Frustrated systems: Ground state properties via combinatorial optimization

  • Heiko Rieger
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 501)


An introduction to the application of combinatorial optimization methods to ground state calculations of frustrated, disordered systems is given. We discuss the interface problem in the random bond Ising ferromagnet, the random field Ising model, the diluted antiferromagnet in an external field, the spin glass problem, the solid-on-solid model with a disordered substrte and other convex cost flow problems occurring in superconducting flux line lattices and traffic flow networks. On the algorithmic side we present a concise introduction to a number of elementary algorithms in combinatorial optimization, in particular network flows: the shortest path algorithm, the maximum-flow algorithms and minimum-cost-flow algorithms. We also take a glance at the minimum weighted matching and branch-and-cut algorithms.


Spin Glass Ground State Property Negative Cycle Short Path Distance Residual Network 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Toulouse, G., Commun. Phys. 2 115 (1977).Google Scholar
  2. [2]
    Villain, J., J. Phys. C 10 1717 (1977)ADSGoogle Scholar
  3. [2a]
    Forgacs, G., Phys. Rev. B 22 4473 (1980)ADSMathSciNetGoogle Scholar
  4. [2b]
    Harris, A. B., Kallin, C., Berlinsky, A. J., Phys. Rev. B 45 2899 (1992)ADSGoogle Scholar
  5. [2c]
    Huse D. A., Rutenberg, A. D., Phys. Rev. B 45 7536 (1992)ADSGoogle Scholar
  6. [2d]
    Chalker, J. T., Holdsworth, P. C. W., Shender, E. F., Phys. Rev. Lett. 68 855 (1992)CrossRefADSGoogle Scholar
  7. [2e]
    Shore, J. D., Holzer, M., Sethna, J. P., Phys. Rev. B 46 11376 (1992)ADSGoogle Scholar
  8. [2f]
    Chandra, P., Coleman, P., Ritchey, I., J. Physique I 3 591 (1993)ADSGoogle Scholar
  9. [2g]
    Reimers, J. N., Berlinsky, A. J., Phys. Rev. B 48 9539 (1993).ADSGoogle Scholar
  10. [3]
    The traveling salesman problem, ed.: Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G., Shmoys, D.,B., Wiley-Interscience series in discrete mathematics, (John Wiley & Sons, Chichester, 1985).zbMATHGoogle Scholar
  11. [4]
    Wilson, R. J., Introduction to graph theory, (Oliver and Boyd, Edinburgh, 1972)zbMATHGoogle Scholar
  12. [4a]
    Bondy, J. A., Murty, U. S. R., Graph theory with applications, (Mac Millan, London, 1976).Google Scholar
  13. [5]
    Lawler, E. L., Combinatorial optimization: Networks and matroids, (Holt, Rinehart and Winston, New York, 1976).zbMATHGoogle Scholar
  14. [6]
    Papadimitriou C. H., Steiglitz, K., Combinatorial optimization: Algorithms and complexity, (Prentice-Hall, Englewood Cliffs NJ, 1982).zbMATHGoogle Scholar
  15. [7]
    Chvátal, V., Linear programming, (Freeman, San Francisco, 1983).zbMATHGoogle Scholar
  16. [8]
    Derigs, U., Programming in networks and graphs, Lecture Notes in Economics and Mathematical Systems 300, (Springer-Verlag, Berlin-Heidelberg, 1988).Google Scholar
  17. [9]
    Grötschel, M., Lovász, L., Schrijver, A., Geometric algorithms and combinatorial optimization, (Springer-Verlag, Berlin-Heidelberg, 1988).zbMATHGoogle Scholar
  18. [10]
    Ahuja, R. K., Magnati, T. L., Orlin, J. B., Network Flows, (Prentice Hall, London, 1993).Google Scholar
  19. [11]
    Middleton, A. A., Phys. Rev. E 52 R3337 (1995).Google Scholar
  20. [12]
    Rieger, H., Monte Carlo simulations of Ising spin glasses and random field systems in Annual Reviews of Computational Physics II, p. 295–341, (World Scientific, Singapore, 1995).Google Scholar
  21. [13]
    Barahona, F., J. Phys. A 18 L673 (1985).Google Scholar
  22. [14]
    Ogielski, A. T. Phys. Rev. Lett. 57 1251 (1986).CrossRefADSGoogle Scholar
  23. [15]
    Fishman, S., Aharony, A., J. Phys. C 12 L729 (1979).Google Scholar
  24. [16]
    Hartmann A. K., Usadel, K. D., Physica A 214 141 (1995).ADSGoogle Scholar
  25. [17]
    Esser, J., private communication.Google Scholar
  26. [18]
    Efros. A. L., Shklovskii, B. I., J. Phys. A 8 L49 (1975).Google Scholar
  27. [19]
    Tenelsen, K., Schreiber, M., Phys. Rev. B 49, 12622 (1994); 52, 13287 (1995).ADSGoogle Scholar
  28. [20]
    Barahona, F. J. Phys. A 15, 3241 (1982)ADSMathSciNetGoogle Scholar
  29. [20a]
    Barahona, F., Maynard, R., Rammal, R., Uhry, J. P., J. Phys. A 15, 673 (1982).ADSMathSciNetGoogle Scholar
  30. [21]
    Kawashima N., Rieger, H., submitted to Europhys. Lett., cond-mat/9612116.Google Scholar
  31. [22]
    Grötschel, M., Jünger, M., Reinelt, G., in: Heidelberg Colloqium on Glassy dynamics and Optimization, ed. L. van Hemmen and I. Morgenstern (Springer-Verlag, Heidelberg 1985).Google Scholar
  32. [23]
    De Simone, C., Diehl, M., Jünger, M., Mutzel, P., Reinelt, G., Rinaldi, G., J. Stat. Phys. 80 487 (1995).zbMATHCrossRefADSGoogle Scholar
  33. [24]
    Klotz, T., Kobe, S., J. Phys. A 27 L95 (1994).Google Scholar
  34. [25]
    Diehl, M., Determination of exact ground states of Ising spin glasses with a branch-and-cut algorithm, (Diploma Thesis, Köln, 1995) unpublished, postscript file available at: Scholar
  35. [26]
    Thienel, S., ABACUS — A Branch-And-CUt System, (Ph.D. thesis, Köln, 1995) unpublished, postscript file available at Scholar
  36. [27]
    Bray, A. J., Moore, M. A., in: Heidelberg Colloqium on Glassy dynamics and Optimization, ed. L. van Hemmen and I. Morgenstern (Springer-Verlag, Berlin-Heidelberg 1985).Google Scholar
  37. [28]
    Rieger, H., Santen, L., Blasum, U., Diehl, M., Jünger, M., Rinaldi, G., J. Phys. A 29 3939 (1996).ADSGoogle Scholar
  38. [29]
    Maynard R., Rammal, R., J. Phys. Lett. (France) 43 L347 (1982)Google Scholar
  39. [29a]
    Ozeki, Y., J. Phys. Soc. Jpn. 59 3531 (1990).CrossRefADSGoogle Scholar
  40. [30]
    Shirakura, T., Matsubara, F., J. Phys. Soc. Jpn. 64 2338 (1995)CrossRefADSGoogle Scholar
  41. [30a]
    Ozeki, Y., Nonomura, Y., J. Phys. Soc. Jpn. 64 3128 (1995).CrossRefADSGoogle Scholar
  42. [31]
    Blasum, U., Hochstättler, W., Moll, C., Rieger, H., J. Phys. A 29 L459 (1996).Google Scholar
  43. [32]
    Rieger, H., Blasum, U., Phys. Rev. Lett., in press.Google Scholar
  44. [33]
    Tsai, Y. C., Shapir, Y., Phys. Rev. Lett. 69 1773 (1992); Phys. Rev. E 40 (194) 3546, 4445.CrossRefADSGoogle Scholar
  45. [34]
    See e. g. Chui, T. S., Weeks, J. D., Phys. Rev. B 14 4978 (1976).ADSGoogle Scholar
  46. [35]
    Zeng, C., Middleton, A. A., Shapir, Y., Phys. Rev. Lett. 77 3204 (1996).CrossRefADSGoogle Scholar
  47. [36]
    Kleinert, H., Gauge Fields in Condensed Matters, (World Scientific, Singapore, 1989).Google Scholar
  48. [37]
    Rieger, H., Kisker, J., in preparation.Google Scholar
  49. [38]
    Wengel, C., Young, A. P., Phys. Rev. B 54 R6869 (1996).Google Scholar
  50. [39]
    Bokil, H. S., and Young, A. P., Phys. Rev. Lett. 74 3021 (1995).CrossRefADSGoogle Scholar
  51. [40]
    Traffic and Granular Flow, ed. D. E. Wolf, M. Schreckenberg and A. Bachem (World Scientific, Singapore 1996).Google Scholar
  52. [41]
    Näher, S., Uhrig, C., The LEDA User Manual, Version R3.4 (Martin-Luther Universität Halle-Wittenberg, Germany, 1996), postscript file available at Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Heiko Rieger
    • 1
  1. 1.HLRZ c/o Forschungszentrum JülichJülichGermany

Personalised recommendations