A derivation of Lovász' theta via augmented Lagrange duality
RAIRO - Operations Research - Recherche Opérationnelle, Tome 37 (2003) no. 1, pp. 17-27.

A recently introduced dualization technique for binary linear programs with equality constraints, essentially due to Poljak et al. [13], and further developed in Lemaréchal and Oustry [9], leads to simple alternative derivations of well-known, important relaxations to two well-known problems of discrete optimization: the maximum stable set problem and the maximum vertex cover problem. The resulting relaxation is easily transformed to the well-known Lovász θ number.

DOI : 10.1051/ro:2003012
Classification : 90C27, 90C27, 90C35
Mots clés : Lagrange duality, stable set, Lovász theta function, semidefinite relaxation
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Pinar, Mustapha Ç. A derivation of Lovász' theta via augmented Lagrange duality. RAIRO - Operations Research - Recherche Opérationnelle, Tome 37 (2003) no. 1, pp. 17-27. doi : 10.1051/ro:2003012. http://archive.numdam.org/articles/10.1051/ro:2003012/

[1] F. Alizadeh, J.-P. Haberly, V. Nayakkankuppam and M.L. Overton, SDPPACK user's guide, Technical Report 734. NYU Computer Science Department (1997).

[2] N. Brixius, R. Sheng and F. Potra, SDPHA user's guide, Technical Report. University of Iowa (1998).

[3] M. Grötschel, L. Lovász and A. Schrijver, Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Berlin (1988). | MR | Zbl

[4] C. Helmberg, Fixing variables in semidefinite relaxations. SIAM J. Matrix Anal. Appl. 21 (2000) 952-969. | MR | Zbl

[5] C. Helmberg, S. Poljak, F. Rendl and H. Wolkowicz, Combining semidefinite and polyhedral relaxations for integer programs, edited by E. Balas and J. Clausen, Integer Programming and Combinatorial Optimization IV. Springer-Verlag, Berlin, Lecture Notes in Comput. Sci. 920 (1995) 124-134. | MR

[6] J. Kleinberg and M. Goemans, The Lovász theta function and a semidefinite programming relaxation of vertex cover. SIAM J. Discrete Math. 11 (1998) 196-204. | MR | Zbl

[7] D. Knuth, The sandwich theorem. Electron. J. Combinatorics 1 (1994); www.combinatorics.org/Volume_1/volume1.html#A1 | MR | Zbl

[8] M. Laurent, S. Poljak and F. Rendl, Connections between semidefinite relaxations of the max-cut and stable set problems. Math. Programming 77 (1997) 225-246. | MR | Zbl

[9] C. Lemaréchal and F. Oustry, Semidefinite relaxation and Lagrangian duality with application to combinatorial optimization, Technical Report 3170. INRIA Rhône-Alpes (1999); http://www.inria.fr/RRRT/RR-3710.html

[10] L. Lovász, On the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25 (1979) 355-381. | MR | Zbl

[11] L. Lovász, Bounding the independence number of a graph. Ann. Discrete Math. 16 (1982). | MR | Zbl

[12] L. Lovász and L. Schrijver, Cones of matrices, and set functions and 0-1 optimization. SIAM J. Optim. 1 (1991) 166-190. | MR | Zbl

[13] S. Poljak, F. Rendl and H. Wolkowicz, A recipe for semidefinite relaxation for (0-1) quadratic programming. J. Global Optim. 7 (1995) 51-73. | MR | Zbl

[14] N. Shor, Nondifferentiable Optimization and Polynomial Problems. Kluwer Academic Publishers, Dordrecht, The Netherlands (1998). | MR | Zbl

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