Is the Valiant-Vazirani Isolation Lemma Improvable?

摘要

The Isolation Lemma of Valiant and Vazirani (1986) provides an efficient procedure for isolating a satisfying assignment of a givensatisfiable circuit: Given a Boolean circuit C on n input variables, the procedure outputs a new circuit C' on the same n input variablessuch 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 successprobability of a randomized procedure to a large constant? 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 has polynomial-size circuits. We establish similarresults 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 blackbox 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.