Complexity separation between approximate sampling and perfect sampling

Determine whether a complexity-theoretic separation exists between approximate sampling and perfect sampling, either for specific problems or in worst-case complexity. Specifically, ascertain whether there exists a distribution that admits an efficient fully polynomial almost uniform sampler (FPAUS) while perfect sampling for that distribution is computationally hard (e.g., NP-hard), or establish that no such separation exists.

Background

The paper surveys the relationship between approximate sampling and perfect sampling, noting that many results connect approximate counting and sampling for self-reducible problems, while perfect sampling has historically required stronger tools. Despite substantial progress on problem-specific perfect sampling algorithms, there is no established complexity-theoretic separation between approximate and perfect sampling, unlike the well-understood separation between approximate and exact counting.

The authors’ results show that, under mild conditions, approximate samplers (e.g., rapidly mixing Markov chains) can be converted into perfect samplers with only small overhead. Nonetheless, whether an inherent complexity gap exists—such as a problem with an efficient approximate sampler but intractable perfect sampler—remains unknown.