Approximate graph coloring by semidefinite programming

We consider the problem of coloring <italic>k</italic>-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on <italic>n</italic> vertices with min{<italic>O</italic>(Δ<supscrpt>1/3</supscrpt> log<supscrpt>1/2</supscrpt> Δ log <italic>n</italic>), <italic>O</italic>(<italic>n</italic><supscrpt>1/4</supscrpt> log<supscrpt>1/2</supscrpt> <italic>n</italic>)} colors where Δ is the maximum degree of any vertex. Besides giving the best known approximation ratio in terms of <italic>n</italic>, this marks the first nontrivial approximation result as a function of the maximum degree Δ. This result can be generalized to <italic>k</italic>-colorable graphs to obtain a coloring using min{<italic>O</italic>(Δ<supscrpt>1-2/<italic>k</italic></supscrpt> log<supscrpt>1/2</supscrpt> Δ log <italic>n</italic>), <italic>O</italic>(<italic>n</italic><supscrpt>1−3/(<italic>k</italic>+1)</supscrpt> log<supscrpt>1/2</supscrpt> <italic>n</italic>)} colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for <italic>semidefinite optimization problems</italic>, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovász &thgr;-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the &thgr;-function.

[1]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[2]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[3]  E. C. Milner.  A Combinatorial Theorem on Systems of Sets , 1968 .

[4]  David C. Wood,et al.  A technique for colouring a graph applicable to large scale timetabling problems , 1969, Computer/law journal.

[5]  Frank E. Grubbs,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[6]  Donald Ervin Knuth,et al.  The Art of Computer Programming, 2nd Ed. (Addison-Wesley Series in Computer Science and Information , 1978 .

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

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

[9]  John Cocke,et al.  Register Allocation Via Coloring , 1981, Comput. Lang..

[10]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[11]  John Cocke,et al.  A methodology for the real world , 1981 .

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

[13]  D. Aldous.  Probability Approximations via the Poisson Clumping Heuristic , 1988 .

[14]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[15]  V. Rich.  Personal communication , 1989, Nature.

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

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

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

[19]  Nathan Linial,et al.  On the Hardness of Approximating the Chromatic Number , 1993, [1993] The 2nd Israel Symposium on Theory and Computing Systems.

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

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

[22]  Mario Szegedy.  A note on the /spl theta/ number of Lovasz and the generalized Delsarte bound , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[23]  David P. Williamson,et al.  .879-approximation algorithms for MAX CUT and MAX 2SAT , 1994, STOC '94.

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

[25]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1994, JACM.

[26]  Donald E. Knuth.  The Sandwich Theorem , 1994, Electron. J. Comb..

[27]  R. Motwani,et al.  On Exact and Approximate Cut Covers of Graphs , 1994 .

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

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

[30]  Noga Alon,et al.  A spectral technique for coloring random 3-colorable graphs (preliminary version) , 1994, STOC '94.

[31]  Rajeev Motwani,et al.  On syntactic versus computational views of approximability , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[32]  Uriel Feige,et al.  Randomized graph products, chromatic numbers, and Lovasz j-function , 1995, STOC '95.

[33]  Ramesh Hariharan,et al.  Derandomizing semidefinite programming based approximation algorithms , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[34]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[35]  Alan M. Frieze,et al.  Improved Approximation Algorithms for MAX k-CUT and MAX BISECTION , 1995, IPCO.

[36]  Farid Alizadeh,et al.  Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization , 1995, SIAM J. Optim..

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

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

[39]  Uriel Feige,et al.  Zero knowledge and the chromatic number , 1996, Proceedings of Computational Complexity (Formerly Structure in Complexity Theory).

[40]  David R. Karger,et al.  An Õ(n^{3/14})-Coloring Algorithm for 3-Colorable Graphs , 1997, Information Processing Letters.

[41]  Uriel Feige.  Randomized graph products, chromatic numbers, and the Lovász ϑ-function , 1997, Comb..

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

[43]  Noga Alon,et al.  Approximating the independence number via theϑ-function , 1998, Math. Program..

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

[45]  Lars Engebretsen,et al.  Clique Is Hard To Approximate Within , 2000 .

[46]  Russ Bubley,et al.  Randomized algorithms , 1995, CSUR.

[47]  Nathan Linial,et al.  On the Hardness of Approximating the Chromatic Number , 2000, Comb..