Abstract
Max-Cut is one of the most studied combinatorial optimization problems because of its wide range of applications and because of its connections with other fields of discrete mathematics (see, e.g., the book by Deza and Laurent [10]). Like other interesting combinatorial optimization problems, Max-Cut is very simple to state.
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Absil, P.-A., Baker, C.G., Gallivan, K.A.: Trust-region methods on Riemannian manifolds. Journal Foundations of Computational Mathematics 7(3), 303–330 (2007)
Anjos, M.F., Wolkowicz, H.: Strengthened semidefinite relaxations via a second lifting for the Max-Cut problem. Discrete Applied Mathematics 119(1-2), 79–106 (2002)
Barahona, F., Mahjoub, A.R.: On the cut polytope. Mathematical Programming 36(2), 157–173 (1986)
Barvinok, A.: Problems of distance geometry and convex properties of quadratic maps. Discrete Computational Geometry 13, 189–202 (1995)
Benson, S.J., Ye, Y., Zhang, X.: Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM Journal on Optimization 10(2), 443–461 (2000)
Boros, E., Hammer, P.L., Tavares, G.: Local search heuristics for quadratic unconstrained binary optimization. Journal of Heuristics 13, 99–132 (2007)
Buchheim, C., Wiegele, A., Zheng, L.: Exact algorithms for the Quadratic Linear Ordering Problem. INFORMS Journal on Computing 22(1), 168–177 (2010)
Burer, S., Monteiro, R.D.C.: A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Mathematical Programming 95(2), 329–357 (2003)
de Klerk, E., Pasechnik, D.V., Warners, J.P.: On approximate graph colouring and Max-k-Cut algorithms based on the \(\vartheta\)-function. Journal of Combinatorial Optimization 8(3), 267–294 (2004)
Deza, M., Laurent, M.: Geometry of Cuts and Metrics, vol. 15 of Algorithms and Combinatorics. Springer-Verlag, Berlin (1997)
Dolan, E.D., Morè, J.J.: Benchmarking optimization software with performance profile. Mathematical Programming 91(2), 201–213 (2001)
Fischer, I., Gruber, G., Rendl, F., Sotirov, R.: Computational experience with a bundle approach for semidefinite cutting plane relaxations of Max-Cut and equipartition. Mathematical Programming 105(2-3, Ser. B), 451–469 (2006)
Frieze, A., Jerrum, M.: Improved approximation algorithms for Max k-Cut and Max Bisection. Algorithmica 18(1), 67–81 (1997)
Ghaddar, B., Anjos, M.F., Liers, F.: A branch-and-cut algorithm based on semidefinite programming for the minimum k-partition problem. Annals of Operations Research, 188(1), 155–174 (2011)
Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the Association for Computing Machinery 42(6), 1115–1145 (1995)
Grippo, L., Palagi, L., Piacentini, M., Piccialli, V.: An unconstrained approach for solving low rank SDP relaxations of { − 1, 1} quadratic problems. Technical Report 1.13, Dip. di Informatica e sistemistica A. Ruberti, Sapienza Università di Roma (2009)
Grippo, L., Palagi, L., Piacentini, M., Piccialli, V., Rinaldi, G.: SpeeDP: A new algorithm to compute the SDP relaxations of Max-Cut for very large graphs. Technical Report 13.10, DII-UTOVRM - Università di Roma Tor Vergata (2010) available on Optimization Online.
Grippo, L., Palagi, L., Piccialli, V.: Necessary and sufficient global optimality conditions for NLP reformulations of linear SDP problems. Journal of Global Optimization 44(3), 339–348 (2009)
Grippo, L., Palagi, L., Piccialli, V.: An unconstrained minimization method for solving low rank SDP relaxations of the Max Cut problem. Mathematical Programming 126(1), 119–146 (2011)
Grippo, L., Sciandrone, M.: Nonmonotone globalization techniques for the Barzilai-Borwein gradient method. Computational Optimization and Applications 23, 143–169 (2002)
Grone, R., Pierce, S., Watkins, W.: Extremal Correlation Matrices. Linear Algebra Application 134, 63–70 (1990)
Grötschel, M., Jünger, M., Reinelt, G.: Facets of the linear ordering polytope. Mathematical Programming 33, 43–60 (1985)
Hammer, P.: Some network flow problems solved with pseudo-boolean programming. Operations Research 13, 388–399 (1965)
Helmberg, C., Rendl, F.: Solving quadratic (0, 1)-problems by semidefinite programs and cutting planes. Mathematical Programming 82(3), 291–315 (1998)
Helmberg, C., Rendl, F.: A spectral bundle method for semidefinite programming. SIAM Journal on Optimization 10, 673–696 (2000)
Hestenes, M.: Multiplier and gradient methods. Journal of Optimization Theory and Application 4, 303–320 (1969)
Homer, S., Peinado, M.: Design and performance of parallel and distributed approximation algorithm for the Maxcut. Journal of Parallel and Distributed Computing 46, 48–61 (1997)
Hungerländer, P., Rendl, F.: Semidefinite relaxations of ordering problems. Accepted for Publication in Mathematical Programming B
Journée, M., Bach, F., Absil, P.A., Sepulchre, R.: Low-rank optimization for semidefinite convex problems. SIAM Journal on Optimization 20(5), 2327–2351 (2010)
Karger, D., Motwani, R., Sudan, M.: Approximate graph colouring by semidefinite programming. Journal of the Association for Computing Machinery 45, 246–265 (1998)
Lasserre, J.B.: An explicit equivalent positive semidefinite program for nonlinear 0-1 programs. SIAM Journal on Optimization 12(3), 756–769 (2002)
Laurent, M., Rendl, F.: Semidefinite programming and integer programming. In: Aardal, K., Nemhauser, G.L., Weismantel, R. (eds.) Handbook in OR & MS, Discrete Optimization, vol. 12, Chap. 8. Elsevier B.V. (2005)
Liers, F., Jünger, M., Reinelt, G., Rinaldi, G.: Computing exact ground states of hard Ising spin glass problems by branch-and-cut. In: Hartmann, A.K., Rieger, H. (eds.) New Optimization Algorithms in Physics, pp. 47–69. Wiley-VCH Verlag (2004)
Lovász, L.: On the Shannon capacity of a graph. IEEE Transactions on Information Theory 25(1), 1–7 (1979)
Pataki, G.: On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Mathematics of Operations Research 23, 339–358 (1998)
Poljak, S., Rendl, F., Wolkowicz, H.: A recipe for semidefinite relaxation for 0-1 quadratic programming. Journal of Global Optimization 7, 51–73 (1995)
Powell, M.J.D.: A method for nonlinear constraints in minimization problem. In: Optimization, pp. 283–298. Academic Press, New York (1969)
Reinelt, G.: The Linear Ordering Problem: Algorithms and Applications. Heldermann Verlag (1985)
Rendl, F., Rinaldi, G., Wiegele, A.: Biq Mac Solver - BInary Quadratic and MAx-Cut Solver. http://biqmac.uni-klu.ac.at/.
Rendl, F., Rinaldi, G., Wiegele, A.: Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations. Mathematical Programming 121(2), 307–335 (2010)
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We thank the two anonymous referees for their careful reading of the chapter, and for their constructive remarks that greatly helped to improve its presentation.
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Palagi, L., Piccialli, V., Rendl, F., Rinaldi, G., Wiegele, A. (2012). Computational Approaches to Max-Cut. In: Anjos, M.F., Lasserre, J.B. (eds) Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, vol 166. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0769-0_28
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