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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

We study the connections between three seemingly different combinatorial structures - uniform brackets in statistics and probability theory, containers in online and distributed learning theory, and combinatorial Macbeath regions, or Mnets in discrete and computational geometry. We show that these three concepts are manifestations of a single combinatorial property that can be expressed under a unified framework along the lines of Vapnik-Chervonenkis type theory for uniform convergence. These new connections help us to bring tools from discrete and computational geometry to prove improved bounds for these objects. Our improved bounds help to get an optimal algorithm for distributed learning of halfspaces, an improved algorithm for the distributed convex set disjointness problem, and improved regret bounds for online algorithms against σ-smoothed adversary for a large class of semi-algebraic threshold functions.

Kunal Dutta, Arijit Ghosh, and Shay Moran. Uniform Brackets, Containers, and Combinatorial Macbeath Regions. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 59:1-59:10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dutta_et_al:LIPIcs.ITCS.2022.59, author = {Dutta, Kunal and Ghosh, Arijit and Moran, Shay}, title = {{Uniform Brackets, Containers, and Combinatorial Macbeath Regions}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {59:1--59:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.59}, URN = {urn:nbn:de:0030-drops-156551}, doi = {10.4230/LIPIcs.ITCS.2022.59}, annote = {Keywords: communication complexity, distributed learning, emperical process theory, online algorithms, discrete geometry, computational geometry} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

"Twenty questions" is a guessing game played by two players: Bob thinks of an integer between 1 and n, and Alice’s goal is to recover it using a minimal number of Yes/No questions. Shannon’s entropy has a natural interpretation in this context. It characterizes the average number of questions used by an optimal strategy in the distributional variant of the game: let μ be a distribution over [n], then the average number of questions used by an optimal strategy that recovers x∼ μ is between H(μ) and H(μ)+1.
We consider an extension of this game where at most k questions can be answered falsely. We extend the classical result by showing that an optimal strategy uses roughly H(μ) + k H_2(μ) questions, where H_2(μ) = ∑_x μ(x)log log 1/μ(x). This also generalizes a result by Rivest et al. (1980) for the uniform distribution.
Moreover, we design near optimal strategies that only use comparison queries of the form "x ≤ c?" for c ∈ [n]. The usage of comparison queries lends itself naturally to the context of sorting, where we derive sorting algorithms in the presence of adversarial noise.

Yuval Dagan, Yuval Filmus, Daniel Kane, and Shay Moran. The Entropy of Lies: Playing Twenty Questions with a Liar. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 1:1-1:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dagan_et_al:LIPIcs.ITCS.2021.1, author = {Dagan, Yuval and Filmus, Yuval and Kane, Daniel and Moran, Shay}, title = {{The Entropy of Lies: Playing Twenty Questions with a Liar}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {1:1--1:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.1}, URN = {urn:nbn:de:0030-drops-135400}, doi = {10.4230/LIPIcs.ITCS.2021.1}, annote = {Keywords: entropy, twenty questions, algorithms, sorting} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We examine connections between combinatorial notions that arise in machine learning and topological notions in cubical/simplicial geometry. These connections enable to export results from geometry to machine learning. Our first main result is based on a geometric construction by H. Tracy Hall (2004) of a partial shelling of the cross-polytope which can not be extended. We use it to derive a maximum class of VC dimension 3 that has no corners. This refutes several previous works in machine learning from the past 11 years. In particular, it implies that the previous constructions of optimal unlabeled compression schemes for maximum classes are erroneous.
On the positive side we present a new construction of an optimal unlabeled compression scheme for maximum classes. We leave as open whether our unlabeled compression scheme extends to ample (a.k.a. lopsided or extremal) classes, which represent a natural and far-reaching generalization of maximum classes. Towards resolving this question, we provide a geometric characterization in terms of unique sink orientations of the 1-skeletons of associated cubical complexes.

Jérémie Chalopin, Victor Chepoi, Shay Moran, and Manfred K. Warmuth. Unlabeled Sample Compression Schemes and Corner Peelings for Ample and Maximum Classes. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 34:1-34:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chalopin_et_al:LIPIcs.ICALP.2019.34, author = {Chalopin, J\'{e}r\'{e}mie and Chepoi, Victor and Moran, Shay and Warmuth, Manfred K.}, title = {{Unlabeled Sample Compression Schemes and Corner Peelings for Ample and Maximum Classes}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {34:1--34:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.34}, URN = {urn:nbn:de:0030-drops-106105}, doi = {10.4230/LIPIcs.ICALP.2019.34}, annote = {Keywords: VC-dimension, sample compression, Sauer-Shelah-Perles lemma, Sandwich lemma, maximum class, ample/extremal class, corner peeling, unique sink orientation} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We show that the Radon number characterizes the existence of weak nets in separable convexity spaces (an abstraction of the Euclidean notion of convexity). The construction of weak nets when the Radon number is finite is based on Helly’s property and on metric properties of VC classes. The lower bound on the size of weak nets when the Radon number is large relies on the chromatic number of the Kneser graph. As an application, we prove an amplification result for weak epsilon-nets.

Shay Moran and Amir Yehudayoff. On Weak epsilon-Nets and the Radon Number. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 51:1-51:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{moran_et_al:LIPIcs.SoCG.2019.51, author = {Moran, Shay and Yehudayoff, Amir}, title = {{On Weak epsilon-Nets and the Radon Number}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {51:1--51:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.51}, URN = {urn:nbn:de:0030-drops-104551}, doi = {10.4230/LIPIcs.SoCG.2019.51}, annote = {Keywords: abstract convexity, weak epsilon nets, Radon number, VC dimension, Haussler packing lemma, Kneser graphs} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Let H be an arbitrary family of hyper-planes in d-dimensions. We show that the point-location problem for H can be solved by a linear decision tree that only uses a special type of queries called generalized comparison queries. These queries correspond to hyperplanes that can be written as a linear combination of two hyperplanes from H; in particular, if all hyperplanes in H are k-sparse then generalized comparisons are 2k-sparse. The depth of the obtained linear decision tree is polynomial in d and logarithmic in |H|, which is comparable to previous results in the literature that use general linear queries.
This extends the study of comparison trees from a previous work by the authors [Kane {et al.}, FOCS 2017]. The main benefit is that using generalized comparison queries allows to overcome limitations that apply for the more restricted type of comparison queries.
Our analysis combines a seminal result of Forster regarding sets in isotropic position [Forster, JCSS 2002], the margin-based inference dimension analysis for comparison queries from [Kane {et al.}, FOCS 2017], and compactness arguments.

Daniel M. Kane, Shachar Lovett, and Shay Moran. Generalized Comparison Trees for Point-Location Problems. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 82:1-82:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kane_et_al:LIPIcs.ICALP.2018.82, author = {Kane, Daniel M. and Lovett, Shachar and Moran, Shay}, title = {{Generalized Comparison Trees for Point-Location Problems}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {82:1--82:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.82}, URN = {urn:nbn:de:0030-drops-90862}, doi = {10.4230/LIPIcs.ICALP.2018.82}, annote = {Keywords: linear decision trees, comparison queries, point location problems} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

We study complexity measures on subsets of the boolean hypercube and exhibit connections between algebra (the Hilbert function) and combinatorics (VC theory). These connections yield results in both directions. Our main complexity-theoretic result demonstrates that a large and natural family of linear program feasibility problems cannot be computed by polynomial-sized constant-depth circuits. Moreover, our result applies to a stronger regime in which the hyperplanes are fixed and only the directions of the inequalities are given as input to the circuit. We derive this result by proving that a rich class of extremal functions in VC theory cannot be approximated by low-degree polynomials. We also present applications of algebra to combinatorics. We provide a new algebraic proof of the Sandwich Theorem, which is a generalization of the well-known Sauer-Perles-Shelah Lemma.
Finally, we prove a structural result about downward-closed sets, related to the Chvatal conjecture in extremal combinatorics.

Shay Moran and Cyrus Rashtchian. Shattered Sets and the Hilbert Function. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 70:1-70:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{moran_et_al:LIPIcs.MFCS.2016.70, author = {Moran, Shay and Rashtchian, Cyrus}, title = {{Shattered Sets and the Hilbert Function}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {70:1--70:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.70}, URN = {urn:nbn:de:0030-drops-64814}, doi = {10.4230/LIPIcs.MFCS.2016.70}, annote = {Keywords: VC dimension, shattered sets, sandwich theorem, Hilbert function, polynomial method, linear programming, Chvatal's conjecture, downward-closed sets} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

We study the complexity of the Hitting Set problem in set systems (hypergraphs) that avoid certain sub-structures. In particular, we characterize the classical and parameterized complexity of the problem when the Vapnik-Chervonenkis dimension (VC-dimension) of the input is small.
VC-dimension is a natural measure of complexity of set systems. Several tractable instances of Hitting Set with a geometric or graph-theoretical flavor are known to have low VC-dimension. In set systems of bounded VC-dimension, Hitting Set is known to admit efficient and almost optimal approximation algorithms (Brönnimann and Goodrich, 1995; Even, Rawitz, and Shahar, 2005; Agarwal and Pan, 2014).
In contrast to these approximation-results, a low VC-dimension does not necessarily imply tractability in the parameterized sense. In fact, we show that Hitting Set is W[1]-hard already on inputs with VC-dimension 2, even if the VC-dimension of the dual set system is also 2. Thus, Hitting Set is very unlikely to be fixed-parameter tractable even in this arguably simple case. This answers an open question raised by King in 2010. For set systems whose (primal or dual) VC-dimension is 1, we show that Hitting Set is solvable in polynomial time.
To bridge the gap in complexity between the classes of inputs with VC-dimension 1 and 2, we use a measure that is more fine-grained than VC-dimension. In terms of this measure, we identify a sharp threshold where the complexity of Hitting Set transitions from polynomial-time-solvable to NP-hard. The tractable class that lies just under the threshold is a generalization of Edge Cover, and thus extends the domain of polynomial-time tractability of Hitting Set.

Karl Bringmann, László Kozma, Shay Moran, and N. S. Narayanaswamy. Hitting Set for Hypergraphs of Low VC-dimension. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 23:1-23:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bringmann_et_al:LIPIcs.ESA.2016.23, author = {Bringmann, Karl and Kozma, L\'{a}szl\'{o} and Moran, Shay and Narayanaswamy, N. S.}, title = {{Hitting Set for Hypergraphs of Low VC-dimension}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {23:1--23:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.23}, URN = {urn:nbn:de:0030-drops-63749}, doi = {10.4230/LIPIcs.ESA.2016.23}, annote = {Keywords: hitting set, VC-dimension} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

We study internal compression of communication protocols to their internal entropy, which is the entropy of the transcript from the players' perspective. We provide two internal compression schemes with error. One of a protocol of Feige et al. for finding the first difference between two strings. The second and main one is an internal compression with error epsilon > 0 of a protocol with internal entropy H^{int} and communication complexity C to a protocol with communication at most order (H^{int}/epsilon)^2 * log(log(C)).
This immediately implies a similar compression to the internal information of public-coin protocols, which provides an exponential improvement over previously known public-coin compressions in the dependence on C. It further shows that in a recent protocol of Ganor, Kol and Raz, it is impossible to move the private randomness to be public without an exponential cost. To the best of our knowledge, No such example was previously known.

Balthazar Bauer, Shay Moran, and Amir Yehudayoff. Internal Compression of Protocols to Entropy. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 481-496, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bauer_et_al:LIPIcs.APPROX-RANDOM.2015.481, author = {Bauer, Balthazar and Moran, Shay and Yehudayoff, Amir}, title = {{Internal Compression of Protocols to Entropy}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {481--496}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.481}, URN = {urn:nbn:de:0030-drops-53198}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.481}, annote = {Keywords: Communication complexity, Information complexity, Compression, Simulation, Entropy} }

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