The Query Complexity of Witness Finding
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(n ^{2}). This matches an O(n ^{2}) upper bound from the ValiantVazirani 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 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(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 }.
Keywords
Monotone Property Search Problem Query Complexity Boolean Formula Satisfying AssignmentPreview
Unable to display preview. Download preview PDF.
References
 1.Alon, N., Spencer, J.: The Probablistic Method, 3rd edn. Wiley (2008)Google Scholar
 2.Bellare, M., Goldwasser, S.: The complexity of decision versus search. SIAM Journal on Computing 23, 97–119 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
 3.BenDavid, S., Chor, B., Goldreich, O., Luby, M.: On the theory of averagecase complexity. Journal of Computer and System Sciences 44(2), 193–219 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
 4.Bollobás, B., Thomason, A.G.: Threshold functions. Combinatorica 7(1), 35–38 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
 5.Chowdhury, A., Patkos, B.: Shadows and intersections in vector spaces. J. of Combinatorial Theory, Ser. A 117, 1095–1106 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
 6.Cover, T., Thomas, J.: Elements of Information Theory. Wiley Interscience, New York (1991)CrossRefzbMATHGoogle Scholar
 7.Dell, H., Kabanets, V., van Melkebeek, D., Watanabe, O.: Is the ValiantVazirani isolation lemma improvable? In: Proc. 27th Conference on Computational Complexity, pp. 10–20 (2012)Google Scholar
 8.Valiant, L., Vazirani, V.: NP is as easy as detecting unique solutions. Theoretical Computer Science 47, 85–93 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
 9.Yao, A.C.: Probabilistic computations: toward a unified measure of complexity. In: Proc. of the 18th IEEE Sympos. on Foundations of Comput. Sci., pp. 222–227. IEEE (1977)Google Scholar