Is Valiant-Vazirani's Isolation Probability Improvable?

摘要

The Valiant-Vazirani Isolation Lemma 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 such that (i) every satisfying assignment of C' also satisfies C, and (ii) if C is satisfiable, then C' has exactly one satisfying assignment. In particular, if C is unsatisfiable, then (i) implies that C' is unsatisfiable. The Valiant-Vazirani procedure is randomized, and when C is satisfiable it produces a uniquely satisfiable circuit C' with probability Omega(1). 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 a large constant? We argue that the answer is likely `No'. More precisely, we prove that there exists a non-uniform randomized polynomial-time witness-isolating procedure with success probability bigger than 2/3 if and only if NP is in P/poly. Thus, an improved witness-isolating procedure would imply the collapse of the polynomial-time hierarchy. We establish similar results for other variants of witness isolation, such as reductions that remove all but an odd number of satisfying assignments of a satisfiable circuit. We also consider a black box setting of witness isolation that generalizes the setting of the Valiant-Vazirani Isolation Lemma, and give an upper bound of O(1) on the success probability for a natural class of randomized witness-isolating procedures.