FP^NP algorithm for Range Avoidance

Determine whether the Range Avoidance problem—given a polynomial-size circuit C: {0,1}^n -> {0,1}^{n+1}, find an output not in the range of C—admits a deterministic polynomial-time algorithm with access to an NP oracle; that is, establish whether Range Avoidance is in FP^NP.

Background

Range Avoidance (Avoid) is a total search problem in TFΣ2^P whose solutions can be verified by a coNP predicate. It has strong connections to derandomization and explicit constructions: a single-valued algorithm for Avoid would yield explicit constructions of hard objects such as rigid matrices and Ramsey graphs.

The best known upper bounds prior to this work place Avoid in FS_2P, while this paper shows Avoid ∈ UEOPL^NP. Whether Avoid admits an FP^NP algorithm remains a central open question with significant implications for circuit lower bounds and explicit constructions.