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Strengthening the Lovász \(\theta(\overline G)\) bound for graph coloring. (English) Zbl 1059.05052
Summary: The Lovász \(\theta\)-number is a way to approximate the independence number of a graph, but also its chromatic number. We express the Lovász bound as the continuous relaxation of a discrete Lovász \(\theta\)-number which we derive from D. Karger et al.’s formulation [J. ACM 45, 246–265 (1998; Zbl 0904.68116)], and which is equal to the chromatic number. We also give another relaxation à la Schrijver-McEliece, which is better than the Lovász \(\theta\)-number.

05C15 Coloring of graphs and hypergraphs
05C35 Extremal problems in graph theory
90C27 Combinatorial optimization
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[1] Alizadeh, F., Haeberly, J.-P., Nayakkankuppam, M.V., Overton, M.L. (1997): SDPPack User?s Guide, version 0.8 beta. Technical report, NYU Computer Science Dpt, March 1997. URL: http://www.cs.nyu.edu/phd_students/madhu/sdppack/sdppack.html
[2] Alon, N. (1994): Explicit Ramsey graphs and orthonormal labelings. Electron. J. Comb. 1, # R12 · Zbl 0814.05056
[3] Alon, N., Kahale, N. (1998): Approximating the independence number via the ?-function. Math. Program. 80, 253-264 · Zbl 0895.90169
[4] Feige, U., Kilian, J. (1996): Zero Knowledge and the Chromatic Number. In: Proceedings of the 11th Annual IEEE Conference in Computing Complexity (preliminary version), pp. 278-287 · Zbl 0921.68089
[5] Frieze, A., Jerrum, M. (1995): Improved approximation algorithms for MAX k-cut and MAX BISECTION. In: Proceedings of the Fourth MPS Conference on Integer Programming and Combinatorial Optimization. Springer · Zbl 1135.90420
[6] Goemans, M.X., Williamson, D.P. (1995): Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming. J. ACM 42, 1115-1145 · Zbl 0885.68088
[7] Horn, R.A., Johnson, C.R. (1985): Matrix Analysis. Cambridge University Press, Cambridge (reedited 1999) · Zbl 0576.15001
[8] Jensen, T.R., Toft, B. (1995): Graph coloring problems. Wiley-Interscience series in discrete mathematics and optimization. Wiley, New York
[9] Karger, D., Motwani, R., Sudan, M. (1998): Approximate graph coloring by semidefinite programming. J. ACM 45 (2), 246-265, March 1998 · Zbl 0904.68116
[10] Knuth, D. (1994): The Sandwich Theorem. Electron. J. Comb. 1, # A1, 48 pp. · Zbl 0810.05065
[11] Lemaréchal, C., Oustry, F. (1999): Semidefinite relaxations and Lagrangian duality with application to combinatorial optimization. Rapport de Recherche Nr. 3710. Inria · Zbl 1160.90639
[12] Lovász, L. (1979): On the Shannon Capacity of a Graph. IEEE Trans. Inf. Theory IT-25 (1), 1-7 · Zbl 0395.94021
[13] Lund, C., Yannakakis, M. (1994): On the hardness of approximating minimization problems. J. ACM 41 (5), 960-981 · Zbl 0814.68064
[14] McEliece, R.J., Rodemich, E.R., Rumsey Jr., H.C. (1978): The Lovász Bound and Some Generalizations. J. Comb. Inf. Syst. Sci. 3 (3), 134-152 · Zbl 0408.05031
[15] Poljak, S., Rendl, F., Wolkowicz, H. (1995): A recipe for semidefinite relaxation for (0-1)-quadratic programming. J. Glob. Optim. 7, 51-73 · Zbl 0843.90088
[16] Schrijver, A. (1979): A Comparison of the Delsarte and Lovász Bounds. IEEE Trans. Inf. Theory IT-25 (4), 425-429 · Zbl 0444.94009
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