# On Approximate Graph Colouring and MAX-*k*-CUT Algorithms Based on the θ-Function

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## Abstract

The problem of colouring a *k*-colourable graph is well-known to be NP-complete, for *k* ≥ 3. The MAX-*k*-CUT approach to approximate *k*-colouring is to assign *k* colours to all of the vertices in polynomial time such that the fraction of `defect edges' (with endpoints of the same colour) is provably small. The best known approximation was obtained by Frieze and Jerrum (1997), using a semidefinite programming (SDP) relaxation which is related to the Lovász θ-function. In a related work, Karger et al. (1998) devised approximation algorithms for colouring *k*-colourable graphs exactly in polynomial time with as few colours as possible. They also used an SDP relaxation related to the θ-function.

In this paper we further explore semidefinite programming relaxations where graph colouring is viewed as a satisfiability problem, as considered in De Klerk et al. (2000). We first show that the approximation to the chromatic number suggested in De Klerk et al. (2000) is bounded from above by the Lovász θ-function. The underlying semidefinite programming relaxation in De Klerk et al. (2000) involves a lifting of the approximation space, which in turn suggests a provably good MAX-*k*-CUT algorithm. We show that of our algorithm is closely related to that of Frieze and Jerrum; thus we can sharpen their approximation guarantees for MAX-*k*-CUT for small fixed values of *k*. For example, if *k* = 3 we can improve their bound from 0.832718 to 0.836008, and for *k* = 4 from 0.850301 to 0.857487. We also give a new asymptotic analysis of the Frieze-Jerrum rounding scheme, that provides a unifying proof of the main results of both Frieze and Jerrum (1997) and Karger et al. (1998) for *k* ≫ 0.

*k*-CUT

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## References

- D.V. Alekseevskij, E.B.Vinberg, and A.S. Solodovnikov, “Geometry of spaces of constant curvature,” in
*Geometry II*, vol. 29 of*Encyclopedia of Mathematical Sciences*, E.B. Vinberg (Ed.), Springer-Verlag, 1993.Google Scholar - N. Alon and N. Kahale, “Approximating the independence number via the
*?*-function,”*Mathematical Programming*, vol. 80, pp. 253–264, 1998.CrossRefGoogle Scholar - G. Andersson, L. Engelbertsen, and J. Hastad, “A new way of using semidefinite programming with applications to linear equations mod
*p*,”*Journal of Algorithms*, vol. 39, pp. 162–204, 2001.CrossRefGoogle Scholar - R. Beigel and D. Eppstein, “3-Coloring in time
*O*(1*.*3446*n*): A no-MIS algorithm,” in*Proc. 36th IEEE Symp. Foundations of Comp. Sci.*, 1995, pp. 444–453.Google Scholar - J.-D. Cho and M. Sarrafzadeh, “Fast approximation algorithms on Maxcut,
*k*-Coloring and*k*-Color ordering for VLSI applications,”*IEEE Transactions on Computers*, vol. 47, no. 11, pp. 1253–1266, 1998.CrossRefGoogle Scholar - E. de Klerk, H. van Maaren, and J.P. Warners, “Relaxations of the satisfiability problem using semidefinite programming,”
*Journal of automated reasoning*, vol. 24, pp. 37–65, 2000.CrossRefGoogle Scholar - U. Feige and M. Goemans, “Approximating the value of two prover proof systems with applications toMAX2SAT and MAX DICUT,” in
*Proc. Third Israel Symposium on Theory of Computing and Systems*, 1995, pp. 182–189.Google Scholar - W. Feller,
*An Introduction to Probability Theory and its Applications*, vol. I, 3rd edition, JohnWiley & Sons Inc.: New York, 1968.Google Scholar - A. Frieze and M. Jerrum, “Improved approximation algorithms for MAX
*k*-cut and MAX BISECTION,”*Algorithmica*, vol. 18, pp. 61–77, 1997.Google Scholar - A. Genz, “Numerical computation of multivariate normal probabilities,”
*J. Comp. Graph. Stat.*, vol. 1, pp. 141–149, 1992.Google Scholar - M. X. Goemans, “Semidefinite programming in combinatorial optimization,”
*Math. Programming*, vol. 79, nos. 1-3, Ser. B, pp. 143–161, 1997. Lectures on mathematical programming (ismp97) (Lausanne, 1997).CrossRefGoogle Scholar - M.X. Goemans and D.P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,”
*Journal of the ACM*, vol. 42, no. 6, pp. 1115–1145, 1995.CrossRefGoogle Scholar - M.X. Goemans and D.P.Williamson, “Approximation algorithms forMAX3-CUT and other problems via complex semidefinite programming,” in
*Proc. 33rd STOC*, ACM, 2001, pp. 443–452.Google Scholar - M. Grötschel, L. Lovász, and A. Schrijver,
*Geometric Algorithms and Combinatorial Optimization*, Springer-Verlag: Berlin, 1988.Google Scholar - E. Halperin and U. Zwick, “Approximation algorithms for MAX 4-SAT and rounding procedures for semidefinite programs,” in
*Proc. 7th IPCO*, 1999, pp. 202–217.Google Scholar - R.A. Horn and C.R. Johnson,
*Matrix Analysis*, Cambridge University Press, 1985.Google Scholar - V. Kann, S. Khanna, J. Lagergren, and A. Panconesi, “On the hardness of approximating Max
*k*-Cut and its dual,”*Chicago Journal of Theoretical Computer Science*, vol. 1997, no. 2, June 1997.Google Scholar - D. Karger, R. Motwani, and M. Sudan, “Approximate graph coloring by semidefinite programming,”
*J. ACM*, vol. 45, no. 2, pp. 246–265, 1998.CrossRefGoogle Scholar - H. Karloff and U. Zwick, “A 7/8-approximation algorithm for MAX 3SAT?” in
*Proc. 38th FOCS*, 1997, pp. 406–415.Google Scholar - S. Khanna, N. Linial, and S. Safra, “On the hardness of approximating the chromatic number,”
*Combinatorica*, vol. 20, pp. 393–415, 2000.CrossRefGoogle Scholar - D.E. Knuth, “The sandwich theorem,”
*The Electronic Journal of Combinatorics*, vol. 1, pp. 1–48, 1994.Google Scholar - S. Kotz, N. Balakrishnan, and Norman L. Johnson,
*Continuous Multivariate Distributions*, vol. 1: Models and Applications of*Wiley Series in Probability and Statistics*, 2nd edition, John Wiley & Sons Inc.: New York, 2000.Google Scholar - L. Lovász, “On the Shannon capacity of a graph,”
*IEEE Trans. on Information Theory*, vol. 25, pp. 1–7, 1979.CrossRefGoogle Scholar - Y. Rinott and V. Rotar, “A remark on quadrant normal probabilities in high dimensions,”
*Statistics and Probability Letters*, vol. 51, no. 1, pp. 47–51, 2001.CrossRefGoogle Scholar - A. Schrijver, “A comparison of the Delsarte and Lovász bounds,”
*IEEE Trans. Inform. Theory*, vol. 25, no. 4, pp. 425–429, 1979.CrossRefGoogle Scholar - U. Zwick, “Outward rotations: A new tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems,” in
*Proc. 31st STOC*, 1999, pp. 679–687.Google Scholar