On 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 approximationpreserving 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 work was supported in part by the EPSRC Research Grant “Sharper Analysis of Randomised Algorithms: a Computational Approach” and by the ESPRIT Projects RANDAPX and ALCOM-FT. †[email protected], School of Computer Studies, University of Leeds, Leeds LS2 9JT, United Kingdom. ‡[email protected], http://www.dcs.warwick.ac.uk/∼leslie/, Department of Computer Science, University of Warwick, Coventry, CV4 7AL, United Kingdom. §[email protected], Department of Mathematics and Statistics, University of Melbourne, Parkville VIC, Australia 3052. Supported by an Australian Research Council Postdoctoral Fellowship. ¶[email protected], http://www.dcs.ed.ac.uk/∼mrj/, Division of Informatics, University of Edinburgh, JCMB, The King’s Buildings, Edinburgh EH9 3JZ, United Kingdom.