The Query Complexity of Witness Finding

Journal Article

© 2016, Springer Science+Business Media New York. 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(n2). This matches an O(n2) upper bound from the Valiant-Vazirani Isolation Lemma [8].•We also prove a tight Ω(n2) 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(n2) 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.

Full Text

Duke Authors

Cited Authors

  • Kawachi, A; Rossman, B; Watanabe, O

Published Date

  • August 1, 2017

Published In

Volume / Issue

  • 61 / 2

Start / End Page

  • 305 - 321

Electronic International Standard Serial Number (EISSN)

  • 1433-0490

International Standard Serial Number (ISSN)

  • 1432-4350

Digital Object Identifier (DOI)

  • 10.1007/s00224-016-9708-y

Citation Source

  • Scopus