Journal of Global Optimization

, Volume 61, Issue 1, pp 183–192 | Cite as

Matrix-power energy-landscape transformation for finding NP-hard spin-glass ground states

  • Markus Manssen
  • Alexander K. Hartmann


A method for solving binary optimization problems was proposed by Karandashev and Kryzhanovsky that can be used for finding ground states of spin glass models. By taking a power of the bond matrix the energy landscape of the system is transformed in such a way, that the global minimum should become easier to find. In this paper we test the combination of the new approach with various algorithms, namely simple random search, a cluster algorithm by Houdayer and Martin, and the common approach of parallel tempering. We apply these approaches to find ground states of the three-dimensional Edwards–Anderson model, which is an NP-hard problem, hence computationally challenging. To investigate whether the power-matrix approach is useful for such hard problems, we use previously computed ground states of this model for systems of size \(10^3\) spins. In particular we try to estimate the difference in needed computation time compared to plain parallel tempering.


Spin glass model Binary minimization Energy landscape transformation Monte Carlo method NP-hardness 



The simulations were performed at the C. v. O. Universität Oldenburg on the HERO cluster funded by the DFG (INST 184/108-1 FUGG) and the ministry of Science and Culture (MWK) of the Lower Saxony State. We would like to thank Simon Knowles for criticially reading the paper.


  1. 1.
    Binder, K., Young, A.: Spin-glasses: experimental facts, theoretical concepts and open questions. Rev. Mod. Phys. 58, 801 (1986)CrossRefGoogle Scholar
  2. 2.
    Mézard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and Beyond. World Scientific, Singapore (1987)MATHGoogle Scholar
  3. 3.
    Young, A.P. (ed.): Spin Glasses and Random Fields. World Scientific, Singapore (1998)Google Scholar
  4. 4.
    Nishimori, H.: Statistical Physics of Spin Glasses and Information Processing: An Introduction. Oxford University Press, Oxford (2001)CrossRefGoogle Scholar
  5. 5.
    Barahona, F.: On the computational complexity of Ising spin glass models. J. Phys. A Math. Gen. 15(10), 3241 (1982)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Swendsen, R.H., Wang, J.S.: Replica Monte Carlo simulation of spin-glasses. Phys. Rev. Lett. 57(21), 2607–2609 (1986)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Hartmann, A.K.: Cluster-exact approximation of spin glass ground states. Phys. A 224, 480–488 (1999)CrossRefGoogle Scholar
  8. 8.
    Hukushima, K., Nemoto, K.: Exchange Monte Carlo method and application to spin glass simulations. J. Phys. Soc. Jpn. 65(6), 1604–1608 (1996)CrossRefGoogle Scholar
  9. 9.
    Hartmann, A.K.: Scaling of stiffness energy for three-dimensional \(\pm \)J Ising spin glasses. Phys. Rev. E 59, 84 (1999)CrossRefGoogle Scholar
  10. 10.
    Hartmann, A.K.: Calculation of ground states of four-dimensional \(\pm \)J Ising spin glasses. Phys. Rev. E 60(5), 5135 (1999)CrossRefGoogle Scholar
  11. 11.
    Wang, F., Landau, D.P.: Determining the density of states for classical statistical models: a random walk algorithm to produce a flat histogram. Phys. Rev. E 64(5), 056101 (2001)CrossRefGoogle Scholar
  12. 12.
    Houdayer, J., Martin, O.C.: Hierarchical approach for computing spin glass ground states. Phys. Rev. E 64(5), 056704 (2001)CrossRefGoogle Scholar
  13. 13.
    Hartmann, A.K., Rieger, H.: Optimization Algorithms in Physics. Wiley-VCH, Weinheim (2001)CrossRefGoogle Scholar
  14. 14.
    Belletti, F., Cotallo, M., Cruz, A., Fernandez, L.A., Gordillo-Guerrero, A., Guidetti, M., Maiorano, A., Mantovani, F., Marinari, E., Martin-Mayor, V., Muñoz-Sudupe, A., Navarro, D., Parisi, G., Perez-Gaviro, S., Rossi, M., Ruiz-Lorenzo, J.J., Schifano, S.F., Sciretti, D., Tarancon, A., Tripiccione, R., Velasco, J.L., Yllanes, D., Zanier, G.: Janus: an FPGA-based system for high-performance scientific computing. Comput. Sci. Eng. 11(1), 48–58 (2009)CrossRefGoogle Scholar
  15. 15.
    Karandashev, Y.M., Kryzhanovsky, B.V.: Transformation of energy landscape in the problem of binary minimization. Doklady Math. 80(3), 927–931 (2009)CrossRefMATHGoogle Scholar
  16. 16.
    Gu, J., Huang, X.: Efficient local search space smoothing: a case study of the traveling salesman problem (tsp). IEEE Trans. Syst. Man Cybern. 24, 728–735 (1994)CrossRefGoogle Scholar
  17. 17.
    Schneider, J.J., Dankesreiter, M., Fettes, W., Morgenstern, I., Schmid, M., Singer, J.M.: Search-space smoothing for combinatorial optimization problems. Phys. A 243, 77–112 (1997)CrossRefGoogle Scholar
  18. 18.
    Zhang, Y., Kihara, D., Skolnick, J.: Local energy landscape flattening: parallel hyperbolic monte carlo sampling of protein folding. Proteins 48, 192–201 (2002)CrossRefGoogle Scholar
  19. 19.
    Pritchard-Bell, A., Shell, M.S.: Smoothing protein energy landscapes by integrating folding models with structure prediction. Biophys. J. 101(9), 2251–2259 (2011)CrossRefGoogle Scholar
  20. 20.
    Cetin, B.C., Barhen, J., Burdick, J.W.: Terminal repeller unconstrained subenergy tunneling (TRUST) for fast global optimization. J. Optim. Theory Appl. 77(1), 97–126 (1993)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Hamacher, K.: Adaptation in stochastic tunneling global optimization of complex potential energy landscapes. Europhys. Lett. 74(6), 944–950 (2006)CrossRefGoogle Scholar
  22. 22.
    Hamacher, K.: A new hybrid metaheuristic—combining stochastic tunneling and energy landscape paving. In: Blesa, M.J., Blum, C., Festa, P., Roli, A., Sampels M. (eds.) Hybrid Metaheuristics, Lecture Notes in Computer Science, 7919:107–117 (2013)Google Scholar
  23. 23.
    Karandashev, I.M., Kryzhanovsky, B.V.: Increasing the attraction area of the global minimum in the binary optimization problem. J. Glob. Optim. 56(3), 1167–1185 (2013)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Edwards, S.F., Anderson, P.W.: Theory of spin glasses. J. Phys. F Met. Phys. 5(5), 965–974 (1975)CrossRefGoogle Scholar
  25. 25.
    Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. PNAS 79(8), 2554–2558 (1982)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Newman, M.E.J., Barkema, G.T.: Monte Carlo Methods in Statistical Physics. Oxford University Press, Oxford (1999)MATHGoogle Scholar
  27. 27.
    Geyer, C.: Markov chain Monte Carlo maximum likelihood. In: Computing Science and Statistics, Proceedings of the 23rd Symposium on the Interface, pp. 156–163. Interface Foundation of North America. (1991)
  28. 28.
    Marinari, E., Parisi, G.: Simulated tempering: a new Monte Carlo scheme. Europhys. Lett. 19(6), 451 (1992)CrossRefGoogle Scholar
  29. 29.
    Guennebaud, G., Jacob, B., et al.: Eigen. (2012)
  30. 30.
    Clauset, A., Shalizi, C.R., Newman, M.E.J.: Power-law distributions in empirical data. SIAM Rev. 51(4), 661–703 (2009)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Baños, A.R., Cruz, A., Fernandez, L.A., Gil-Narvion, J.M., Gordillo-Guerrero, A., Guidetti, M., Maiorano, A., Mantovani, F., Marinari, E., Martin-Mayor, V., Monforte-Garcia, J., Muñoz Sudupe, A., Navarro, D., Parisi, G., Perez-Gaviro, S., Ruiz-Lorenzo, J.J., Schifano, S.F., Seoane, B., Tarancon, A., Tripiccione, R., Yllanes, D.: Nature of the spin-glass phase at experimental length scales. J. Stat. Mech. 2010(06), P06026 (2010)Google Scholar
  32. 32.
    Katzgraber, H.G., Palassini, M., Young, A.P.: Monte carlo simulations of spin glasses at low temperatures. Phys. Rev. B 63(18), 184422 (2001)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute of PhysicsCarl von Ossietzky University of OldenburgOldenburgGermany

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