Journal of Statistical Physics

, Volume 48, Issue 1, pp 121–134

Two-dimensional monomer-dimer systems are computationally intractable

  • Mark Jerrum

DOI: 10.1007/BF01010403

Cite this article as:
Jerrum, M. J Stat Phys (1987) 48: 121. doi:10.1007/BF01010403


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 

Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

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

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