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Larger Corner-Free Sets from Combinatorial Degenerations

Authors Matthias Christandl, Omar Fawzi, Hoang Ta, Jeroen Zuiddam

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Author Details

Matthias Christandl
  • Department of Mathematical Sciences, University of Copenhagen, Denmark
Omar Fawzi
  • Univ. Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, France
Hoang Ta
  • Univ. Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, France
Jeroen Zuiddam
  • Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands

Funding

  • Christandl, Matthias: European Research Council (ERC Grant Agreement No. 818761), the VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059) and the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union’s Horizon 2020 Programme (QuantAlgo project) via the Innovation Fund Denmark.
  • Fawzi, Omar: European Research Council (ERC Grant Agreement No. 851716).
  • Ta, Hoang: European Research Council (ERC Grant Agreement No. 851716), LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d’Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

Cite As Get BibTex

Matthias Christandl, Omar Fawzi, Hoang Ta, and Jeroen Zuiddam. Larger Corner-Free Sets from Combinatorial Degenerations. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 48:1-48:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.48

Abstract

There is a large and important collection of Ramsey-type combinatorial problems, closely related to central problems in complexity theory, that can be formulated in terms of the asymptotic growth of the size of the maximum independent sets in powers of a fixed small hypergraph, also called the Shannon capacity. An important instance of this is the corner problem studied in the context of multiparty communication complexity in the Number On the Forehead (NOF) model. Versions of this problem and the NOF connection have seen much interest (and progress) in recent works of Linial, Pitassi and Shraibman (ITCS 2019) and Linial and Shraibman (CCC 2021).
We introduce and study a general algebraic method for lower bounding the Shannon capacity of directed hypergraphs via combinatorial degenerations, a combinatorial kind of "approximation" of subgraphs that originates from the study of matrix multiplication in algebraic complexity theory (and which play an important role there) but which we use in a novel way. 
Using the combinatorial degeneration method, we make progress on the corner problem by explicitly constructing a corner-free subset in F₂ⁿ × F₂ⁿ of size Ω(3.39ⁿ/poly(n)), which improves the previous lower bound Ω(2.82ⁿ) of Linial, Pitassi and Shraibman (ITCS 2019) and which gets us closer to the best upper bound 4^{n - o(n)}. Our new construction of corner-free sets implies an improved NOF protocol for the Eval problem. In the Eval problem over a group G, three players need to determine whether their inputs x₁, x₂, x₃ ∈ G sum to zero. We find that the NOF communication complexity of the Eval problem over F₂ⁿ is at most 0.24n + 𝒪(log n), which improves the previous upper bound 0.5n + 𝒪(log n).

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Algebraic complexity theory
  • Mathematics of computing → Discrete mathematics
Keywords
  • Corner-free sets
  • communication complexity
  • number on the forehead
  • combinatorial degeneration
  • hypergraphs
  • Shannon capacity
  • eval problem

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