We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a k-colorable graph with k colors so that a maximum fraction of edges are properly colored (i.e. their endpoints receive different colors). A random k-coloring properly colors an expected fraction \(1-\frac{1}{k}\) of edges. We prove that given a graph promised to be k-colorable, it is NP-hard to find a k-coloring that properly colors more than a fraction \(\approx 1-\frac{1}{33 k}\) of edges. Previously, only a hardness factor of \(1- O\bigl(\frac{1}{k^2}\bigr)\) was known. Our result pins down the correct asymptotic dependence of the approximation factor on k. Along the way, we prove that approximating the Maximum 3-colorable subgraph problem within a factor greater than \(\frac{32}{33}\) is NP-hard.

Using semidefinite programming, it is known that one can do better than a random coloring and properly color a fraction \(1-\frac{1}{k} +\frac{2 \ln k}{k^2}\) of edges in polynomial time. We show that, assuming the 2-to-1 conjecture, it is hard to properly color (using k colors) more than a fraction \(1-\frac{1}{k} + O\left(\frac{\ln k}{k^2}\right)\) of edges of a k-colorable graph.


Edge Weight Constraint Satisfaction Problem Noise Operator Hardness Result Markov Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Frieze, A.M., Jerrum, M.: Improved approximation algorithms for max k-cut and max bisection. Algorithmica 18(1), 67–81 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Khot, S., Kindler, G., Mossel, E., O’Donnell, R.: Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM J. Comput. 37(1), 319–357 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Petrank, E.: The hardness of approximation: Gap location. Computational Complexity 4, 133–157 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43(3), 425–440 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kann, V., Khanna, S., Lagergren, J., Panconesi, A.: On the hardness of approximating max k-cut and its dual. Chicago J. Theor. Comput. Sci. 1997 (1997)Google Scholar
  6. 6.
    Guruswami, V., Lewin, D., Sudan, M., Trevisan, L.: A tight characterization of NP with 3 query PCPs. In: Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, pp. 8–17 (1998)Google Scholar
  7. 7.
    Dinur, I., Mossel, E., Regev, O.: Conditional hardness for approximate coloring. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 344–353 (2006)Google Scholar
  8. 8.
    Mossel, E., O’Donnell, R., Oleszkiewicz, K.: Noise stability of functions with low influences: invariance and optimality. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pp. 21–30 (2005)Google Scholar
  9. 9.
    O’Donnell, R., Wu, Y.: Conditional hardness for satisfiable CSPs. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing (to appear, 2009)Google Scholar
  10. 10.
    Raghavendra, P.: Optimal algorithms and inapproximability results for every CSP? In: Proceedings of the 40th ACM Symposium on Theory of Computing, pp. 245–254 (2008)Google Scholar
  11. 11.
    Crescenzi, P., Silvestri, R., Trevisan, L.: On weighted vs unweighted versions of combinatorial optimization problems. Inf. Comput. 167(1), 10–26 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Weichsel, P.M.: The kronecker product of graphs. Proceedings of the American Mathematical Society 13(1), 47–52 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Khot, S.: On the power of unique 2-prover 1-round games. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pp. 767–775 (2002)Google Scholar
  14. 14.
    Sinclair, A.: Improved bounds for mixing rates of markov chains and multicommodity flow. Combinatorics, Probability and Computing 1, 351–370 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Venkatesan Guruswami
    • 1
  • Ali Kemal Sinop
    • 1
  1. 1.Computer Science Department, School of Computer ScienceCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations