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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2026.106

Spiky Rank and Its Applications to Rigidity and Circuits

Authors: Lianna Hambardzumyan, Konstantin Myasnikov, Artur Riazanov, Morgan Shirley, and Adi Shraibman

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract

We introduce spiky rank, a new matrix parameter that enhances blocky rank by combining the combinatorial structure of the latter with linear-algebraic flexibility. A spiky matrix is block-structured with diagonal blocks that are arbitrary rank-one matrices, and the spiky rank of a matrix is the minimum number of such matrices required to express it as a sum. This measure extends blocky rank to real matrices and is more robust for problems with both combinatorial and algebraic character. Our conceptual contribution is as follows: we propose spiky rank as a well-behaved candidate matrix complexity measure and demonstrate its potential through applications. We show that large spiky rank implies high matrix rigidity, and that spiky rank lower bounds yield lower bounds for depth-2 ReLU circuits, the basic building blocks of neural networks. On the technical side, we establish tight bounds for random matrices and develop a framework for explicit lower bounds, applying it to Hamming distance matrices and spectral expanders. Finally, we relate spiky rank to other matrix parameters, including blocky rank, sparsity, and the γ₂-norm.

Cite as

Lianna Hambardzumyan, Konstantin Myasnikov, Artur Riazanov, Morgan Shirley, and Adi Shraibman. Spiky Rank and Its Applications to Rigidity and Circuits. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 106:1-106:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{hambardzumyan_et_al:LIPIcs.ICALP.2026.106,
  author =	{Hambardzumyan, Lianna and Myasnikov, Konstantin and Riazanov, Artur and Shirley, Morgan and Shraibman, Adi},
  title =	{{Spiky Rank and Its Applications to Rigidity and Circuits}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{106:1--106:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael 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.2026.106},
  URN =		{urn:nbn:de:0030-drops-264954},
  doi =		{10.4230/LIPIcs.ICALP.2026.106},
  annote =	{Keywords: blocky rank, matrix rigidity, ReLU circuits, spiky rank}
}

Document

DOI: 10.4230/LIPIcs.STACS.2026.35

A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity

Authors: Pavel Dvořák, Bruno Loff, and Suhail Sherif

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract

We are interested in what happens when we take a Π₁ combinatorial statement, write its negation as a homogeneous quadratic feasibility problem (HQFP), and relax the problem into a positive semidefinite feasibility problem. This question is particularly interesting owing to the fact that any statement written as a PSD feasibility problem can be proven or disproven using a short proof. We investigate this for one very simple and one very complicated statement. The simple statement we look at is the pigeonhole principle. We prove that the relaxed negation of the PHP remains unsatisfiable and we thus obtain a new "quantum" pigeonhole principle (QPHP) which is a stronger statement than the vanilla PHP. It states that if we take n copies of the same state, and measure each copy using a measurement with only n-1 outcomes (the measurement can be different for different copies), then there will be an outcome j and two copies i₁, i₂ where the resulting states, obtained when the outcome is j for both copies, are not orthogonal. We then look at the statement "the deterministic communication complexity of f is ≤ k", where f could be either a function or a relation. We write this statement in two equivalent ways, using two different HQFPs. By relaxing to PSD feasibility, we increase the set of available protocols, and thus we always get a communication model which is stronger than deterministic communication complexity. An argument from proof complexity shows that any model obtained in this way will solve all Karchmer-Wigderson games efficiently. However, the argument is very indirect and does not give us an explicit protocol that solves the Karchmer-Wigderson games. We then work to find such protocols in the two communication models obtained by relaxing our two formulations. When relaxing the first of the two formulations we obtain a structured variant of the γ₂ norm. This communication model is to subunit γ₂ norm matrices like deterministic protocols are to rectangles, and so we call the protocols in this model γ₂ protocols. We show that log-inverse-discrepancy is a lower-bound for this model. We then show how to compute equality (deterministically) using O(1) bits of γ₂-communication, which implies that KW games are easy in the model. When relaxing the second of the two formulations we obtain what we call quantum lab protocols. This model happens to have a functional description, wherein Alice and Bob communicate solely via the outcomes of binary measurements of a shared quantum state (whose initial state is independent of the inputs). They are required to give the correct output with zero error probability. We use our QPHP to prove a lower-bound of n against two-round quantum lab protocols for equality. However we also show that any Boolean function f can be computed in three rounds and four measurements.

Cite as

Pavel Dvořák, Bruno Loff, and Suhail Sherif. A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dvorak_et_al:LIPIcs.STACS.2026.35,
  author =	{Dvo\v{r}\'{a}k, Pavel and Loff, Bruno and Sherif, Suhail},
  title =	{{A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{35:1--35:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.35},
  URN =		{urn:nbn:de:0030-drops-255243},
  doi =		{10.4230/LIPIcs.STACS.2026.35},
  annote =	{Keywords: Proofs, Semidefinite Programs, Quantum Pigeonhole Principle, Communication Complexity}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2025.37

Communication Complexity of Equality and Error-Correcting Codes

Authors: Dale Jacobs, John Jeang, Vladimir Podolskii, Morgan Prior, and Ilya Volkovich

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract

We study the public-coin randomized communication complexity of the equality function. The communication complexity of this function is known to be low when the error probability is constant and the players have access to many random bits. The complexity grows, however, if the allowed error probability and the amount of randomness are restricted. We show that public-coin randomized protocols for equality and error-correcting codes are essentially the same object. That is, given a protocol for equality, we can construct a code, and vice versa. We substantially extend the protocol-implies-code direction: any protocol computing a function with a large fooling set can be converted into an error-correcting code. As a corollary, we show that among functions with a fooling set of size s, equality on log s bits has the least randomized communication complexity, regardless of the restrictions on the error probability and the amount of randomness. Finally, we use the connection to error-correcting codes to analyze the randomized communication complexity of equality for varying restrictions on the error probability and the amount of randomness. In most cases, we provide tight bounds. We pinpoint the setting in which tight bounds are still unknown.

Cite as

Dale Jacobs, John Jeang, Vladimir Podolskii, Morgan Prior, and Ilya Volkovich. Communication Complexity of Equality and Error-Correcting Codes. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 37:1-37:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jacobs_et_al:LIPIcs.FSTTCS.2025.37,
  author =	{Jacobs, Dale and Jeang, John and Podolskii, Vladimir and Prior, Morgan and Volkovich, Ilya},
  title =	{{Communication Complexity of Equality and Error-Correcting Codes}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{37:1--37:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.37},
  URN =		{urn:nbn:de:0030-drops-251175},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.37},
  annote =	{Keywords: communication complexity, randomized communication complexity, error-correcting codes}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.39

Consumable Data via Quantum Communication

Authors: Dar Gilboa, Siddhartha Jain, and Jarrod R. McClean

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract

Classical data can be copied and re-used for computation, with adverse consequences economically and in terms of data privacy. Motivated by this, we formulate problems in one-way communication complexity where Alice holds some data x and Bob holds m inputs y_1, …, y_m. They want to compute m instances of a bipartite relation R(⋅,⋅) on every pair (x, y_1), …, (x, y_m). We call this the asymmetric direct sum question for one-way communication. We give examples where the quantum communication complexity of such problems scales polynomially with m, while the classical communication complexity depends at most logarithmically on m. Thus, for such problems, data behaves like a consumable resource that is effectively destroyed upon use when the owner stores and transmits it as quantum states, but not when transmitted classically. We show an application to a strategic data-selling game, and discuss other potential economic implications.

Cite as

Dar Gilboa, Siddhartha Jain, and Jarrod R. McClean. Consumable Data via Quantum Communication. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 39:1-39:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gilboa_et_al:LIPIcs.APPROX/RANDOM.2025.39,
  author =	{Gilboa, Dar and Jain, Siddhartha and McClean, Jarrod R.},
  title =	{{Consumable Data via Quantum Communication}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{39:1--39:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.39},
  URN =		{urn:nbn:de:0030-drops-244059},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.39},
  annote =	{Keywords: quantum communication, one-time programs, data markets}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.58

Equality Is Far Weaker Than Constant-Cost Communication

Authors: Mika Göös, Nathaniel Harms, and Artur Riazanov

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract

We exhibit an n-bit communication problem with a constant-cost randomized protocol but which requires n^Ω(1) deterministic (or even non-deterministic) queries to an Equality oracle. Therefore, even constant-cost randomized protocols cannot be efficiently "derandomized" using Equality oracles. This improves on several recent results and answers a question from the survey of Hatami and Hatami (SIGACT News 2024). It also gives a significantly simpler and quantitatively superior proof of the main result of Fang, Göös, Harms, and Hatami (STOC 2025), that constant-cost communication does not reduce to the k-Hamming Distance hierarchy.

Cite as

Mika Göös, Nathaniel Harms, and Artur Riazanov. Equality Is Far Weaker Than Constant-Cost Communication. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{goos_et_al:LIPIcs.APPROX/RANDOM.2025.58,
  author =	{G\"{o}\"{o}s, Mika and Harms, Nathaniel and Riazanov, Artur},
  title =	{{Equality Is Far Weaker Than Constant-Cost Communication}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{58:1--58:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.58},
  URN =		{urn:nbn:de:0030-drops-244246},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.58},
  annote =	{Keywords: Equality oracle, constant-cost communication, gamma-2 norm, spectral norm}
}

Document

DOI: 10.4230/LIPIcs.CCC.2025.16

Direct Sums for Parity Decision Trees

Authors: Tyler Besselman, Mika Göös, Siyao Guo, Gilbert Maystre, and Weiqiang Yuan

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract

Direct sum theorems state that the cost of solving k instances of a problem is at least Ω(k) times the cost of solving a single instance. We prove the first such results in the randomised parity decision tree model. We show that a direct sum theorem holds whenever (1) the lower bound for parity decision trees is proved using the discrepancy method; or (2) the lower bound is proved relative to a product distribution.

Cite as

Tyler Besselman, Mika Göös, Siyao Guo, Gilbert Maystre, and Weiqiang Yuan. Direct Sums for Parity Decision Trees. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 16:1-16:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{besselman_et_al:LIPIcs.CCC.2025.16,
  author =	{Besselman, Tyler and G\"{o}\"{o}s, Mika and Guo, Siyao and Maystre, Gilbert and Yuan, Weiqiang},
  title =	{{Direct Sums for Parity Decision Trees}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{16:1--16:38},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.16},
  URN =		{urn:nbn:de:0030-drops-237105},
  doi =		{10.4230/LIPIcs.CCC.2025.16},
  annote =	{Keywords: direct sum, parity decision trees, query complexity}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2025.79

Catalytic Communication

Authors: Edward Pyne, Nathan S. Sheffield, and William Wang

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract

The study of space-bounded computation has drawn extensively from ideas and results in the field of communication complexity. Catalytic Computation (Buhrman, Cleve, Koucký, Loff and Speelman, STOC 2013) studies the power of bounded space augmented with a pre-filled hard drive that can be used non-destructively during the computation. Presently, many structural questions in this model remain open. Towards a better understanding of catalytic space, we define a model of catalytic communication complexity and prove new upper and lower bounds. In our model, Alice and Bob share a blackboard with a tiny number of free bits, and a larger section with an arbitrary initial configuration. They must jointly compute a function of their inputs, communicating only via the blackboard, and must always reset the blackboard to its initial configuration. We prove several upper and lower bounds: 1) We characterize the simplest nontrivial model, that of one bit of free space and three rounds, in terms of 𝔽₂ rank. In particular, we give natural problems that are solvable with a minimal-sized blackboard that require near-maximal (randomized) communication complexity, and vice versa. 2) We show that allowing constantly many free bits, as opposed to one, allows an exponential improvement on the size of the blackboard for natural problems. To do so, we connect the problem to existence questions in extremal graph theory. 3) We give tight connections between our model and standard notions of non-uniform catalytic computation. Using this connection, we show that with an arbitrary constant number of rounds and bits of free space, one can compute all functions in TC⁰. We view this model as a step toward understanding the value of filled space in computation.

Cite as

Edward Pyne, Nathan S. Sheffield, and William Wang. Catalytic Communication. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 79:1-79:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{pyne_et_al:LIPIcs.ITCS.2025.79,
  author =	{Pyne, Edward and Sheffield, Nathan S. and Wang, William},
  title =	{{Catalytic Communication}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{79:1--79:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.79},
  URN =		{urn:nbn:de:0030-drops-227076},
  doi =		{10.4230/LIPIcs.ITCS.2025.79},
  annote =	{Keywords: Catalytic computation, Branching programs, Communication complexity}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2025.37

A Lower Bound on the Trace Norm of Boolean Matrices and Its Applications

Authors: Tsun-Ming Cheung, Hamed Hatami, Kaave Hosseini, Aleksandar Nikolov, Toniann Pitassi, and Morgan Shirley

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract

We present a simple method based on a variant of Hölder’s inequality to lower-bound the trace norm of Boolean matrices. As the main result, we obtain an exponential separation between the randomized decision tree depth and the spectral norm (i.e. the Fourier L₁-norm) of a Boolean function. This answers an open question of Cheung, Hatami, Hosseini and Shirley (CCC 2023). As immediate consequences, we obtain the following results. - We give an exponential separation between the logarithm of the randomized and the deterministic parity decision tree size. This is in sharp contrast with the standard binary decision tree setting where the logarithms of randomized and deterministic decision tree size are essentially polynomially related, as shown recently by Chattopadhyay, Dahiya, Mande, Radhakrishnan, and Sanyal (STOC 2023). - We give an exponential separation between the approximate and the exact spectral norm for Boolean functions. - We give an exponential separation for XOR functions between the deterministic communication complexity with oracle access to Equality function (D^EQ) and randomized communication complexity. Previously, such a separation was known for general Boolean matrices by Chattopadhyay, Lovett, and Vinyals (CCC 2019) using the Integer Inner Product (IIP) function. - Finally, our method gives an elementary and short proof for the mentioned exponential D^EQ lower bound of Chattopadhyay, Lovett, and Vinyals for Integer Inner Product (IIP).

Cite as

Tsun-Ming Cheung, Hamed Hatami, Kaave Hosseini, Aleksandar Nikolov, Toniann Pitassi, and Morgan Shirley. A Lower Bound on the Trace Norm of Boolean Matrices and Its Applications. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cheung_et_al:LIPIcs.ITCS.2025.37,
  author =	{Cheung, Tsun-Ming and Hatami, Hamed and Hosseini, Kaave and Nikolov, Aleksandar and Pitassi, Toniann and Shirley, Morgan},
  title =	{{A Lower Bound on the Trace Norm of Boolean Matrices and Its Applications}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{37:1--37:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.37},
  URN =		{urn:nbn:de:0030-drops-226654},
  doi =		{10.4230/LIPIcs.ITCS.2025.37},
  annote =	{Keywords: Boolean function complexity, parity decision trees, randomized communication complexity}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2024.58

An Improved Protocol for ExactlyN with More Than 3 Players

Authors: Lianna Hambardzumyan, Toniann Pitassi, Suhail Sherif, Morgan Shirley, and Adi Shraibman

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

Abstract

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.

Cite as

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}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2023.89

The Strength of Equality Oracles in Communication

Authors: Toniann Pitassi, Morgan Shirley, and Adi Shraibman

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

Abstract

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.

Cite as

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}
}

Document

DOI: 10.4230/LIPIcs.CCC.2021.2

An Improved Protocol for the Exactly-N Problem

Authors: Nati Linial and Adi Shraibman

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

Abstract

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].

Cite as

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}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2019.54

On the Communication Complexity of High-Dimensional Permutations

Authors: Nati Linial, Toniann Pitassi, and Adi Shraibman

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

Abstract

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.

Cite as

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}
}

Document

DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.34

The Cover Number of a Matrix and its Algorithmic Applications

Authors: Noga Alon, Troy Lee, and Adi Shraibman

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

Abstract

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}.

Cite as

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}
}

Document

DOI: 10.4230/DagSemProc.08381.3

Approximation norms and duality for communication complexity lower bounds

Authors: Troy Lee and Adi Shraibman

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

Abstract

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.

Cite as

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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