Math @ Duke

Publications [#348674] of Benjamin Rossman
Papers Published
 Kawachi, A; Rossman, B; Watanabe, O, The Query Complexity of Witness Finding,
Theory of Computing Systems, vol. 61 no. 2
(August, 2017),
pp. 305321 [doi]
(last updated on 2022/05/19)
Abstract: We study the following informationtheoretic witness finding problem: for a hidden nonempty subset W of {0,1}n, how many nonadaptive 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 ValiantVazirani 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 searchtodecision reduction of BenDavid, Chor, Goldreich and Luby [3] is optimal in a certain blackbox 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.


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