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RANDOM

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

We study a simple and general template for constructing affine extractors by composing a linear transformation with resilient functions. Using this we show that good affine extractors can be computed by non-explicit circuits of various types, including AC0-Xor circuits: AC0 circuits with a layer of parity gates at the input. We also show that one-sided extractors can be computed by small DNF-Xor circuits, and separate these circuits from other well-studied classes. As a further motivation for studying DNF-Xor circuits we show that if they can approximate inner product then small AC0-Xor circuits can compute it exactly - a long-standing open problem.

Xuangui Huang, Peter Ivanov, and Emanuele Viola. Affine Extractors and AC0-Parity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{huang_et_al:LIPIcs.APPROX/RANDOM.2022.9, author = {Huang, Xuangui and Ivanov, Peter and Viola, Emanuele}, title = {{Affine Extractors and AC0-Parity}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {9:1--9:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.9}, URN = {urn:nbn:de:0030-drops-171313}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.9}, annote = {Keywords: affine extractor, resilient function, constant-depth circuit, parity gate, inner product} }

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**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Space-efficient Laplacian solvers are closely related to derandomization of space-bound randomized computations. We show that if the probabilistic logarithmic-space solver or the deterministic nearly logarithmic-space solver for undirected Laplacian matrices can be extended to solve slightly larger subclasses of linear systems, then they can be used to solve all linear systems with similar space complexity. Previously Kyng and Zhang [Rasmus Kyng and Peng Zhang, 2017] proved such results in the time complexity setting using reductions between approximate solvers. We prove that their reductions can be implemented using constant-depth, polynomial-size threshold circuits.

Xuangui Huang. Space Hardness of Solving Structured Linear Systems. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{huang:LIPIcs.ISAAC.2020.56, author = {Huang, Xuangui}, title = {{Space Hardness of Solving Structured Linear Systems}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {56:1--56:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.56}, URN = {urn:nbn:de:0030-drops-134001}, doi = {10.4230/LIPIcs.ISAAC.2020.56}, annote = {Keywords: linear system solver, logarithmic space, threshold circuit} }

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**Published in:** LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

For every natural number q let FO_q denote the class of sentences of
first-order logic FO of quantifier rank at most q. If a graph property can be defined in FO_q, then it can be decided in time O(n^q). Thus, minimizing q has favorable algorithmic consequences. Many graph properties amount to the existence of a certain set of vertices of size k. Usually this can only be expressed by a sentence of quantifier rank at least k. We use the color coding method to demonstrate that some (hyper)graph problems can be defined in FO_q where q is independent of k. This property of a graph problem is equivalent to the question of whether the corresponding parameterized problem is in the class para-AC^0.
It is crucial for our results that the FO-sentences have access to built-in addition and multiplication (and constants for an initial segment of natural numbers whose length depends only on k). It is known that then FO corresponds to the circuit complexity class uniform AC^0. We explore the connection between the quantifier rank of FO-sentences and the depth of AC^0-circuits, and prove that FO_q is strictly contained in FO_{q+1} for structures with built-in addition and multiplication.

Yijia Chen, Jörg Flum, and Xuangui Huang. Slicewise Definability in First-Order Logic with Bounded Quantifier Rank. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chen_et_al:LIPIcs.CSL.2017.19, author = {Chen, Yijia and Flum, J\"{o}rg and Huang, Xuangui}, title = {{Slicewise Definability in First-Order Logic with Bounded Quantifier Rank}}, booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)}, pages = {19:1--19:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-045-3}, ISSN = {1868-8969}, year = {2017}, volume = {82}, editor = {Goranko, Valentin and Dam, Mads}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.19}, URN = {urn:nbn:de:0030-drops-76742}, doi = {10.4230/LIPIcs.CSL.2017.19}, annote = {Keywords: first-order logic, quantifier rank, parameterized AC^0, circuit depth} }

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