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A Composition Theorem for Randomized Query Complexity via Max-Conflict Complexity

Authors Dmitry Gavinsky, Troy Lee, Miklos Santha, Swagato Sanyal

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Author Details

Dmitry Gavinsky
  • Institute of Mathematics, Czech Academy of Sciences, 115 67 Žitna 25, Praha 1, Czech Republic
Troy Lee
  • Centre for Quantum Software and Information, Faculty of Engineering and Information Technology, University of Technology Sydney, Australia
Miklos Santha
  • CNRS, IRIF, Université de Paris, 75205 Paris, France
  • Centre for Quantum Technologies, National University of Singapore, Singapore 117543
  • MajuLab, UMI 3654, Singapore
Swagato Sanyal
  • Indian Institute of Technology Kharagpur, India

Funding

Part of this work was conducted while T.L. and S.S. were at the Nanyang Technological University and the Centre for Quantum Technologies, supported by the Singapore National Research Foundation under NRF RF Award No. NRF-NRFF2013-13. This work was additionally supported by the Singapore National Research Foundation, the Prime Minister’s Office, Singapore and the Ministry of Education, Singapore under the Research Centres of Excellence programme under research grant R 710-000-012-135. This research was supported in part by the QuantERA ERA-NET Cofund project QuantAlgo. D.G. is partially funded by the grant 19-27871X of GA ČR.

Acknowledgements

We thank Rahul Jain for useful discussions. We thank Srijita Kundu and Jevgēnijs Vihrovs for their helpful comments on the manuscript. We thank Yuval Filmus for suggesting to look at the min-max version of conflict complexity, which led to the development of max-conflict complexity. We thank the anonymous reviewers for their helpful comments.

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Dmitry Gavinsky, Troy Lee, Miklos Santha, and Swagato Sanyal. A Composition Theorem for Randomized Query Complexity via Max-Conflict Complexity. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 64:1-64:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.64

Abstract

For any relation f subseteq {0,1}^n x S and any partial Boolean function g:{0,1}^m -> {0,1,*}, we show that R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * sqrt{R_{1/3}(g)}) , where R_epsilon(*) stands for the bounded-error randomized query complexity with error at most epsilon, and f o g^n subseteq ({0,1}^m)^n x S denotes the composition of f with n instances of g.
The new composition theorem is optimal, at least, for the general case of relational problems: A relation f_0 and a partial Boolean function g_0 are constructed, such that R_{4/9}(f_0) in Theta(sqrt n), R_{1/3}(g_0)in Theta(n) and R_{1/3}(f_0 o g_0^n) in Theta(n).
The theorem is proved via introducing a new complexity measure, max-conflict complexity, denoted by bar{chi}(*). Its investigation shows that bar{chi}(g) in Omega(sqrt{R_{1/3}(g)}) for any partial Boolean function g and R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * bar{chi}(g)) for any relation f, which readily implies the composition statement. It is further shown that bar{chi}(g) is always at least as large as the sabotage complexity of g.

Subject Classification

ACM Subject Classification
  • Theory of computation → Oracles and decision trees
Keywords
  • query complexity
  • lower bounds

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References

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