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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

In a recent work, Chen, Hoza, Lyu, Tal and Wu (FOCS 2023) showed an improved error reduction framework for the derandomization of regular read-once branching programs (ROBPs). Their result is based on a clever modification to the inverse Laplacian perspective of space-bounded derandomization, which was originally introduced by Ahmadinejad, Kelner, Murtagh, Peebles, Sidford and Vadhan (FOCS 2020).
In this work, we give an alternative error reduction framework for regular ROBPs. Our new framework is based on a binary recursive formula from the work of Chattopadhyay and Liao (CCC 2020), that they used to construct weighted pseudorandom generators (WPRGs) for general ROBPs.
Based on our new error reduction framework, we give alternative proofs to the following results for regular ROBPs of length n and width w, both of which were proved in the work of Chen et al. using their error reduction:
- There is a WPRG with error ε that has seed length Õ(log(n)(√{log(1/ε)}+log(w))+log(1/ε)).
- There is a (non-black-box) deterministic algorithm which estimates the expectation of any such program within error ±ε with space complexity Õ(log(nw)⋅log log(1/ε)). This was first proved in the work of Ahmadinejad et al., but the proof by Chen et al. is simpler. Because of the binary recursive nature of our new framework, both of our proofs are based on a straightforward induction that is arguably simpler than the Laplacian-based proof in the work of Chen et al.
In fact, because of its simplicity, our proof of the second result directly gives a slightly stronger claim: our algorithm computes a ε-singular value approximation (a notion of approximation introduced in a recent work by Ahmadinejad, Peebles, Pyne, Sidford and Vadhan (FOCS 2023)) of the random walk matrix of the given ROBP in space Õ(log(nw)⋅log log(1/ε)). It is not clear how to get this stronger result from the previous proofs.

Eshan Chattopadhyay and Jyun-Jie Liao. Recursive Error Reduction for Regular Branching Programs. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chattopadhyay_et_al:LIPIcs.ITCS.2024.29, author = {Chattopadhyay, Eshan and Liao, Jyun-Jie}, title = {{Recursive Error Reduction for Regular Branching Programs}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {29:1--29:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.29}, URN = {urn:nbn:de:0030-drops-195571}, doi = {10.4230/LIPIcs.ITCS.2024.29}, annote = {Keywords: read-once branching program, regular branching program, weighted pseudorandom generator, derandomization} }

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**Published in:** LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

In a recent work, Gryaznov, Pudlák and Talebanfard (CCC '22) introduced a linear variant of read-once branching programs, with motivations from circuit and proof complexity. Such a read-once linear branching program is a branching program where each node is allowed to make 𝔽₂-linear queries, and is read-once in the sense that the queries on each path is linearly independent. As their main result, they constructed an explicit function with average-case complexity 2^{n/3-o(n)} against a slightly restricted model, which they call strongly read-once linear branching programs. The main tool in their lower bound result is a new type of extractor, called directional affine extractors, that they introduced.
Our main result is an explicit function with 2^{n-o(n)} average-case complexity against the strongly read-once linear branching program model, which is almost optimal. This result is based on a new connection from this problem to sumset extractors, which is a randomness extractor model introduced by Chattopadhyay and Li (STOC '16) as a generalization of many other well-studied models including two-source extractors, affine extractors and small-space extractors. With this new connection, our lower bound naturally follows from a recent construction of sumset extractors by Chattopadhyay and Liao (STOC '22). In addition, we show that directional affine extractors imply sumset extractors in a restricted setting. We observe that such restricted sumset sources are enough to derive lower bounds, and obtain an arguably more modular proof of the lower bound by Gryaznov, Pudlák and Talebanfard.
We also initiate a study of pseudorandomness against linear branching programs. Our main result here is a hitting set generator construction against regular linear branching programs with constant width. We derive this result based on a connection to Kakeya sets over finite fields.

Eshan Chattopadhyay and Jyun-Jie Liao. Hardness Against Linear Branching Programs and More. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 9:1-9:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2023.9, author = {Chattopadhyay, Eshan and Liao, Jyun-Jie}, title = {{Hardness Against Linear Branching Programs and More}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {9:1--9:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-282-2}, ISSN = {1868-8969}, year = {2023}, volume = {264}, editor = {Ta-Shma, Amnon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.9}, URN = {urn:nbn:de:0030-drops-182794}, doi = {10.4230/LIPIcs.CCC.2023.9}, annote = {Keywords: linear branching programs, circuit lower bound, sumset extractors, hitting sets} }

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**Published in:** LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

In a seminal work, Nisan (Combinatorica'92) constructed a pseudorandom generator for length n and width w read-once branching programs with seed length O(log n⋅ log(nw)+log n⋅log(1/ε)) and error ε. It remains a central question to reduce the seed length to O(log (nw/ε)), which would prove that 𝐁𝐏𝐋 = 𝐋. However, there has been no improvement on Nisan’s construction for the case n = w, which is most relevant to space-bounded derandomization.
Recently, in a beautiful work, Braverman, Cohen and Garg (STOC'18) introduced the notion of a pseudorandom pseudo-distribution (PRPD) and gave an explicit construction of a PRPD with seed length Õ(log n⋅ log(nw)+log(1/ε)). A PRPD is a relaxation of a pseudorandom generator, which suffices for derandomizing 𝐁𝐏𝐋 and also implies a hitting set. Unfortunately, their construction is quite involved and complicated. Hoza and Zuckerman (FOCS'18) later constructed a much simpler hitting set generator with seed length O(log n⋅ log(nw)+log(1/ε)), but their techniques are restricted to hitting sets.
In this work, we construct a PRPD with seed length O(log n⋅ log (nw)⋅ log log(nw)+log(1/ε)). This improves upon the construction by Braverman, Cogen and Garg by a O(log log(1/ε)) factor, and is optimal in the small error regime. In addition, we believe our construction and analysis to be simpler than the work of Braverman, Cohen and Garg.

Eshan Chattopadhyay and Jyun-Jie Liao. Optimal Error Pseudodistributions for Read-Once Branching Programs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 25:1-25:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2020.25, author = {Chattopadhyay, Eshan and Liao, Jyun-Jie}, title = {{Optimal Error Pseudodistributions for Read-Once Branching Programs}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {25:1--25:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-156-6}, ISSN = {1868-8969}, year = {2020}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.25}, URN = {urn:nbn:de:0030-drops-125779}, doi = {10.4230/LIPIcs.CCC.2020.25}, annote = {Keywords: Derandomization, explicit constructions, space-bounded computation} }

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