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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

The ExactlyN problem in the number-on-forehead (NOF) communication setting asks k players, each of whom can see every input but their own, if the k input numbers add up to N. Introduced by Chandra, Furst and Lipton in 1983, ExactlyN is important for its role in understanding the strength of randomness in communication complexity with many players. It is also tightly connected to the field of combinatorics: its k-party NOF communication complexity is related to the size of the largest corner-free subset in [N]^{k-1}.
In 2021, Linial and Shraibman gave more efficient protocols for ExactlyN for 3 players. As an immediate consequence, this also gave a new construction of larger corner-free subsets in [N]². Later that year Green gave a further refinement to their argument. These results represent the first improvements to the highest-order term for k = 3 since the famous work of Behrend in 1946. In this paper we give a corresponding improvement to the highest-order term for k > 3, the first since Rankin in 1961. That is, we give a more efficient protocol for ExactlyN as well as larger corner-free sets in higher dimensions.
Nearly all previous results in this line of research approached the problem from the combinatorics perspective, implicitly resulting in non-constructive protocols for ExactlyN. Approaching the problem from the communication complexity point of view and constructing explicit protocols for ExactlyN was key to the improvements in the k = 3 setting. As a further contribution we provide explicit protocols for ExactlyN for any number of players which serves as a base for our improvement.

Lianna Hambardzumyan, Toniann Pitassi, Suhail Sherif, Morgan Shirley, and Adi Shraibman. An Improved Protocol for ExactlyN with More Than 3 Players. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 58:1-58:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hambardzumyan_et_al:LIPIcs.ITCS.2024.58, author = {Hambardzumyan, Lianna and Pitassi, Toniann and Sherif, Suhail and Shirley, Morgan and Shraibman, Adi}, title = {{An Improved Protocol for ExactlyN with More Than 3 Players}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {58:1--58:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.58}, URN = {urn:nbn:de:0030-drops-195868}, doi = {10.4230/LIPIcs.ITCS.2024.58}, annote = {Keywords: Corner-free sets, number-on-forehead communication} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

It is well-known that randomized communication protocols are more powerful than deterministic protocols. In particular the Equality function requires Ω(n) deterministic communication complexity but has efficient randomized protocols. Previous work of Chattopadhyay, Lovett and Vinyals shows that randomized communication is strictly stronger than what can be solved by deterministic protocols equipped with an Equality oracle. Despite this separation, we are far from understanding the exact strength of Equality oracles in the context of communication complexity.
In this work we focus on nondeterminisic communication equipped with an Equality oracle, which is a subclass of Merlin-Arthur communication. We show that this inclusion is strict by proving that the previously-studied Integer Inner Product function, which can be efficiently computed even with bounded-error randomness, cannot be computed using sublinear communication in the nondeterministic Equality model. To prove this we give a new matrix-theoretic characterization of the nondeterministic Equality model: specifically, there is a tight connection between this model and a covering number based on the blocky matrices of Hambardzumyan, Hatami, and Hatami, as well as a natural variant of the Gamma-2 factorization norm. Similar equivalences are shown for the unambiguous nondeterministic model with Equality oracles. A bonus result arises from these proofs: for the studied communication models, a single Equality oracle call suffices without loss of generality.
Our results allow us to prove a separation between deterministic and unambiguous nondeterminism in the presence of Equality oracles. This stands in contrast to the result of Yannakakis which shows that these models are polynomially-related without oracles. We suggest a number of intriguing open questions along this direction of inquiry, as well as others that arise from our work.

Toniann Pitassi, Morgan Shirley, and Adi Shraibman. The Strength of Equality Oracles in Communication. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 89:1-89:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{pitassi_et_al:LIPIcs.ITCS.2023.89, author = {Pitassi, Toniann and Shirley, Morgan and Shraibman, Adi}, title = {{The Strength of Equality Oracles in Communication}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {89:1--89:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.89}, URN = {urn:nbn:de:0030-drops-175927}, doi = {10.4230/LIPIcs.ITCS.2023.89}, annote = {Keywords: Factorization norm, blocky rank, Merlin-Arthur} }

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**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

In the 3-players exactly-N problem the players need to decide whether x+y+z = N for inputs x,y,z and fixed N. This is the first problem considered in the multiplayer Number On the Forehead (NOF) model. Even though this is such a basic problem, no progress has been made on it throughout the years. Only recently have explicit protocols been found for the first time, yet no improvement in complexity has been achieved to date. The present paper offers the first improved protocol for the exactly-N problem. This improved protocol has also interesting consequences in additive combinatorics. As we explain below, it yields a higher lower bound on the possible density of corner-free sets in [N]×[N].

Nati Linial and Adi Shraibman. An Improved Protocol for the Exactly-N Problem. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 2:1-2:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{linial_et_al:LIPIcs.CCC.2021.2, author = {Linial, Nati and Shraibman, Adi}, title = {{An Improved Protocol for the Exactly-N Problem}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {2:1--2:8}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.2}, URN = {urn:nbn:de:0030-drops-142760}, doi = {10.4230/LIPIcs.CCC.2021.2}, annote = {Keywords: Communication complexity, Number-On-the-Forehead, Corner-free sets} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

We study the multiparty communication complexity of high dimensional permutations in the Number On the Forehead (NOF) model. This model is due to Chandra, Furst and Lipton (CFL) who also gave a nontrivial protocol for the Exactly-n problem where three players receive integer inputs and need to decide if their inputs sum to a given integer n. There is a considerable body of literature dealing with the same problem, where (N,+) is replaced by some other abelian group. Our work can be viewed as a far-reaching extension of this line of research. We show that the known lower bounds for that group-theoretic problem apply to all high dimensional permutations. We introduce new proof techniques that reveal new and unexpected connections between NOF communication complexity of permutations and a variety of well-known problems in combinatorics. We also give a direct algorithmic protocol for Exactly-n. In contrast, all previous constructions relied on large sets of integers without a 3-term arithmetic progression.

Nati Linial, Toniann Pitassi, and Adi Shraibman. On the Communication Complexity of High-Dimensional Permutations. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 54:1-54:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{linial_et_al:LIPIcs.ITCS.2019.54, author = {Linial, Nati and Pitassi, Toniann and Shraibman, Adi}, title = {{On the Communication Complexity of High-Dimensional Permutations}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {54:1--54:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.54}, URN = {urn:nbn:de:0030-drops-101470}, doi = {10.4230/LIPIcs.ITCS.2019.54}, annote = {Keywords: High dimensional permutations, Number On the Forehead model, Additive combinatorics} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Given a matrix A, we study how many epsilon-cubes are required to cover the convex hull of the columns of A. We show bounds on this cover number in terms of VC dimension and the gamma_2 norm and give algorithms for enumerating elements of a cover. This leads to algorithms for computing approximate Nash equilibria that unify and extend several previous results in the literature. Moreover, our approximation algorithms can be applied quite generally to a family of quadratic optimization problems that also includes finding the k-by-k combinatorial rectangle of a matrix. In particular, for this problem we give the first quasi-polynomial time additive approximation algorithm that works for any matrix A in [0,1]^{m x n}.

Noga Alon, Troy Lee, and Adi Shraibman. The Cover Number of a Matrix and its Algorithmic Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 34-47, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{alon_et_al:LIPIcs.APPROX-RANDOM.2014.34, author = {Alon, Noga and Lee, Troy and Shraibman, Adi}, title = {{The Cover Number of a Matrix and its Algorithmic Applications}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {34--47}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.34}, URN = {urn:nbn:de:0030-drops-46865}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.34}, annote = {Keywords: Approximation algorithms, Approximate Nash equilibria, Cover number, VC dimension} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 8381, Computational Complexity of Discrete Problems (2008)

Abstract: We will discuss a general norm based framework for showing
lower bounds on communication complexity. An advantage of this approach is that one can use duality theory to obtain a lower bound quantity phrased as a
maximization problem, which can be more convenient to work with in showing lower bounds.
We discuss two applications of this approach.
1. The approximation rank of a matrix A is the minimum rank of a
matrix close to A in ell_infty norm. The logarithm of approximation rank lower bounds quantum communication complexity and is one of the most powerful techniques available, albeit difficult to compute in practice. We
show that an approximation norm known as gamma_2 is polynomially
related to approximation rank.
This results in a polynomial time algorithm to approximate
approximation rank, and also shows that the logarithm of approximation rank lower bounds quantum communication complexity even with entanglement which was previously not known.
2. By means of an approximation norm which lower bounds multiparty
number-on-the-forehead complexity, we show non-trivial lower bounds on the complexity of the disjointness function for up to c log log n players, c <1.

Troy Lee and Adi Shraibman. Approximation norms and duality for communication complexity lower bounds. In Computational Complexity of Discrete Problems. Dagstuhl Seminar Proceedings, Volume 8381, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{lee_et_al:DagSemProc.08381.3, author = {Lee, Troy and Shraibman, Adi}, title = {{Approximation norms and duality for communication complexity lower bounds}}, booktitle = {Computational Complexity of Discrete Problems}, pages = {1--9}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8381}, editor = {Peter Bro Miltersen and R\"{u}diger Reischuk and Georg Schnitger and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08381.3}, URN = {urn:nbn:de:0030-drops-17768}, doi = {10.4230/DagSemProc.08381.3}, annote = {Keywords: Communication complexity, lower bounds} }

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