## Abstract

Given an alphabet size m inN thought of as a constant, and vec{k}=(k1,.., km}) whose entries sum of up n, the vec{k-multi-slice is the set of vectors x in[m] n in which each symbol i in[m] appears precisely ki times. We show an invariance principle for low-degree functions over the multi-slice, to functions over the product space ([m] n, μ n) in which μ(i)=ki}/n. This answers a question raised by [21]. As applications of the invariance principle, we show: 1)An analogue of the 'dictatorship test implies computational hardness' paradigm for problems with perfect completeness, for a certain class of dictatorship tests. Our computational hardness is proved assuming a recent strengthening of the Unique-Games Conjecture, called the Rich 2-to-1 Games Conjecture. Using this analogue, we show that assuming the Rich 2-to-1 Games Conjecture, (a) there is an r-ary CSP P}r for which it is NP-hard to distinguish satisfiable instances of the CSP and instances that are at most 2r+12r}}+o(1) satisfiable, and (b) hardness of distinguishing 3-colorable graphs, and graphs that do not contain an independent set of size o(1). 2)A reduction of the problem of studying expectations of products of functions on the multi-slice to studying expectations of products of functions on correlated, product spaces. In particular, we are able to deduce analogues of the Gaussian bounds from [38] for the multi-slice. 3)In a companion paper, we show further applications of our invariance principle in extremal combinatorics, and more specifically to proving removal lemmas of a wide family of hypergraphs H called ζ-forests, which is a natural extension of the well-studied case of matchings.

Original language | English |
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Title of host publication | Proceedings - 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science, FOCS 2021 |

Publisher | IEEE Computer Society |

Pages | 228-236 |

Number of pages | 9 |

ISBN (Electronic) | 9781665420556 |

DOIs | |

State | Published - 2022 |

Externally published | Yes |

Event | 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021 - Virtual, Online, United States Duration: 7 Feb 2022 → 10 Feb 2022 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2022-February |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021 |
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Country/Territory | United States |

City | Virtual, Online |

Period | 7/02/22 → 10/02/22 |

### Bibliographical note

Funding Information:Author Braverman was supported by the NSF Alan T. Waterman Award, Grant No. 1933331, a Packard Fellowship in Science and Engineering, and the Simons Collaboration on Algorithms and Geometry. Author Khot was supported by NSF Award CCF-1422159, the Simons Collaboration on Algorithms and Geometry, and the Simons Investigator Award. Author Minzer was supported by by a Sloan Research Fellowship.

Publisher Copyright:

© 2022 IEEE.

## Keywords

- Analysis of Boolean functions
- Hardness of approximation
- PCP