Approximating the Independence Number and the Chromatic Number in Expected Polynomial Time

The independence number of a graph and its chromatic number are known to be hard to approximate. Due to recent complexity results, unless coRP = NP, there is no polynomial time algorithm which approximates any of these quantities within a factor of n1−∈ for graphs on n vertices.We show that the situation is significantly better for the average case. For every edge probability p = p(n) in the range n−1/2+∈ ≤ p ≤ 3/4, we present an approximation algorithm for the independence number of graphs on n vertices, whose approximation ratio is O((np)1/2/log n) and whose expected running time over the probability space G(n, p) is polynomial. An algorithm with similar features is described also for the chromatic number.A key ingredient in the analysis of both algorithms is a new large deviation inequality for eigenvalues of random matrices, obtained through an application of Talagrand's inequality.

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

[2]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[3]  János Komlós,et al.  The eigenvalues of random symmetric matrices , 1981, Comb..

[4]  Ravi B. Boppana,et al.  Eigenvalues and graph bisection: An average-case analysis , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[5]  Béla Bollobás,et al.  The chromatic number of random graphs , 1988, Comb..

[6]  Martin E. Dyer,et al.  The Solution of Some Random NP-Hard Problems in Polynomial Expected Time , 1989, J. Algorithms.

[7]  Ravi B. Boppana,et al.  Approximating maximum independent sets by excluding subgraphs , 1990, BIT.

[8]  Michel Talagrand.  A new isoperimetric inequality for product measure and the tails of sums of independent random variables , 1991 .

[9]  Ludek Kucera,et al.  The Greedy Coloring Is a Bad Probabilistic Algorithm , 1991, J. Algorithms.

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

[11]  M. Halldórsson.  A Still Better Performance Guarantee for Approximate Graph Coloring , 1993, Inf. Process. Lett..

[12]  C. R. Subramanian,et al.  Coloring Random Graphs in Polynomial Expected Time , 1993, ISAAC.

[13]  D. Welsh,et al.  A Spectral Technique for Coloring Random 3-Colorable Graphs , 1994 .

[14]  J. Håstad.  Clique is hard to approximate within n 1-C , 1996 .

[15]  Johan Håstad,et al.  Clique is hard to approximate within n/sup 1-/spl epsiv// , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[16]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[17]  Dorit S. Hochba,et al.  Approximation Algorithms for NP-Hard Problems , 1997, SIGA.

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

[19]  Noga Alon,et al.  Finding a large hidden clique in a random graph , 1998, SODA '98.

[20]  Noga Alon,et al.  Spectral Techniques in Graph Algorithms , 1998, LATIN.

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

[22]  U. Feige,et al.  Finding and certifying a large hidden clique in a semirandom graph , 2000, Random Struct. Algorithms.