., Monteiro, R.D.C., Zhang, Y.: Rank-two relaxation heuristics for max-cut and other binary quadratic programs. SIAM Journal on Optimization 12(2), 503–521 (2002)
Caramia, M., Dell’Olmo, P.: Bounding vertex coloring by trancated Multistage branch and bound. Networks 44(4), 231–242 (2004)
Dueck, G., Scheuer, T.: Threshold accepting: A generel purpose optimization algorithm appearing superior to simulated annealing. Journal of Computational Physics 90, 161–175 (1990)
Eiben, A., van der Hauw, J.: Grpah coloring with adaptive genetic algorithms. Technical Report TR96-11, Leiden University (August 1996)
Hertz, A., de Werra, D.: Using tabu search techniques for graph coloring. Computing 39, 345–351 (1987)
Hertz, A., Plumettaz, M., Zufferey, N.: Variable space search for graph coloring, Working Paper (2007)
Hsu, W.L., Tsai, K.H.: Linear time algorithms on circular-arc graphs. Information Processing Letters 40, 123–129 (1991)
Karger, D., Motwani, R., Sudan, M.: Approximate graph coloring by semidefinite programming. Journal of the ACM 45(2), 246–265, 3 (1998)
Katz, M.J., Nielsen, F., Segal, M.: Maintenance of a piercing set for intervals with applications. Algorithmica 36(1), 59–73 (2003)
Kochenberger, G.A., Glover, F., Alidaee, B., Rego, C.: An unconstrained quadratic binary programming approach to the vertex coloring problem. Annals of Operations Research 139(1), 229–241 (2005)
Laguna, M., Martí, R.: A GRASP for coloring sparse graphs. Computational Optimization and Applications 19(2), 165–178 (2001)
Mladenović, N., Plastria, F., Urošević, D.: Formulation space search for circle packing problems. In: Stützle, T., Birattari, M., Hoos, H.H. (eds.) SLS 2007. LNCS, vol. 4638, pp. 212–216. Springer, Heidelberg (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ono, T., Yagiura, M., Hirata, T. (2008). A Vector Assignment Approach for the Graph Coloring Problem. In: Maniezzo, V., Battiti, R., Watson, JP. (eds) Learning and Intelligent Optimization. LION 2007. Lecture Notes in Computer Science, vol 5313. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92695-5_13
Download citation
DOI: https://doi.org/10.1007/978-3-540-92695-5_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-92694-8
Online ISBN: 978-3-540-92695-5
eBook Packages: Computer ScienceComputer Science (R0)