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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of the matrix equals the Fourier sparsity of the function, which is the number of its nonzero Fourier coefficients.
In this note, we refute two conjectures. The first has origins in Montanaro and Osborne (arXiv'09) and is considered in Tsang, Wong, Xie, and Zhang (FOCS'13), and the second is due to Mande and Sanyal (FSTTCS'20). These conjectures were proposed in order to improve the best-known bound of Lovett (STOC'14) regarding the log-rank conjecture in the special case of XOR functions. Both conjectures speculate that the set of nonzero Fourier coefficients of the boolean function has some strong additive structure. We refute these conjectures by constructing two specific boolean functions tailored to each.

Hamed Hatami, Kaave Hosseini, Shachar Lovett, and Anthony Ostuni. Refuting Approaches to the Log-Rank Conjecture for XOR Functions. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 82:1-82:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hatami_et_al:LIPIcs.ICALP.2024.82, author = {Hatami, Hamed and Hosseini, Kaave and Lovett, Shachar and Ostuni, Anthony}, title = {{Refuting Approaches to the Log-Rank Conjecture for XOR Functions}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {82:1--82:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.82}, URN = {urn:nbn:de:0030-drops-202252}, doi = {10.4230/LIPIcs.ICALP.2024.82}, annote = {Keywords: Communication complexity, log-rank conjecture, XOR functions, additive structure} }

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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

We study the discrepancy of the following communication problem. Alice receives a halfplane, and Bob receives a point in the plane, and their goal is to determine whether Bob’s point belongs to Alice’s halfplane. This communication task corresponds to determining whether x₁y₁+y₂ ≥ x₂, where the first player knows (x₁,x₂) and the second player knows (y₁,y₂).
Denoting n = m³, we show that when the inputs are chosen from [m] × [m²], the communication discrepancy of the above problem is O(n^{-1/6} log^{3/2} n).
On the other hand, through the connections to the notion of hereditary discrepancy by Matoušek, Nikolov, and Tawler (IMRN 2020) and a classical result of Matoušek (Discrete Comput. Geom. 1995), we show that the communication discrepancy of every set of n points and n halfplanes is at least Ω(n^{-1/4} log^{-1} n).

Manasseh Ahmed, Tsun-Ming Cheung, Hamed Hatami, and Kusha Sareen. Communication Complexity and Discrepancy of Halfplanes. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 5:1-5:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ahmed_et_al:LIPIcs.SoCG.2024.5, author = {Ahmed, Manasseh and Cheung, Tsun-Ming and Hatami, Hamed and Sareen, Kusha}, title = {{Communication Complexity and Discrepancy of Halfplanes}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {5:1--5:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.5}, URN = {urn:nbn:de:0030-drops-199504}, doi = {10.4230/LIPIcs.SoCG.2024.5}, annote = {Keywords: Randomized communication complexity, Discrepancy theory, factorization norm} }

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

In an influential paper, Linial and Shraibman (STOC '07) introduced the factorization norm as a powerful tool for proving lower bounds against randomized and quantum communication complexities. They showed that the logarithm of the approximate γ₂-factorization norm is a lower bound for these parameters and asked whether a stronger lower bound that replaces approximate γ₂ norm with the γ₂ norm holds.
We answer the question of Linial and Shraibman in the negative by exhibiting a 2ⁿ×2ⁿ Boolean matrix with γ₂ norm 2^Ω(n) and randomized communication complexity O(log n).
As a corollary, we recover the recent result of Chattopadhyay, Lovett, and Vinyals (CCC '19) that deterministic protocols with access to an Equality oracle are exponentially weaker than (one-sided error) randomized protocols. In fact, as a stronger consequence, our result implies an exponential separation between the power of unambiguous nondeterministic protocols with access to Equality oracle and (one-sided error) randomized protocols, which answers a question of Pitassi, Shirley, and Shraibman (ITSC '23).
Our result also implies a conjecture of Sherif (Ph.D. thesis) that the γ₂ norm of the Integer Inner Product function (IIP) in dimension 3 or higher is exponential in its input size.

Tsun-Ming Cheung, Hamed Hatami, Kaave Hosseini, and Morgan Shirley. Separation of the Factorization Norm and Randomized Communication Complexity. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cheung_et_al:LIPIcs.CCC.2023.1, author = {Cheung, Tsun-Ming and Hatami, Hamed and Hosseini, Kaave and Shirley, Morgan}, title = {{Separation of the Factorization Norm and Randomized Communication Complexity}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {1:1--1:16}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.1}, URN = {urn:nbn:de:0030-drops-182714}, doi = {10.4230/LIPIcs.CCC.2023.1}, annote = {Keywords: Factorization norms, randomized communication complexity} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

In a recent article, Alon, Hanneke, Holzman, and Moran (FOCS '21) introduced a unifying framework to study the learnability of classes of partial concepts. One of the central questions studied in their work is whether the learnability of a partial concept class is always inherited from the learnability of some "extension" of it to a total concept class.
They showed this is not the case for PAC learning but left the problem open for the stronger notion of online learnability.
We resolve this problem by constructing a class of partial concepts that is online learnable, but no extension of it to a class of total concepts is online learnable (or even PAC learnable).

Tsun-Ming Cheung, Hamed Hatami, Pooya Hatami, and Kaave Hosseini. Online Learning and Disambiguations of Partial Concept Classes. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cheung_et_al:LIPIcs.ICALP.2023.42, author = {Cheung, Tsun-Ming and Hatami, Hamed and Hatami, Pooya and Hosseini, Kaave}, title = {{Online Learning and Disambiguations of Partial Concept Classes}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {42:1--42:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.42}, URN = {urn:nbn:de:0030-drops-180946}, doi = {10.4230/LIPIcs.ICALP.2023.42}, annote = {Keywords: Online learning, Littlestone dimension, VC dimension, partial concept class, clique vs independent set, Alon-Saks-Seymour conjecture, Standard Optimal Algorithm, PAC learning} }

Document

RANDOM

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

The sign-rank of a matrix A with ±1 entries is the smallest rank of a real matrix with the same sign pattern as A. To the best of our knowledge, there are only three known methods for proving lower bounds on the sign-rank of explicit matrices: (i) Sign-rank is at least the VC-dimension; (ii) Forster’s method, which states that sign-rank is at least the inverse of the largest possible average margin among the representations of the matrix by points and half-spaces; (iii) Sign-rank is at least a logarithmic function of the density of the largest monochromatic rectangle.
We prove several results regarding the limitations of these methods.
- We prove that, qualitatively, the monochromatic rectangle density is the strongest of these three lower bounds. If it fails to provide a super-constant lower bound for the sign-rank of a matrix, then the other two methods will fail as well.
- We show that there exist n × n matrices with sign-rank n^Ω(1) for which none of these methods can provide a super-constant lower bound.
- We show that sign-rank is at most an exponential function of the deterministic communication complexity with access to an equality oracle. We combine this result with Green and Sanders' quantitative version of Cohen’s idempotent theorem to show that for a large class of sign matrices (e.g., xor-lifts), sign-rank is at most an exponential function of the γ₂ norm of the matrix. We conjecture that this holds for all sign matrices.
- Towards answering a question of Linial, Mendelson, Schechtman, and Shraibman regarding the relation between sign-rank and discrepancy, we conjecture that sign-ranks of the ±1 adjacency matrices of hypercube graphs can be arbitrarily large. We prove that none of the three lower bound techniques can resolve this conjecture in the affirmative.

Hamed Hatami, Pooya Hatami, William Pires, Ran Tao, and Rosie Zhao. Lower Bound Methods for Sign-Rank and Their Limitations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 22:1-22:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hatami_et_al:LIPIcs.APPROX/RANDOM.2022.22, author = {Hatami, Hamed and Hatami, Pooya and Pires, William and Tao, Ran and Zhao, Rosie}, title = {{Lower Bound Methods for Sign-Rank and Their Limitations}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {22:1--22:24}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.22}, URN = {urn:nbn:de:0030-drops-171445}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.22}, annote = {Keywords: Average Margin, Communication complexity, margin complexity, monochromatic rectangle, Sign-rank, Unbounded-error communication complexity, VC-dimension} }

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

Sign-rank and discrepancy are two central notions in communication complexity. The seminal work of Babai, Frankl, and Simon from 1986 initiated an active line of research that investigates the gap between these two notions. In this article, we establish the strongest possible separation by constructing a boolean matrix whose sign-rank is only 3, and yet its discrepancy is 2^{-Ω(n)}. We note that every matrix of sign-rank 2 has discrepancy n^{-O(1)}.
Our result in particular implies that there are boolean functions with O(1) unbounded error randomized communication complexity while having Ω(n) weakly unbounded error randomized communication complexity.

Hamed Hatami, Kaave Hosseini, and Shachar Lovett. Sign Rank vs Discrepancy. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hatami_et_al:LIPIcs.CCC.2020.18, author = {Hatami, Hamed and Hosseini, Kaave and Lovett, Shachar}, title = {{Sign Rank vs Discrepancy}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {18:1--18:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.18}, URN = {urn:nbn:de:0030-drops-125700}, doi = {10.4230/LIPIcs.CCC.2020.18}, annote = {Keywords: Discrepancy, sign rank, Unbounded-error communication complexity, weakly unbounded error communication complexity} }

Document

Track A: Algorithms, Complexity and Games

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

The seminal result of Kahn, Kalai and Linial shows that a coalition of O(n/(log n)) players can bias the outcome of any Boolean function {0,1}^n -> {0,1} with respect to the uniform measure. We extend their result to arbitrary product measures on {0,1}^n, by combining their argument with a completely different argument that handles very biased input bits.
We view this result as a step towards proving a conjecture of Friedgut, which states that Boolean functions on the continuous cube [0,1]^n (or, equivalently, on {1,...,n}^n) can be biased using coalitions of o(n) players. This is the first step taken in this direction since Friedgut proposed the conjecture in 2004.
Russell, Saks and Zuckerman extended the result of Kahn, Kalai and Linial to multi-round protocols, showing that when the number of rounds is o(log^* n), a coalition of o(n) players can bias the outcome with respect to the uniform measure. We extend this result as well to arbitrary product measures on {0,1}^n.
The argument of Russell et al. relies on the fact that a coalition of o(n) players can boost the expectation of any Boolean function from epsilon to 1-epsilon with respect to the uniform measure. This fails for general product distributions, as the example of the AND function with respect to mu_{1-1/n} shows. Instead, we use a novel boosting argument alongside a generalization of our first result to arbitrary finite ranges.

Yuval Filmus, Lianna Hambardzumyan, Hamed Hatami, Pooya Hatami, and David Zuckerman. Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 58:1-58:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{filmus_et_al:LIPIcs.ICALP.2019.58, author = {Filmus, Yuval and Hambardzumyan, Lianna and Hatami, Hamed and Hatami, Pooya and Zuckerman, David}, title = {{Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {58:1--58:13}, 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.2019.58}, URN = {urn:nbn:de:0030-drops-106340}, doi = {10.4230/LIPIcs.ICALP.2019.58}, annote = {Keywords: Boolean function analysis, coin flipping} }

Document

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

We consider the standard two-party communication model. The central problem studied in this article is how much can one save in information complexity by allowing a certain error.
* For arbitrary functions, we obtain lower bounds and upper bounds indicating a gain that is of order Omega(h(epsilon)) and O(h(sqrt{epsilon})). Here h denotes the binary entropy function.
* We analyze the case of the two-bit AND function in detail to show that for this function the gain is Theta(h(epsilon)). This answers a question of Braverman et al. [Braverman, STOC 2013].
* We obtain sharp bounds for the set disjointness function of order n. For the case of the distributional error, we introduce a new protocol that achieves a gain of Theta(sqrt{h(epsilon)}) provided that n is sufficiently large. We apply these results to answer another of question of Braverman et al. regarding the randomized communication complexity of the set disjointness function.
* Answering a question of Braverman [Braverman, STOC 2012], we apply our analysis of the set disjointness function to establish a gap between the two different notions of the prior-free information cost. In light of [Braverman, STOC 2012], this implies that amortized randomized communication complexity is not necessarily equal to the amortized distributional communication complexity with respect to the hardest distribution.
As a consequence, we show that the epsilon-error randomized communication complexity of the set disjointness function of order n is n[C_{DISJ} - Theta(h(epsilon))] + o(n), where C_{DISJ} ~ 0.4827$ is the constant found by Braverman et al. [Braverman, STOC 2012].

Yuval Dagan, Yuval Filmus, Hamed Hatami, and Yaqiao Li. Trading Information Complexity for Error. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 16:1-16:59, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dagan_et_al:LIPIcs.CCC.2017.16, author = {Dagan, Yuval and Filmus, Yuval and Hatami, Hamed and Li, Yaqiao}, title = {{Trading Information Complexity for Error}}, booktitle = {32nd Computational Complexity Conference (CCC 2017)}, pages = {16:1--16:59}, 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.2017.16}, URN = {urn:nbn:de:0030-drops-75179}, doi = {10.4230/LIPIcs.CCC.2017.16}, annote = {Keywords: communication complexity, information complexity} }

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