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Erschienen in:
19.09.2016
The Query Complexity of Witness Finding
Erschienen in: Theory of Computing Systems | Ausgabe 2/2017
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Abstract
We 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 “\(W \cap S \ne \emptyset \)?” 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
.