Abstract
The Lovász theta number of a graph G can be viewed as a semidefinite programming relaxation of the stability number of G. It has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of G. We introduce a related strengthening of the Lovász theta number toward the chromatic number of G, which is shown to be equal to the fractional chromatic number of G. Solving copositive programs is NP-hard. This motivates the study of tractable approximations of the copositive cone. We investigate the Parrilo hierarchy to approximate this cone and provide computational simplifications for the approximation of the chromatic number of vertex transitive graphs. We provide some computational results indicating that the Lovász theta number can be strengthened significantly toward the fractional chromatic number of G on some Hamming graphs.
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References
Bomze I.M., de Klerk E.: Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J. Global Optim. 24, 163–185 (2001)
Bomze I.M., Dür M., de Klerk E., Roos C., Quist A.J., Terlaky T.: On copositive programming and standard quadratic optimization problems. J. Global Optim. 18, 301–320 (2000)
Burer, S.: On the copositive representation of binary and continuos nonconvex quadratic Programs. Working paper (2007)
Busygin S., Pasechnik D.V.: On NP-hardness of the clique partition—independence number gap recognition and related problems. Discrete Math. 306(4), 460–463 (2006)
de Klerk E.: Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications. Kluwer, Dordrecht (2002)
de Klerk E., Pasechnik D.V.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12, 875–892 (2002)
de Klerk E., Pasechnik D.V., Schrijver A.: Reduction of symmetric semidefinite programs using the regular *-representation. Math. Program. 109(2–3), 613–624 (2007)
Dukanovic, I.: Semidefinite programming applied to graph coloring problem. Ph.D. thesis, University of Klagenfurt, Austria (2008, forthcoming)
Dukanovic, I., Rendl, F.: Reductions of group symmetric semidefinite programs. Working paper (2005)
Dukanovic I., Rendl F.: Semidefinite programming relaxations for graph coloring and maximal clique problems. Math. Program. B 109(2), 345–366 (2007)
Gatermann K., Parrilo P.A.: Symmetry groups, semidefinite programs, and sums of squares. J. Pure Appl. Algebra 192, 95–128 (2004)
Gvozdenović, N., Laurent, M.: Personal communication (2005)
Gvozdenović N., Laurent M.: Semidefinite bounds for the stability number of a graph via sums of squares of polynomials. Lect. Notes Comput. Sci. 3509, 136–151 (2005)
Gvozdenović, N., Laurent, M.: Computing semidefinite programming lower bounds for the (fractional) chromatic number via block-diagonalization. SIAM J. Optim. (2008, to appear)
Gvozdenović, N., Laurent, M.: The operator Ψ for the chromatic number of a graph. SIAM J. Optim. (2008, to appear)
Halperin E., Nathaniel R., Zwick U.: Coloring k-colorable graphs using relatively small palettes. J. Algorithms 45, 72–90 (2002)
Karger D., Motwani R., Sudan M.: Approximate graph coloring by semidefinite programming. J. Assoc. Comput. Machinery 45, 246–265 (1998)
Knuth D.E.: The sandwich theorem. Electron. J. Combinatorics 1, 1–48 (1994)
Lasserre, J.B.: On explicit exact SDP relaxation for nonlinear 0-1 programs. In: Ardal, K., Gerards, A.M.H. (eds.) Lecture Notes in Computer Science, vol. 2081, pp. 293–303 (2001)
Laurent, M.: Strengthened semidefinite bounds for codes. Preprint, p. 17 (2005)
Laurent M., Rendl F.: Semidefinite programming and integer programming. In: Weismantel, R., Aardal, K., Nemhauser, G.(eds) Handbook on Discrete Optimization., pp. 393–514. Elsevier B. V., Amsterdam (2005)
Lovász L.: On the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25, 1–7 (1979)
Lovász L., Schrijver A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1, 166–190 (1991)
McKay B.D.: Practical graph isomorphism. Congressus Numerantium 30, 45–87 (1981)
Murty K.G., Kabadi S.N.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39, 117–129 (1987)
Parrilo, P.A.: Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. Ph.D. thesis, California Institute of Technology (2000)
Peña J., Vera J., Zuluaga L.F.: Computing the stability number of a graph via linear and semidefinite programming. SIAM J. Optim. 18, 87–105 (2007)
Povh, J., Rendl, F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Working paper (2006)
Quist A.J., de Klerk E., Roos C., Terlaky T.: Copositive relaxation for general quadratic programming. Optim. Methods Softw. 9, 185–209 (1998)
Schrijver A.: A comparison of the Delsarte and Lovász bounds. IEEE Trans. Inform. Theory IT 25, 425–429 (1979)
Schrijver A.: Combinatorial Optimization—Polyhedra and Efficiency, vol. B. Springer, New York (2003)
Szegedy, M.: A note on the theta number of Lovász and the generalized Delsarte bound. In: 35th Annual Symposium on Foundations of Computer Science, pp. 36–39 (1994)
Todd M.J.: Semidefinite optimization. Acta Numerica 10, 515–560 (2001)
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Partial support by the EU project Algorithmic Discrete Optimization (ADONET), MRTN-CT-2003-504438, is gratefully acknowledged.
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Dukanovic, I., Rendl, F. Copositive programming motivated bounds on the stability and the chromatic numbers. Math. Program. 121, 249–268 (2010). https://doi.org/10.1007/s10107-008-0233-x
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DOI: https://doi.org/10.1007/s10107-008-0233-x