Complexity separation between approximate and perfect sampling

Determine whether there exists a computational complexity separation between approximate sampling and perfect sampling, either for specific problems or in terms of worst-case complexity. Concretely, ascertain whether there are problem families that admit efficient approximate sampling algorithms but for which perfect sampling is computationally harder, or prove that no such separation exists.

Background

The paper surveys the relationships between counting, approximate sampling, and perfect sampling, noting that there are well-understood separations between exact and approximate counting but no analogous separations are known between approximate and perfect sampling. Establishing such a separation (or proving none exists) would clarify the relative power of approximate versus perfect sampling and address a central complexity-theoretic question raised by the results in the paper.

The authors’ main results show broad black-box reductions from approximate to perfect sampling under mild conditions, strengthening the belief that perfect sampling may not be harder than approximate sampling. However, the absence of proven separations remains a fundamental unresolved issue.