Nearly Bounded Error Probabilistic Sets

Abstract

We study polynomial-time randomized algorithms that solve problems on “most” inputs with “small” error probability. The sets that have such algorithms are called nearly BPP sets, which naturally expand BPP sets. Notably, sparse sets and average BPP sets are typical examples of nearly BPP sets. It is, however, open whether all NP sets are nearly BPP. The nearly BPP sets can be captured by Nisan-Wigderson’s approximation scheme as well as viewed as a special case of promise BPP problems. Moreover, nearly BPP sets are precisely described in terms of Sipser’s distinguishing complexity. These sets have a connection to average-case complexity and cryptography. Nevertheless, unlike BPP, the class of nearly BPP sets is not closed even under honest polynomial-time one-one reductions. In this paper, we study a more general notion of nearly ${\text{BP}}\left[ \mathcal{C} \right]$ sets, analogous to Schöning’s probabilistic class ${\text{BP}}\left[ \mathcal{C} \right]$ for any complexity class $\mathcal{C}$ . The “infinitely-often” version of nearly BPP sets shows a direct connection to cryptographic one-way partial functions.