In my gift to Shafi, I noted a relation between subclasses of the class of search problems that have efficiently recognizable solutions and BPP in its traditional (decision) and promise problem versions. In particular:
The Acceptance Probability Estimation Problem (APEP) is to additively approximate the acceptance probability of a Boolean circuit. This problem admits a probabilistic approximation scheme. A central question is whether we can design a pseudodeterministic approximation algorithm for this problem: a probabilistic polynomial-time algorithm that outputs a canonical approximation with high probability. Recently, it was shown that such an algorithm would imply that every approximation algorithm can be made pseudodeterministic (Dixon, Pavan, Vinodchandran; (ITCS 2021).
The main conceptual contribution of this work is to establish that the existence of a pseudodeterministic algorithm for APEP is fundamentally connected to the relationship between probabilistic promise classes and the corresponding standard complexity classes. In particular, we show the following equivalence: every promise problem in PromiseBPP has a solution in BPP if and only if APEP has a pseudodeterministic algorithm. Based on this intuition, we show that pseudodeterministic algorithms for APEP can shed light on a few central topics in complexity theory such as circuit lowerbounds, probabilistic hierarchy theorems, and multi-pseudodeterminism.
See ECCC TR21-043