Journal of Statistical Physics

, Volume 48, Issue 1–2, pp 121–134 | Cite as

Two-dimensional monomer-dimer systems are computationally intractable

  • Mark Jerrum


The classic problem of counting monomer-dimer arrangements on a two-dimensional lattice is analyzed using techniques from theoretical computer science. Under a certain assumption, made precise in the text, it can be shown that the general problem is computationally intractable. This negative result contrasts with the special case of a system with monomer density zero, for which efficient solutions have been known for some time. A second, much easier result, obtained under the same assumption, is that the partition function of a three-dimensional Ising system is computationally intractable. Again, the negative result contrasts with known efficient techniques for evaluating the partition function of a two-dimensional system.

Key words

Computational complexity Ising model monomer-dimer system #P-completeness 


  1. 1.
    M. E. Fisher, Statistical mechanics of dimers on a plane lattice,Phys. Rev. 124:1664–1672 (1961).Google Scholar
  2. 2.
    M. E. Fisher, On the dimer solution of planar Ising models,J. Math. Phys. 7:1776–1781 (1966).Google Scholar
  3. 3.
    R. H. Fowler and G. S. Rushbrooke, Statistical theory of perfect solutions,Trans. Faraday Soc. 33:1272–1294 (1937).Google Scholar
  4. 4.
    M. R. Garey and D. S. Johnson,Computers and Intractability—A Guide to the Theory of NP-Completeness (Freeman, 1979).Google Scholar
  5. 5.
    O. J. Heilmann and E. H. Lieb, Theory of monomer-dimer systems,Commun. Math. Phys. 25:190–232 (1972).Google Scholar
  6. 6.
    J. E. Hopcroft and J. D. Ullman,Introduction to Automata Theory, Languages and Computation (Addison-Wesley, 1979).Google Scholar
  7. 7.
    M. R. Jerrum, The complexity of evaluating multivariate polynomials, Ph. D. Thesis CST-11–81, Department of Computer Science, University of Edinburgh, (1981).Google Scholar
  8. 8.
    P. W. Kasteleyn, Dimer statistics and phase transitions,J. Math. Phys. 4:287–293 (1963).Google Scholar
  9. 9.
    P. W. Kasteleyn, Graph theory and crystal physics, inGraph Theory and Theoretical Physics, F. Harary, ed. (Academic Press, 1967), pp. 43–110.Google Scholar
  10. 10.
    J. K. Percus,Combinatorial Methods (Applied Mathematical Sciences 4, Springer-Verlag, 1971).Google Scholar
  11. 11.
    J. E. Savage,The Complexity of Computing (Wiley, 1976).Google Scholar
  12. 12.
    H. N. V. Temperley and M. E. Fisher, Dimer problem in statistical mechanics—An exact result,Phil. Mag. 6:1061–1063 (1961).Google Scholar
  13. 13.
    L. G. Valiant, The complexity of computing the permanent,Theor. Computer Sci. 8:189–201 (1979).Google Scholar
  14. 14.
    L. G. Valiant, The complexity of enumeration and reliability problems,SIAM J. Computing 8: 410–421 (1979).Google Scholar

Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

  • Mark Jerrum
    • 1
  1. 1.Department of Computer ScienceUniversity of EdinburghEdinburghScotland

Personalised recommendations