Optimal tiling of the euclidean space using permutation-symmetric bodies

Mark Braverman, Dor Minzer

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

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.

Original languageEnglish (US)
Title of host publication36th Computational Complexity Conference, CCC 2021
EditorsValentine Kabanets
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771931
DOIs
StatePublished - Jul 1 2021
Event36th Computational Complexity Conference, CCC 2021 - Virtual, Toronto, Canada
Duration: Jul 20 2021Jul 23 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume200
ISSN (Print)1868-8969

Conference

Conference36th Computational Complexity Conference, CCC 2021
Country/TerritoryCanada
CityVirtual, Toronto
Period7/20/217/23/21

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Parallel repetition
  • PCP
  • Tilings

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