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RANDOM

**Published in:** LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Consider the model where we can access a parity function through random uniform labeled examples in the presence of random classification noise. In this paper, we show that approximating the number of relevant variables in the parity function is as hard as properly learning parities.
More specifically, let γ:ℝ^+ → ℝ^+, where γ(x) ≥ x, be any strictly increasing function. In our first result, we show that from any polynomial-time algorithm that returns a γ-approximation, D (i.e., γ^{-1}(d(f)) ≤ D ≤ γ(d(f))), of the number of relevant variables d(f) for any parity f, we can, in polynomial time, construct a solution to the long-standing open problem of polynomial-time learning k(n)-sparse parities (parities with k(n) ≤ n relevant variables), where k(n) = ω_n(1).
In our second result, we show that from any T(n)-time algorithm that, for any parity f, returns a γ-approximation of the number of relevant variables d(f) of f, we can, in polynomial time, construct a poly(Γ(n))T(Γ(n)²)-time algorithm that properly learns parities, where Γ(x) = γ(γ(x)).
If T(Γ(n)²) = exp({o(n/log n)}), this would resolve another long-standing open problem of properly learning parities in the presence of random classification noise in time exp(o(n/log n)).

Nader H. Bshouty and George Haddad. Approximating the Number of Relevant Variables in a Parity Implies Proper Learning. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bshouty_et_al:LIPIcs.APPROX/RANDOM.2024.38, author = {Bshouty, Nader H. and Haddad, George}, title = {{Approximating the Number of Relevant Variables in a Parity Implies Proper Learning}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {38:1--38:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.38}, URN = {urn:nbn:de:0030-drops-210316}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.38}, annote = {Keywords: PAC Learning, Random Classification Noise, Uniform Distribution, Parity, Sparcity Approximation} }

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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Koch, Strassle, and Tan [SODA 2023], show that, under the randomized exponential time hypothesis, there is no distribution-free PAC-learning algorithm that runs in time n^Õ(log log s) for the classes of n-variable size-s DNF, size-s Decision Tree, and log s-Junta by DNF (that returns a DNF hypothesis). Assuming a natural conjecture on the hardness of set cover, they give the lower bound n^Ω(log s). This matches the best known upper bound for n-variable size-s Decision Tree, and log s-Junta.
In this paper, we give the same lower bounds for PAC-learning of n-variable size-s Monotone DNF, size-s Monotone Decision Tree, and Monotone log s-Junta by DNF. This solves the open problem proposed by Koch, Strassle, and Tan and subsumes the above results.
The lower bound holds, even if the learner knows the distribution, can draw a sample according to the distribution in polynomial time, and can compute the target function on all the points of the support of the distribution in polynomial time.

Nader H. Bshouty. Superpolynomial Lower Bounds for Learning Monotone Classes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 34:1-34:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bshouty:LIPIcs.APPROX/RANDOM.2023.34, author = {Bshouty, Nader H.}, title = {{Superpolynomial Lower Bounds for Learning Monotone Classes}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {34:1--34:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.34}, URN = {urn:nbn:de:0030-drops-188594}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.34}, annote = {Keywords: PAC Learning, Monotone DNF, Monotone Decision Tree, Monotone Junta, Lower Bound} }

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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Let M be an n × m (0,1)-matrix. We define the s-binary rank, denoted as br_s(M), of M as the minimum integer d such that there exist d monochromatic rectangles covering all the 1-entries in the matrix, with each 1-entry being covered by at most s rectangles. When s = 1, this corresponds to the binary rank, denoted as br(M), which is well-known in the literature and has many applications.
Let R(M) and C(M) denote the sets of rows and columns of M, respectively. Using the result of Sgall [Jiří Sgall, 1999], we establish that if M has an s-binary rank at most d, then |R(M)| ⋅ |C(M)| ≤ binom(d, ≤ s)2^d, where binom(d, ≤ s) = ∑_{i=0}^s binom(d,i). This bound is tight, meaning that there exists a matrix M' with an s-binary rank of d, for which |R(M')| ⋅ |C(M')| = binom(d, ≤ s)2^d.
Using this result, we present novel one-sided adaptive and non-adaptive testers for (0,1)-matrices with an s-binary rank at most d (and exactly d). These testers require Õ(binom(d, ≤ s)2^d/ε) and Õ(binom(d, ≤ s)2^d/ε²) queries, respectively.
For a fixed s, this improves upon the query complexity of the tester proposed by Parnas et al. in [Michal Parnas et al., 2021] by a factor of Θ(2^d).

Nader H. Bshouty. On Property Testing of the Binary Rank. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bshouty:LIPIcs.MFCS.2023.27, author = {Bshouty, Nader H.}, title = {{On Property Testing of the Binary Rank}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {27:1--27:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.27}, URN = {urn:nbn:de:0030-drops-185616}, doi = {10.4230/LIPIcs.MFCS.2023.27}, annote = {Keywords: Property testing, binary rank, Boolean rank} }

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**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

We give the first polynomial-time non-adaptive proper learning algorithm of Boolean sparse multivariate polynomial under the uniform distribution. Our algorithm, for s-sparse polynomial over n variables, makes q = (s/ε)^{γ(s,ε)}log n queries where 2.66 ≤ γ(s,ε) ≤ 6.922 and runs in Õ(n)⋅ poly(s,1/ε) time. We also show that for any ε = 1/s^{O(1)} any non-adaptive learning algorithm must make at least (s/ε)^{Ω(1)}log n queries. Therefore, the query complexity of our algorithm is also polynomial in the optimal query complexity and optimal in n.

Nader H. Bshouty. Non-Adaptive Proper Learning Polynomials. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 16:1-16:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bshouty:LIPIcs.STACS.2023.16, author = {Bshouty, Nader H.}, title = {{Non-Adaptive Proper Learning Polynomials}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {16:1--16:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.16}, URN = {urn:nbn:de:0030-drops-176689}, doi = {10.4230/LIPIcs.STACS.2023.16}, annote = {Keywords: Polynomial, Learning, Testing} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

In this paper, we study testing decision tree of size and depth that are significantly smaller than the number of attributes n.
Our main result addresses the problem of poly(n,1/ε) time algorithms with poly(s,1/ε) query complexity (independent of n) that distinguish between functions that are decision trees of size s from functions that are ε-far from any decision tree of size ϕ(s,1/ε), for some function ϕ > s. The best known result is the recent one that follows from Blanc, Lange and Tan, [Guy Blanc et al., 2020], that gives ϕ(s,1/ε) = 2^{O((log³s)/ε³)}. In this paper, we give a new algorithm that achieves ϕ(s,1/ε) = 2^{O(log² (s/ε))}.
Moreover, we study the testability of depth-d decision tree and give a distribution free tester that distinguishes between depth-d decision tree and functions that are ε-far from depth-d² decision tree.

Nader H. Bshouty and Catherine A. Haddad-Zaknoon. On Testing Decision Tree. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bshouty_et_al:LIPIcs.STACS.2022.17, author = {Bshouty, Nader H. and Haddad-Zaknoon, Catherine A.}, title = {{On Testing Decision Tree}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {17:1--17:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.17}, URN = {urn:nbn:de:0030-drops-158273}, doi = {10.4230/LIPIcs.STACS.2022.17}, annote = {Keywords: Testing decision trees} }

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RANDOM

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

We give improved and almost optimal testers for several classes of Boolean functions on n variables that have concise representation in the uniform and distribution-free model. Classes, such as k-Junta, k-Linear, s-Term DNF, s-Term Monotone DNF, r-DNF, Decision List, r-Decision List, size-s Decision Tree, size-s Boolean Formula, size-s Branching Program, s-Sparse Polynomial over the binary field and functions with Fourier Degree at most d.
The approach is new and combines ideas from Diakonikolas et al. [Ilias Diakonikolas et al., 2007], Bshouty [Nader H. Bshouty, 2018], Goldreich et al. [Oded Goldreich et al., 1998], and learning theory. The method can be extended to several other classes of functions over any domain that can be approximated by functions with a small number of relevant variables.

Nader H. Bshouty. Almost Optimal Testers for Concise Representations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bshouty:LIPIcs.APPROX/RANDOM.2020.5, author = {Bshouty, Nader H.}, title = {{Almost Optimal Testers for Concise Representations}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {5:1--5:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.5}, URN = {urn:nbn:de:0030-drops-126087}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.5}, annote = {Keywords: Property Testing, Boolean function, Junta} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

We study the problem of determining the exact number of defective items in an adaptive group testing by using a minimum number of tests. We improve the existing algorithm and prove a lower bound that shows that the number of tests in our algorithm is optimal up to small additive terms.

Nader H. Bshouty, Catherine A. Haddad-Zaknoon, Raghd Boulos, Foad Moalem, Jalal Nada, Elias Noufi, and Yara Zaknoon. Optimal Randomized Group Testing Algorithm to Determine the Number of Defectives. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 18:1-18:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bshouty_et_al:LIPIcs.SWAT.2020.18, author = {Bshouty, Nader H. and Haddad-Zaknoon, Catherine A. and Boulos, Raghd and Moalem, Foad and Nada, Jalal and Noufi, Elias and Zaknoon, Yara}, title = {{Optimal Randomized Group Testing Algorithm to Determine the Number of Defectives}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {18:1--18:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.18}, URN = {urn:nbn:de:0030-drops-122658}, doi = {10.4230/LIPIcs.SWAT.2020.18}, annote = {Keywords: Group Testing, Randomized Algorithm} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

We prove that to estimate within a constant factor the number of defective items in a non-adaptive randomized group testing algorithm we need at least Omega~(log n) tests. This solves the open problem posed by Damaschke and Sheikh Muhammad in [Peter Damaschke and Azam Sheikh Muhammad, 2010; Peter Damaschke and Azam Sheikh Muhammad, 2010].

Nader H. Bshouty. Lower Bound for Non-Adaptive Estimation of the Number of Defective Items. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 2:1-2:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bshouty:LIPIcs.ISAAC.2019.2, author = {Bshouty, Nader H.}, title = {{Lower Bound for Non-Adaptive Estimation of the Number of Defective Items}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {2:1--2:9}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.2}, URN = {urn:nbn:de:0030-drops-114983}, doi = {10.4230/LIPIcs.ISAAC.2019.2}, annote = {Keywords: Group Testing, Estimation, Defective Items} }

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**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

We consider the problem of testing whether an unknown n-variable Boolean function is a k-junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over {0,1}^n. Chen, Liu, Servedio, Sheng and Xie [Zhengyang Liu et al., 2018] showed that the distribution-free k-junta testing can be performed, with one-sided error, by an adaptive algorithm that makes O~(k^2)/epsilon queries. In this paper, we give a simple two-sided error adaptive algorithm that makes O~(k/epsilon) queries.

Nader H. Bshouty. Almost Optimal Distribution-Free Junta Testing. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 2:1-2:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bshouty:LIPIcs.CCC.2019.2, author = {Bshouty, Nader H.}, title = {{Almost Optimal Distribution-Free Junta Testing}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {2:1--2:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.2}, URN = {urn:nbn:de:0030-drops-108249}, doi = {10.4230/LIPIcs.CCC.2019.2}, annote = {Keywords: Distribution-free property testing, k-Junta} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

In this paper we consider the problem of reconstructing a hidden
weighted hypergraph of constant rank using additive queries. We
prove the following: Let $G$ be a weighted hidden hypergraph of
constant rank with~$n$ vertices and $m$ hyperedges. For any $m$
there exists a non-adaptive algorithm that finds the edges of the
graph and their weights using
$$
O\left(\frac{m\log n}{\log m}\right)
$$
additive queries. This solves the open problem in [S. Choi, J. H.
Kim. Optimal Query Complexity Bounds for Finding Graphs. {\em
STOC}, 749--758, 2008].
When the weights of the hypergraph are integers that are less than
$O(poly(n^d/m))$ where $d$ is the rank of the hypergraph (and
therefore for unweighted hypergraphs) there exists a non-adaptive
algorithm that finds the edges of the graph and their weights using
$$
O\left(\frac{m\log \frac{n^d}{m}}{\log m}\right).
$$
additive queries.
Using the information theoretic bound the above query complexities
are tight.

Nader H. Bshouty and Hanna Mazzawi. Optimal Query Complexity for Reconstructing Hypergraphs. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 143-154, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{bshouty_et_al:LIPIcs.STACS.2010.2496, author = {Bshouty, Nader H. and Mazzawi, Hanna}, title = {{Optimal Query Complexity for Reconstructing Hypergraphs}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {143--154}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2496}, URN = {urn:nbn:de:0030-drops-24968}, doi = {10.4230/LIPIcs.STACS.2010.2496}, annote = {Keywords: Query complexity, hypergraphs} }

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