Improved bounds and algorithms for hypergraph 2-coloring

We show that for all large n, every n-uniform hypergraph with at most edges can be 2-colored. This makes progress on a problem of Erdos [Nordisk Mat. Tidskrift 11, 5–10 (1963)], improving the previous-best bound of n1/3−o(1)×2n due to Beck [Discrete Math. 24, 127–137 (1978)]. We further generalize this to a “local” version, improving on one of the first applications of the Lovasz local lemma. We also present fast randomized algorithms that output a proper 2-coloring with high probability for n-uniform hypergraphs with at most edges, for all large n. In addition, we derandomize and parallelize these algorithms, to derive NC1 versions of these results. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16: 4–32, 2000

[1]  J. Beck,et al.  On 3-chromatic hypergraphs , 1978, Discret. Math..

[2]  Alan M. Frieze,et al.  Coloring Bipartite Hypergraphs , 1996, IPCO.

[3]  P. Erdos.  On a combinatorial problem. II , 1964 .

[4]  Colin McDiarmid.  Hypergraph colouring and the Lovász Local Lemma , 1997, Discret. Math..

[5]  Noga Alon,et al.  Every 8-uniform 8-regular hypergraph is 2-colorable , 1988, Graphs Comb..

[6]  Zoltán Szabó.  An Application of Lovász's Local Lemma - A New Lower Bound for the van der Waerden Number , 1990, Random Struct. Algorithms.

[7]  Moni Naor,et al.  Small-Bias Probability Spaces: Efficient Constructions and Applications , 1993, SIAM J. Comput..

[8]  Noga Alon,et al.  A Parallel Algorithmic Version of the Local Lemma , 1991, Random Struct. Algorithms.

[9]  Joel H. Spencer,et al.  Coloring n-Sets Red and Blue , 1981, J. Comb. Theory, Ser. A.

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

[11]  Colin McDiarmid,et al.  A Random Recolouring Method for Graphs and Hypergraphs , 1993, Combinatorics, Probability and Computing.

[12]  Oded Goldreich,et al.  Efficient approximation of product distributions , 1998 .

[13]  Noam Nisan,et al.  Approximations of general independent distributions , 1992, STOC '92.

[14]  Bruce A. Reed,et al.  Further algorithmic aspects of the local lemma , 1998, STOC '98.

[15]  David R. Karger,et al.  Approximate graph coloring by semidefinite programming , 1998, JACM.

[16]  József Beck,et al.  An Algorithmic Approach to the Lovász Local Lemma. I , 1991, Random Struct. Algorithms.

[17]  Carsten Thomassen.  The even cycle problem for directed graphs , 1992 .

[18]  Lars Engebretsen,et al.  Better Approximation Algorithms for SET SPLITTING and NOT-ALL-EQUAL SAT , 1998, Inf. Process. Lett..

[19]  C. Fortuin,et al.  Correlation inequalities on some partially ordered sets , 1971 .