Understanding parallel repetition requires understanding foams

Uriel Feige; Guy Kindler; Ryan O'Donnell

doi:10.1109/CCC.2007.39

Understanding parallel repetition requires understanding foams

Uriel Feige^*, Guy Kindler, Ryan O'Donnell

^*Corresponding author for this work

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

34 Scopus citations

Abstract

Motivated by the study of Parallel Repetition and also by the Unique Games Conjecture, we investigate the value of the "Odd Cycle Games" under parallel repetition. Using tools from discrete harmonic analysis, we show that after d rounds on the cycle of length m, the value of the game is at most 1 - (1/m) · Ω̃U(√d) (for d ≤ m², say). This beats the natural barrier of 1 - ⊖(1/m)² · d for Raz-style proofs [31, 21 j (see [11]) and also the SDP bound of Feige-Lovász [14, 17]; however, it just barely fails to have implications for Unique Games. On the other hand, we also show that improving our bound would require proving nontrivial lower bounds on the surface area of high-dimensional foams. Specifically, one would need to answer: What is the least surface area of a cell that tiles ℝ^d by the lattice ℤ^d ?

Original language	English
Title of host publication	Proceedings - Twenty-Second Annual IEEE Conference on Computational Complexity, CCC 2007
Pages	179-192
Number of pages	14
DOIs	https://doi.org/10.1109/CCC.2007.39
State	Published - 2007
Externally published	Yes
Event	22nd Annual IEEE Conference on Computational Complexity, CCC 2007 - San Diego, CA, United States Duration: 13 Jun 2007 → 16 Jun 2007

Publication series

Name	Proceedings of the Annual IEEE Conference on Computational Complexity
ISSN (Print)	1093-0159

Conference

Conference	22nd Annual IEEE Conference on Computational Complexity, CCC 2007
Country/Territory	United States
City	San Diego, CA
Period	13/06/07 → 16/06/07

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10.1109/CCC.2007.39

Cite this

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title = "Understanding parallel repetition requires understanding foams",

abstract = "Motivated by the study of Parallel Repetition and also by the Unique Games Conjecture, we investigate the value of the {"}Odd Cycle Games{"} under parallel repetition. Using tools from discrete harmonic analysis, we show that after d rounds on the cycle of length m, the value of the game is at most 1 - (1/m) · {\~Ω}U(√d) (for d ≤ m2, say). This beats the natural barrier of 1 - ⊖(1/m)2 · d for Raz-style proofs [31, 21 j (see [11]) and also the SDP bound of Feige-Lov{\'a}sz [14, 17]; however, it just barely fails to have implications for Unique Games. On the other hand, we also show that improving our bound would require proving nontrivial lower bounds on the surface area of high-dimensional foams. Specifically, one would need to answer: What is the least surface area of a cell that tiles ℝd by the lattice ℤd ?",

author = "Uriel Feige and Guy Kindler and Ryan O'Donnell",

year = "2007",

doi = "10.1109/CCC.2007.39",

language = "אנגלית",

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series = "Proceedings of the Annual IEEE Conference on Computational Complexity",

pages = "179--192",

booktitle = "Proceedings - Twenty-Second Annual IEEE Conference on Computational Complexity, CCC 2007",

note = "22nd Annual IEEE Conference on Computational Complexity, CCC 2007 ; Conference date: 13-06-2007 Through 16-06-2007",

}

Feige, U, Kindler, G & O'Donnell, R 2007, Understanding parallel repetition requires understanding foams. in Proceedings - Twenty-Second Annual IEEE Conference on Computational Complexity, CCC 2007., 4262762, Proceedings of the Annual IEEE Conference on Computational Complexity, pp. 179-192, 22nd Annual IEEE Conference on Computational Complexity, CCC 2007, San Diego, CA, United States, 13/06/07. https://doi.org/10.1109/CCC.2007.39

Understanding parallel repetition requires understanding foams. / Feige, Uriel; Kindler, Guy; O'Donnell, Ryan.
Proceedings - Twenty-Second Annual IEEE Conference on Computational Complexity, CCC 2007. 2007. p. 179-192 4262762 (Proceedings of the Annual IEEE Conference on Computational Complexity).

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

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Understanding parallel repetition requires understanding foams

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