Is the Valiant-Vazirani Isolation Lemma Improvable?

The Valiant-Vazirani Isolation Lemma [TCS, vol. 47, pp. 85–93, 1986] provides an efficient procedure for isolating a satisfying assignment of a given satisfiable circuit: given a Boolean circuit C on n input variables, the procedure outputs a new circuit C ′ on the same n input variables with the property that the set of satisfying assignments for C ′ is a subset of those for C, and moreover, if C is satisfiable then C ′ has exactly one satisfying assignment. The Valiant-Vazirani procedure is randomized, and it produces a uniquely satisfiable circuit C ′ with probability Ω(1/n). Is it possible to have an efficient deterministic witness-isolating procedure? Or, at least, is it possible to improve the success probability of a randomized procedure to Ω(1)? We argue that the answer is likely ‘No’. More precisely, we prove that 1. a non-uniform deterministic polynomial-time witness-isolating procedure exists if and only if NP ⊆ P/poly, and 2. if there is a randomized polynomial-time witness-isolating procedure with success probability bigger than 2/3, then coNP ⊆ NP/poly. Thus, an improved witness-isolating procedure would imply the collapse of the Polynomial-Time Hierarchy. Finally, we consider a black-box setting of witness isolation (generalizing the setting of the Valiant-Vazirani Isolation Lemma), and give the upper bound O(1/n) on the success probability for a natural class of randomized witness-isolating procedures.