On the Hardness of Approximating the Chromatic Number

k-colorable for some fixed . Our main result is that it is NP-hard to find a 4-coloring of a 3-chromatic graph. As an immediate corollary we obtain that it is NP-hard to color a k-chromatic graph with at most colors. We also give simple proofs of two results of Lund and Yannakakis [20]. The first result shows that it is NP-hard to approximate the chromatic number to within for some fixed ε > 0. We point here that this hardness result applies only to graphs with large chromatic numbers. The second result shows that for any positive constant h, there exists an integer , such that it is NP-hard to decide whether a given graph G is -chromatic or any coloring of G requires colors.

[1]  Mihir Bellare,et al.  Improved non-approximability results , 1994, STOC '94.

[2]  A. Blum.  ALGORITHMS FOR APPROXIMATE GRAPH COLORING , 1991 .

[3]  J. Håstad.  Clique is hard to approximate withinn1−ε , 1999 .

[4]  Magnús M. Hallórsson.  A still better performance guarantee for approximate graph coloring , 1993 .

[5]  Avrim Blum,et al.  An O(n0.4)-approximation algorithm for 3-coloring , 1989, STOC '89.

[6]  David S. Johnson,et al.  The Complexity of Near-Optimal Graph Coloring , 1976, J. ACM.

[7]  Avi Wigderson,et al.  Improving the performance guarantee for approximate graph coloring , 1983, JACM.

[8]  Avrim Blum,et al.  Some tools for approximate 3-coloring , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[9]  Nathan Linial,et al.  Graph products and chromatic numbers , 1989, 30th Annual Symposium on Foundations of Computer Science.

[10]  Sanjeev Arora,et al.  Probabilistic checking of proofs; a new characterization of NP , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[11]  Magnús M. Halldórsson,et al.  A Still Better Performance Guarantee for Approximate Graph Coloring , 1993, Information Processing Letters.

[12]  Richard M. Karp,et al.  Reducibility among combinatorial problems" in complexity of computer computations , 1972 .

[13]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[14]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[15]  Uriel Feige,et al.  Zero Knowledge and the Chromatic Number , 1998, J. Comput. Syst. Sci..

[16]  László Lovász,et al.  Interactive proofs and the hardness of approximating cliques , 1996, JACM.

[17]  M. Furer.  Improved hardness results for approximating the chromatic number , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[18]  Avrim Blum,et al.  New approximation algorithms for graph coloring , 1994, JACM.

[19]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1993, STOC.

[20]  László Lovász,et al.  Approximating clique is almost NP-complete , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.