Skip to main content

The Relative Complexity of Approximate Counting Problems


Two natural classes of counting problems that are interreducible under approximation-preserving reductions are: (i) those that admit a particular kind of efficient approximation algorithm known as an “FPRAS”, and (ii) those that are complete for #P with respect to approximation-preserving reducibility. We describe and investigate not only these two classes but also a third class, of intermediate complexity, that is not known to be identical to (i) or (ii). The third class can be characterised as the hardest problems in a logically defined subclass of #P.

This is a preview of subscription content, access via your institution.

Author information



Corresponding authors

Correspondence to Martin Dyer or Leslie Ann Goldberg or Catherine Greenhill or Mark Jerrum.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Dyer, M., Goldberg, L., Greenhill, C. et al. The Relative Complexity of Approximate Counting Problems. Algorithmica 38, 471–500 (2004).

Download citation

  • Approximate counting
  • Computational complexity