Promise Problems Meet Pseudodeterminism

TL;DR

This paper explores the connection between pseudodeterministic algorithms for acceptance probability estimation and fundamental questions in complexity theory, revealing equivalences with promise class containments and implications for circuit lower bounds.

Contribution

It establishes a fundamental equivalence between pseudodeterministic algorithms for APEP and the relationship between PromiseBPP and BPP, linking algorithmic and complexity class questions.

Findings

Pseudodeterministic APEP algorithms relate to promise class containment.

Equivalence between PromiseBPP in BPP and pseudodeterministic APEP.

Implications for circuit lower bounds and probabilistic hierarchy theorems.

Abstract

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.