The Query Complexity of Witness Finding

Akinori Kawachi; Benjamin Rossman; Osamu Watanabe

doi:10.1007/s00224-016-9708-y

Theory of Computing Systems

August 2017, Volume 61, Issue 2, pp 305–321 | Cite as

The Query Complexity of Witness Finding

Authors
Authors and affiliations

Akinori Kawachi
Benjamin RossmanEmail author
Osamu Watanabe

Article

First Online: 19 September 2016

139 Downloads

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.

Keywords

Query complexity Witness finding problem Search-to-decision reduction

This is a preview of subscription content, to check access.

Notes

Acknowledgments

We thank Oded Goldreich for feedback on an earlier manuscript. We are also grateful to the anonymous reviewers for their detailed and extremely helpful comments.

Appendix A: Proof of Lemma 1

In order to apply Yao’s minimax principle [8], we express $m(\mathcal W,\mathcal Q)$ in terms of a particular matrix M. Let $\mathcal {F}$ be the set of functions {⊤,⊥}^m→{0,1}ⁿ. Let $\mathcal {A} := \mathcal {Q}^{m} \times \mathcal {F}$ (representing the set of deterministic witness finding algorithms). Let $\mathcal {W}_{0} := \mathcal {W} \setminus \{\emptyset \}$. Finally, let M be the $\mathcal {A} \times \mathcal {W}_{0}$-matrix defined by

$$M_{(Q_{1},\dots,Q_{m};f),W} := \left\{\begin{array}{ll} 1 &\text{if } f(Q_{1}(W),\dots,Q_{m}(W)) \in W,\\ 0 &\text{otherwise}. \end{array}\right. $$

In this context, Yao’s minimax principle states that for all random variables W on $\mathcal W_{0}$ and $(\mathbf {Q}_{1},\dots ,\mathbf {Q}_{m};\mathbf {f})$ on $\mathcal {A}$,

$$\min_{(Q_{1},\dots,Q_{m};f) \in \mathcal A} \mathbb{E}[M_{(Q_{1},\dots,Q_{m};f),\mathbf{W}}] \le \max_{W \in \mathcal W_{0}} \mathbb{E}[M_{(\mathbf{Q}_{1},\dots,\mathbf{Q}_{m};\mathbf f),W}]. $$

It follows that, if $\mathbb {P}[f(Q_{1}(\mathbf {W}),\dots ,Q_{m}(\mathbf {W})) \in \mathbf {W}] \le 1/2$ for all $Q_{1},\dots ,Q_{m} \in \mathcal Q$ and every function f:{⊤,⊥}^m→{0,1}ⁿ, then for all $(\mathbf {Q}_{1},\dots ,\mathbf {Q}_{m};\mathbf {f}) \in \mathcal {A}$ (including the special case where f is deterministic, as in the definition of witness finding procedures), there exists $W \in \mathcal {W}_{0}$ such that $\mathbb {P}[\mathbf {f}(\mathbf {Q}_{1}(W),\dots ,\mathbf {Q}_{m}(W)) \in W] \le 1/2$. Therefore, the $\mathcal {Q}$-query complexity of $\mathcal {W}$-witness finding is greater than m.

B: Proof of Lemma 2

For inequality (1), let $\mathbf Y_{1},\dots ,\mathbf Y_{2^{i}}$ be independent copies of W _𝜃−i. For each x∈{0,1}ⁿ, we have

$$\mathbb{P}[x \in (\mathbf Y_{1} \cup {\dots} \cup \mathbf Y_{2^{i}})] = 1 - (1 - 2^{\theta-i-n})^{2^{i}} < 2^{\theta-n} = \mathbb{P}[x \in \mathbf{W}_{\theta}]. $$

Moreover, events $\{x \in \mathbf Y_{1} \cup {\dots } \cup \mathbf Y_{2^{i}}\}$ are independent over x∈{0,1}ⁿ. Thus, $\mathbf {Y}_{1} \cup {\dots } \cup \mathbf {Y}_{2^{i}}$ and W _𝜃 are both product distributions on the lattice of subsets of {0,1}ⁿ with biases $p = 1 - (1 - 2^{\theta -i-n})^{2^{i}}$ and $q = 2^{\theta _{n}}$ respectively. Since $p < q, \mathbf {Y}_{1} \cup {\dots } \cup \mathbf {Y}_{2^{i}}$ is stochastically dominated by W _𝜃. (That is, $\mathbb {E}[f(\mathbf {Y}_{1} \cup {\dots } \cup \mathbf {Y}_{2^{i}})] \le \mathbb {E}[f(\mathbf {W}_{\theta })]$ for every monotone increasing function f of subsets of {0,1}ⁿ). Since Q is a monotone query, we have

$$\mathbb{P}[Q(\mathbf Y_{1}) \vee {\dots} \vee Q(\mathbf Y_{2^{i}})] \le \mathbb{P}[Q(\mathbf Y_{1} \cup {\dots} \cup \mathbf Y_{2^{i}})] \le \mathbb{P}[Q(\mathbf{W}_{\theta})]. $$

By independence of $\mathbf Y_{1},\dots ,\mathbf Y_{2^{i}}$, it follows that

$$\begin{array}{@{}rcl@{}} 1/2 \ge \mathbb{P}[Q(\mathbf{W}_{\theta})] \ge \mathbb{P}[\textstyle\bigvee_{j=1}^{2^{i}} Q(\mathbf Y_{j})] = 1 - \mathbb{P}[\neg Q(\mathbf{W}_{\theta-i})]^{2^{i}} = 1 - (1 - p_{\theta-i})^{2^{i}}. \end{array} $$

Therefore, $p_{\theta -i} \le 1 - (1/2)^{1/2^{i}} < (\ln 2)/2^{i}$.

For inequality (2), let $\mathbf Z_{1},\dots ,\mathbf Z_{2^{i}}$ be independent copies of W _𝜃+1. By a similar argument, we have

$$p_{\theta+i+1} = \mathbb{P}[Q(\mathbf{W}_{\theta+i+1})] \ge \mathbb{P}[{\textstyle\bigvee_{j=1}^{2^{i}}} Q(\mathbf Z_{j})] = 1 - \mathbb{P}[\neg Q(\mathbf{W}_{\theta+1})]^{2^{i}} > 1 - \frac{1}{2^{2^{i}}}. $$

Finally, for inequality (3), note that for all p, q∈[0,1],

$$0 \le \min(p,1-p) \le q \le 1/2 \,\Longrightarrow\, H(p) \le H(q) \le 2q\log(1/q). $$

By this observation, together with (1) and (2), we have

$$H(p_{\theta-i-1}) \le 2\frac{\ln 2}{2^{i+1}}\log(\frac{2^{i+1}}{\ln 2}) < \frac{i+2}{2^{i}},\! \quad H(p_{\theta+i+1}) \le 2\frac{1}{2^{2^{i}}}\log(2^{2^{i}}) = \frac{1}{2^{2^{i} - i - 1}}. $$

From these two inequalities, it follows that H(p _k)≤(|𝜃−k|+1)/2^{|𝜃−k|−1}.

References

1.
Alon, N., Spencer, J.: The Probablistic Method, 3rd edn. Wiley (2008)Google Scholar
2.
Ben-David, S., Chor, B., Goldreich, O., Luby, M.: On the theory of average-case complexity. J. Comput. Syst. Sci. 44(2), 193–219 (1992)MathSciNetCrossRefMATHGoogle Scholar
3.
Bollobás, B., Thomason, A.G.: Threshold functions. Combinatorica 7(1), 35–38 (1987)MathSciNetCrossRefMATHGoogle Scholar
4.
Chowdhury, A., Patkos, B.: Shadows and intersections in vector spaces. J. Comb. Theory Ser. A 117, 1095–1106 (2010)MathSciNetCrossRefMATHGoogle Scholar
5.
Cover, T., Thomas, J.: Elements of Information Theory. Wiley-Interscience, New York, NY (1991)CrossRefMATHGoogle Scholar
6.
Dell, H., Kabanets, V., Van Melkebeek, D., Watanabe, O.: Is the Valiant-Vazirani isolation lemma improvable?. In: Proceedings 27th Conference on Computational Complexity, pp 10–20 (2012)Google Scholar
7.
Valiant, L., Vazirani, V.: NP is as easy as detecting unique solutions. Theor. Comput. Sci. 47, 85–93 (1986)MathSciNetCrossRefMATHGoogle Scholar
8.
Yao, A.C.: Probabilistic computations: toward a unified measure of complexity. Proceedings of the 18th IEEE Symposium on Foundations of Computer Science, IEEE, 222–227 (1977)Google Scholar

Copyright information

Authors and Affiliations

Akinori Kawachi
- 1
Benjamin Rossman
- 2
Email author
Osamu Watanabe
- 1

1.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyMeguro-kuJapan
2.National Institute of InformaticsChiyoda-kuJapan

The Query Complexity of Witness Finding

Abstract

Keywords

Notes

Acknowledgments

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article