The Query Complexity of Witness Finding [chapter]

Akinori Kawachi, Benjamin Rossman, Osamu Watanabe
2014 Lecture Notes in Computer Science  
We study the following information-theoretic witness finding problem: for a hidden nonempty subset W of {0, 1} n , how many nonadaptive randomized queries (yes/no questions about W ) are required to guess an element x ∈ {0, 1} n such that x ∈ W with probability > 1/2? Motivated by questions in complexity theory, our results are 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
more » ... upper bound coming from the Valiant-Vazirani Isolation Lemma [8]. • We also give a tight Ω(n 2 ) lower bound for the class of NP queries (queries defined by NP machines 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 . 1 We initiate a study of the following information-theoretic search problem, parameterized by a family W of subsets of {0, 1} n and a family Q of functions W → { , ⊥} (i.e. yes/no questions about elements of W, which we refer to as "queries"). Question 1.1. What is the fewest number of nonadaptive randomized queries in Q required to guess an element x ∈ {0, 1} n such that P[x ∈ W ] > 1/2 for every nonempty W ∈ W? Formally, Question 1.1 asks for a joint distribution (Q 1 , . . . , Q m ) on Q m together with a function f : { , ⊥} m → {0, 1} n such that P[f (Q 1 (W ), . . . , Q m (W )) ∈ W ] > 1/2 for every nonempty W ∈ W. Note that randomized queries Q 1 , . . . , Q m are nonadaptive, though not necessarily independent. 1 We refer to Question 1.1 as the witness finding problem and to its answer, m = m(W, Q), as the Q-query complexity of W-witness finding. (We introduce the terminology "witness finding" to distinguish this information-theoretic problem from traditional computational search problems where the solution space is determined by an input, such as a boolean formula ϕ in the case of the search problem for SAT.) Note that m(W, Q) is monotone increasing with respect to W and monotone decreasing with respect to Q. In this paper, we mainly study the setting where W is the set of all subsets of {0, 1} n . Here, to simplify notation, we simply write m(Q) and speak of the Q-query complexity of witness finding. Our main results are tight lower bounds on m(Q) for a few specific classes of queries (namely, intersection queries, monotone queries and NP queries). However, before defining these classes and stating our results formally, let us first dispense with the trivial cases where Q is the class All of all possible queries or the class Direct of direct queries of the form "x ∈ W ?" where x ∈ {0, 1} n . It is easy to see that m(All) = n and m(Direct) = 2 n − 1. Both lower bounds m(All) ≥ n and m(Direct) ≥ 2 n − 1 follow from considering the random singleton witness set {x} where x is uniform in {0, 1} n . The upper bound m(Direct) ≤ 2 n − 1 is obvious, while the upper bound m(All) ≤ n comes via deterministic queries Q 1 , . . . , Q n where Q i (W ) asks for the ith coordinate in the lexicographically minimal element of W .
doi:10.1007/978-3-319-06686-8_17 fatcat:pgbbrcawmvdo7bqc27hnokcc5m