Intermediate witness-isolation success scales

Determine whether randomized nonuniform polynomial-size pruning transformations can achieve witness-isolation success probabilities strictly between the affine-hashing scale \(\Theta(1/n)\) and the inverse-logarithmic scale \(O(1/\log n)\), specifically in the regime \(p_{\mathrm{wit}}(n)=\omega(1/n)\) and \(p_{\mathrm{wit}}(n)=o(1/\log n)\), without implying \(\mathrm{NP}\subseteq\mathrm{P}\).

Background

The paper studies randomized nonuniform pruning procedures that retain a subset of a circuit’s satisfying assignments and aim to retain exactly one witness. Affine hashing provides a general success guarantee of order $\Theta(1/n)$, where $n$ is the witness length. The paper proves that success of order $1/\log L$, measured using the circuit-description length $L$, already yields a complexity collapse $\mathrm{NP}\subseteq\mathrm{P}$.

Consequently, the authors leave unresolved the intermediate regime between these two scales, including success probabilities that grow asymptotically faster than $1/n$ but decay asymptotically faster than $1/\log n$. The open range concerns whether such guarantees can exist under the assumption $\mathrm{NP}\not\subseteq\mathrm{P}$.