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**Published in:** LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

We construct improved hitting set generators (HSGs) for ordered (read-once) regular branching programs in two parameter regimes. First, we construct an explicit ε-HSG for unbounded-width regular branching programs with a single accept state with seed length Õ(log n ⋅ log(1/ε)), where n is the length of the program. Second, we construct an explicit ε-HSG for width-w length-n regular branching programs with seed length Õ(log n ⋅ (√{log(1/ε)} + log w) + log(1/ε)). For context, the "baseline" in this area is the pseudorandom generator (PRG) by Nisan (Combinatorica 1992), which fools ordered (possibly non-regular) branching programs with seed length O(log(wn/ε) ⋅ log n). For regular programs, the state-of-the-art PRG, by Braverman, Rao, Raz, and Yehudayoff (FOCS 2010, SICOMP 2014), has seed length Õ(log(w/ε) ⋅ log n), which beats Nisan’s seed length when log(w/ε) = o(log n). Taken together, our two new constructions beat Nisan’s seed length in all parameter regimes except when log w and log(1/ε) are both Ω(log n) (for the construction of HSGs for regular branching programs with a single accept vertex).
Extending work by Reingold, Trevisan, and Vadhan (STOC 2006), we furthermore show that an explicit HSG for regular branching programs with a single accept vertex with seed length o(log² n) in the regime log w = Θ(log(1/ε)) = Θ(log n) would imply improved HSGs for general ordered branching programs, which would be a major breakthrough in derandomization. Pyne and Vadhan (CCC 2021) recently obtained such parameters for the special case of permutation branching programs.

Andrej Bogdanov, William M. Hoza, Gautam Prakriya, and Edward Pyne. Hitting Sets for Regular Branching Programs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 3:1-3:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bogdanov_et_al:LIPIcs.CCC.2022.3, author = {Bogdanov, Andrej and Hoza, William M. and Prakriya, Gautam and Pyne, Edward}, title = {{Hitting Sets for Regular Branching Programs}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {3:1--3:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-241-9}, ISSN = {1868-8969}, year = {2022}, volume = {234}, editor = {Lovett, Shachar}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.3}, URN = {urn:nbn:de:0030-drops-165658}, doi = {10.4230/LIPIcs.CCC.2022.3}, annote = {Keywords: Pseudorandomness, hitting set generators, space-bounded computation} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

A function f(x₁, … , x_n) from a product domain 𝒟₁ × ⋯ × 𝒟_n to an abelian group 𝒢 is a direct sum if it is of the form f₁(x₁) + ⋯ + f_n(x_n). We present a new 4-query direct sum test with optimal (up to constant factors) soundness error. This generalizes a result of Dinur and Golubev (RANDOM 2019) which is tailored to the target group 𝒢 = ℤ₂. As a special case, we obtain an optimal affinity test for 𝒢-valued functions on domain {0, 1}ⁿ under product measure. Our analysis relies on the hypercontractivity of the binary erasure channel.
We also study the testability of function partitionability over product domains into disjoint components. A 𝒢-valued f(x₁, … , x_n) is k-direct sum partitionable if it can be written as a sum of functions over k nonempty disjoint sets of inputs. A function f(x₁, … , x_n) with unstructured product range ℛ^k is direct product partitionable if its outputs depend on disjoint sets of inputs.
We show that direct sum partitionability and direct product partitionability are one-sided error testable with O((n - k)(log n + 1/ε) + 1/ε) adaptive queries and O((n/ε) log²(n/ε)) nonadaptive queries, respectively. Both bounds are tight up to the logarithmic factors for constant ε even with respect to adaptive, two-sided error testers. We also give a non-adaptive one-sided error tester for direct sum partitionability with query complexity O(kn² (log n)² / ε).

Andrej Bogdanov and Gautam Prakriya. Direct Sum and Partitionability Testing over General Groups. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bogdanov_et_al:LIPIcs.ICALP.2021.33, author = {Bogdanov, Andrej and Prakriya, Gautam}, title = {{Direct Sum and Partitionability Testing over General Groups}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {33:1--33:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.33}, URN = {urn:nbn:de:0030-drops-141028}, doi = {10.4230/LIPIcs.ICALP.2021.33}, annote = {Keywords: Direct Sum Test, Function Partitionability, Hypercontractive Inequality, Property Testing} }

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**Published in:** LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

We study the possibility of deterministic and randomness-efficient isolation in space-bounded models of computation: Can one efficiently reduce instances of computational problems to equivalent instances that have at most one solution? We present results for the NL-complete problem of reachability on digraphs, and for the LogCFL-complete problem of certifying acceptance on shallow semi-unbounded circuits.
A common approach employs small weight assignments that make the solution of minimum weight unique. The Isolation Lemma and other known procedures use Omega(n) random bits to generate weights of individual bitlength O(log(n)). We develop a derandomized version for both settings that uses O(log(n)^{3/2}) random bits and produces weights of bitlength O(log(n)^{3/2}) in logarithmic space. The construction allows us to show that every language in NL can be accepted by a nondeterministic machine that runs in polynomial time and O(log(n)^{3/2}) space, and has at most one accepting computation path on every input. Similarly, every language in LogCFL can be accepted by a nondeterministic machine equipped with a stack that does not count towards the space bound, that runs in polynomial time and O(log(n)^{3/2}) space, and has at most one accepting computation path on every input.
We also show that the existence of somewhat more restricted isolations for reachability on digraphs implies that NL can be decided in logspace with polynomial advice. A similar result holds for certifying acceptance on shallow semi-unbounded circuits and LogCFL.

Dieter van Melkebeek and Gautam Prakriya. Derandomizing Isolation in Space-Bounded Settings. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 5:1-5:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{vanmelkebeek_et_al:LIPIcs.CCC.2017.5, author = {van Melkebeek, Dieter and Prakriya, Gautam}, title = {{Derandomizing Isolation in Space-Bounded Settings}}, booktitle = {32nd Computational Complexity Conference (CCC 2017)}, pages = {5:1--5:32}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-040-8}, ISSN = {1868-8969}, year = {2017}, volume = {79}, editor = {O'Donnell, Ryan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.5}, URN = {urn:nbn:de:0030-drops-75297}, doi = {10.4230/LIPIcs.CCC.2017.5}, annote = {Keywords: Isolation lemma, derandomization, unambiguous nondeterminism, graph reachability, circuit certification} }

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