The Query Complexity of Witness Finding

Abstract

AbstractWe study the following information-theoretic witness finding problem: for a hidden nonempty subset W of {0,1}n, how many non-adaptive randomized queries (yes/no questions about W) are needed to guess an element x ∈ {0,1}n such that x ∈ W with probability > 1/2? Motivated by questions in complexity theory, we prove tight lower bounds with respect to a few different classes of queries: We show that the monotone query complexity of witness finding is Ω(n 2). This matches an O(n 2) upper bound from the Valiant-Vazirani Isolation Lemma [8]. We also prove a tight Ω(n 2) lower bound for the class of NP queries (queries defined by an NP machine with an oracle to W). This shows that the classic search-to-decision reduction of Ben-David, Chor, Goldreich and Luby [3] is optimal in a certain black-box model. Finally, we consider the setting where W is an affine subspace of {0,1}n and prove an Ω(n 2) lower bound for the class of intersection queries (queries of the form “W ∩ S ≠ ∅?” where S is a fixed subset of {0,1}n). Along the way, we show that every monotone property defined by an intersection query has an exponentially sharp threshold in the lattice of affine subspaces of {0,1}n. KeywordsMonotone PropertySearch ProblemQuery ComplexityBoolean FormulaSatisfying AssignmentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.