Hardness Condensation by Restriction

Mika Göös; Ilan Newman; Artur Riazanov; Dmitry Sokolov

doi:10.1145/3618260.3649711

Hardness Condensation by Restriction

Mika Göös, Ilan Newman, Artur Riazanov, Dmitry Sokolov

Department of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

Can every n-bit boolean function with deterministic query complexity k≪ n be restricted to O(k) variables such that the query complexity remains ω(k)? That is, can query complexity be condensed via restriction? We study such hardness condensation questions in both query and communication complexity, proving two main results. Negative: Query complexity cannot be condensed in general: There is a function f with query complexity k such that any restriction of f to O(k) variables has query complexity O(k^3/4). Positive: Randomised communication complexity can be condensed for the sink-of-xor function. This yields a quantitatively improved counterexample to the log-approximate-rank conjecture, achieving parameters conjectured by Chattopadhyay, Garg, and Sherif (2021). Along the way we show the existence of Shearer extractors - a new type of seeded extractor whose output bits satisfy prescribed dependencies across distinct seeds.

Original language	English
Title of host publication	STOC 2024 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing
Editors	Bojan Mohar, Igor Shinkar, Ryan O�Donnell
Publisher	Association for Computing Machinery
Pages	2016-2027
Number of pages	12
ISBN (Electronic)	9798400703836
DOIs	https://doi.org/10.1145/3618260.3649711
State	Published - 10 Jun 2024
Event	56th Annual ACM Symposium on Theory of Computing, STOC 2024 - Vancouver, Canada Duration: 24 Jun 2024 → 28 Jun 2024

Publication series

Name	Proceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)	0737-8017

Conference

Conference	56th Annual ACM Symposium on Theory of Computing, STOC 2024
Country/Territory	Canada
City	Vancouver
Period	24/06/24 → 28/06/24

Bibliographical note

Publisher Copyright:
© 2024 Copyright is held by the owner/author(s). Publication rights licensed to ACM.

Keywords

communication complexity
hardness condensation
query complexity

ASJC Scopus subject areas

Software

Access to Document

10.1145/3618260.3649711

Cite this

Göös, M., Newman, I., Riazanov, A., & Sokolov, D. (2024). Hardness Condensation by Restriction. In B. Mohar, I. Shinkar, & R. O�Donnell (Eds.), STOC 2024 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing (pp. 2016-2027). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/3618260.3649711

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abstract = "Can every n-bit boolean function with deterministic query complexity k≪ n be restricted to O(k) variables such that the query complexity remains ω(k)? That is, can query complexity be condensed via restriction? We study such hardness condensation questions in both query and communication complexity, proving two main results. Negative: Query complexity cannot be condensed in general: There is a function f with query complexity k such that any restriction of f to O(k) variables has query complexity O(k3/4). Positive: Randomised communication complexity can be condensed for the sink-of-xor function. This yields a quantitatively improved counterexample to the log-approximate-rank conjecture, achieving parameters conjectured by Chattopadhyay, Garg, and Sherif (2021). Along the way we show the existence of Shearer extractors - a new type of seeded extractor whose output bits satisfy prescribed dependencies across distinct seeds.",

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Göös, M, Newman, I, Riazanov, A & Sokolov, D 2024, Hardness Condensation by Restriction. in B Mohar, I Shinkar & R O�Donnell (eds), STOC 2024 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing. Proceedings of the Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, pp. 2016-2027, 56th Annual ACM Symposium on Theory of Computing, STOC 2024, Vancouver, Canada, 24/06/24. https://doi.org/10.1145/3618260.3649711

Hardness Condensation by Restriction. / Göös, Mika; Newman, Ilan; Riazanov, Artur et al.
STOC 2024 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing. ed. / Bojan Mohar; Igor Shinkar; Ryan O�Donnell. Association for Computing Machinery, 2024. p. 2016-2027 (Proceedings of the Annual ACM Symposium on Theory of Computing).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - Can every n-bit boolean function with deterministic query complexity k≪ n be restricted to O(k) variables such that the query complexity remains ω(k)? That is, can query complexity be condensed via restriction? We study such hardness condensation questions in both query and communication complexity, proving two main results. Negative: Query complexity cannot be condensed in general: There is a function f with query complexity k such that any restriction of f to O(k) variables has query complexity O(k3/4). Positive: Randomised communication complexity can be condensed for the sink-of-xor function. This yields a quantitatively improved counterexample to the log-approximate-rank conjecture, achieving parameters conjectured by Chattopadhyay, Garg, and Sherif (2021). Along the way we show the existence of Shearer extractors - a new type of seeded extractor whose output bits satisfy prescribed dependencies across distinct seeds.

AB - Can every n-bit boolean function with deterministic query complexity k≪ n be restricted to O(k) variables such that the query complexity remains ω(k)? That is, can query complexity be condensed via restriction? We study such hardness condensation questions in both query and communication complexity, proving two main results. Negative: Query complexity cannot be condensed in general: There is a function f with query complexity k such that any restriction of f to O(k) variables has query complexity O(k3/4). Positive: Randomised communication complexity can be condensed for the sink-of-xor function. This yields a quantitatively improved counterexample to the log-approximate-rank conjecture, achieving parameters conjectured by Chattopadhyay, Garg, and Sherif (2021). Along the way we show the existence of Shearer extractors - a new type of seeded extractor whose output bits satisfy prescribed dependencies across distinct seeds.

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Hardness Condensation by Restriction

Abstract

Publication series

Conference

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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