TY - GEN

T1 - Optimal tiling of the euclidean space using permutation-symmetric bodies

AU - Braverman, Mark

AU - Minzer, Dor

N1 - Publisher Copyright:
© Mark Braverman and Dor Minzer; licensed under Creative Commons License CC-BY 4.0

PY - 2021/7/1

Y1 - 2021/7/1

N2 - What is the least surface area of a permutation-symmetric body B whose Zn translations tile Rn? Since any such body must have volume 1, the isoperimetric inequality implies that its surface area must be at least Ω(√n). Remarkably, Kindler et al. showed that for general bodies B this is tight, i.e. that there is a tiling body of Rn whose surface area is O(√n). In theoretical computer science, the tiling problem is intimately related to the study of parallel repetition theorems (which are an important component in PCPs), and more specifically in the question of whether a “strong version” of the parallel repetition theorem holds. Raz showed, using the odd cycle game, that strong parallel repetition fails in general, and subsequently these ideas were used in order to construct non-trivial tilings of Rn. In this paper, motivated by the study of a symmetric parallel repetition, we consider the permutation-symmetric variant of the tiling problem in Rn. We show that any permutation-symmetric body that tiles Rn must have surface area at least Ω(n/√log n), and that this bound is tight, i.e. that there is a permutation-symmetric tiling body of Rn with surface area O(n/√log n). We also give matching bounds for the value of the symmetric parallel repetition of Raz's odd cycle game. Our result suggests that while strong parallel repetition fails in general, there may be important special cases where it still applies.

AB - What is the least surface area of a permutation-symmetric body B whose Zn translations tile Rn? Since any such body must have volume 1, the isoperimetric inequality implies that its surface area must be at least Ω(√n). Remarkably, Kindler et al. showed that for general bodies B this is tight, i.e. that there is a tiling body of Rn whose surface area is O(√n). In theoretical computer science, the tiling problem is intimately related to the study of parallel repetition theorems (which are an important component in PCPs), and more specifically in the question of whether a “strong version” of the parallel repetition theorem holds. Raz showed, using the odd cycle game, that strong parallel repetition fails in general, and subsequently these ideas were used in order to construct non-trivial tilings of Rn. In this paper, motivated by the study of a symmetric parallel repetition, we consider the permutation-symmetric variant of the tiling problem in Rn. We show that any permutation-symmetric body that tiles Rn must have surface area at least Ω(n/√log n), and that this bound is tight, i.e. that there is a permutation-symmetric tiling body of Rn with surface area O(n/√log n). We also give matching bounds for the value of the symmetric parallel repetition of Raz's odd cycle game. Our result suggests that while strong parallel repetition fails in general, there may be important special cases where it still applies.

KW - PCP

KW - Parallel repetition

KW - Tilings

UR - http://www.scopus.com/inward/record.url?scp=85115312441&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.CCC.2021.5

DO - 10.4230/LIPIcs.CCC.2021.5

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AN - SCOPUS:85115312441

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 36th Computational Complexity Conference, CCC 2021

A2 - Kabanets, Valentine

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 36th Computational Complexity Conference, CCC 2021

Y2 - 20 July 2021 through 23 July 2021

ER -