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# Hitting Sets Give Two-Sided Derandomization of Small Space

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LIPIcs.CCC.2020.10.pdf
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## Acknowledgements

We thank David Zuckerman, Dean Doron, and Pooya Hatami for helpful discussions and for comments on drafts of this paper. We thank Adam Klivans for bringing his work with Gopalan and Meka [Parikshit Gopalan et al., 2012] to our attention.

## Cite As

Kuan Cheng and William M. Hoza. Hitting Sets Give Two-Sided Derandomization of Small Space. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 10:1-10:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.CCC.2020.10

## Abstract

A hitting set is a "one-sided" variant of a pseudorandom generator (PRG), naturally suited to derandomizing algorithms that have one-sided error. We study the problem of using a given hitting set to derandomize algorithms that have two-sided error, focusing on space-bounded algorithms. For our first result, we show that if there is a log-space hitting set for polynomial-width read-once branching programs (ROBPs), then not only does 𝐋 = 𝐑𝐋, but 𝐋 = 𝐁𝐏𝐋 as well. This answers a question raised by Hoza and Zuckerman [William M. Hoza and David Zuckerman, 2018]. Next, we consider constant-width ROBPs. We show that if there are log-space hitting sets for constant-width ROBPs, then given black-box access to a constant-width ROBP f, it is possible to deterministically estimate 𝔼[f] to within ± ε in space O(log(n/ε)). Unconditionally, we give a deterministic algorithm for this problem with space complexity O(log² n + log(1/ε)), slightly improving over previous work. Finally, we investigate the limits of this line of work. Perhaps the strongest reduction along these lines one could hope for would say that for every explicit hitting set, there is an explicit PRG with similar parameters. In the setting of constant-width ROBPs over a large alphabet, we prove that establishing such a strong reduction is at least as difficult as constructing a good PRG outright. Quantitatively, we prove that if the strong reduction holds, then for every constant α > 0, there is an explicit PRG for constant-width ROBPs with seed length O(log^{1 + α} n). Along the way, unconditionally, we construct an improved hitting set for ROBPs over a large alphabet.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Pseudorandomness and derandomization
• Theory of computation → Complexity classes
##### Keywords
• hitting sets
• derandomization

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