Abstract
An errorless circuit for a boolean function is one that outputs the correct answer or “don’t know” on each input (and never outputs the wrong answer). The goal of errorless hardness amplification is to show that if f has no size s errorless circuit that outputs “don’t know” on at most a δ fraction of inputs, then some f′ related to f has no size s′ errorless circuit that outputs “don’t know” on at most a 1 − ε fraction of inputs. Thus the hardness is “amplified” from δ to 1 − ε. Unfortunately, this amplification comes at the cost of a loss in circuit size. This is because such results are proven by reductions which show that any size s′ errorless circuit for f′ that outputs “don’t know” on at most a 1 − ε fraction of inputs could be used to construct a size s errorless circuit for f that outputs “don’t know” on at most a δ fraction of inputs. If the reduction makes q queries to the hypothesized errorless circuit for f′, then plugging in a size s′ circuit yields a circuit of size ≥ qs′, and thus we must have s′ ≤ s/q. Hence it is desirable to keep the query complexity to a minimum. The first results on errorless hardness amplification were obtained by Bogdanov and Safra (FOCS 2007). They achieved query complexity \(O\big((\frac{1}{\delta}\log\frac{1}{\epsilon})^2\cdot\frac{1}{\epsilon}\log\frac{1}{\delta}\big)\) when f′ is the XOR of several independent copies of f. We improve the query complexity (and hence the loss in circuit size) to \(O\big(\frac{1}{\epsilon}\log\frac{1}{\delta}\big)\), which is optimal up to constant factors for nonadaptive black-box errorless hardness amplification. Bogdanov and Safra also proved a result that allows for errorless hardness amplification within NP. They achieved query complexity \(O\big(k^3\cdot\frac{1}{\epsilon^2}\log\frac{1}{\delta}\big)\) when f′ consists of any monotone function applied to the outputs of k independent copies of f, provided the monotone function satisfies a certain combinatorial property parameterized by δ and ε. We improve the query complexity to \(O\big(\frac{k}{t}\cdot\frac{1}{\epsilon}\log\frac{1}{\delta}\big)\), where t ≥ 1 is a certain parameter of the monotone function.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Artemenko, S., Shaltiel, R.: Lower Bounds on the Query Complexity of Non-Uniform and Adaptive Reductions Showing Hardness Amplification. In: To appear in Proceedings of the 15th International Workshop on Randomization and Computation (2011)
Bogdanov, A., Safra, M.: Hardness Amplification for Errorless Heuristics. In: Proceedings of the 48th IEEE Symposium on Foundations of Computer Science, pp. 418–426 (2007)
Goldreich, O., Nisan, N., Wigderson, A.: On Yao’s XOR Lemma. Electronic Colloquium on Computational Complexity, Technical Report TR95-050 (1995)
Healy, A., Vadhan, S., Viola, E.: Using Nondeterminism to Amplify Hardness. SIAM Journal on Computing 35(4), 903–931 (2006)
Impagliazzo, R.: Hard-Core Distributions for Somewhat Hard Problems. In: Proceedings of the 36th IEEE Symposium on Foundations of Computer Science, pp. 538–545 (1995)
Klivans, A., Servedio, R.: Boosting and Hard-Core Sets. Machine Learning 53(3), 217–238 (2003)
Levin, L.: Average Case Complete Problems. SIAM Journal on Computing 15(1), 285–286 (1986)
Levin, L.: One-Way Functions and Pseudorandom Generators. Combinatorica 7(4), 357–363 (1987)
O’Donnell, R.: Hardness Amplification Within NP. Journal of Computer and System Sciences 69(1), 68–94 (2004)
Shaltiel, R., Viola, E.: Hardness Amplification Proofs Require Majority. SIAM Journal on Computing 39(7), 3122–3154 (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Watson, T. (2011). Query Complexity in Errorless Hardness Amplification. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2011 2011. Lecture Notes in Computer Science, vol 6845. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22935-0_58
Download citation
DOI: https://doi.org/10.1007/978-3-642-22935-0_58
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22934-3
Online ISBN: 978-3-642-22935-0
eBook Packages: Computer ScienceComputer Science (R0)