## Abstract

We initiate the study of property testing problems concerning relations between permutations. In such problems, the input is a tuple (σ1,.., σd) of permutations on\{1,.., n\}, and one wishes to determine whether this tuple satisfies a certain system of relations E, or is far from every tuple that satisfies E. If this computational problem can be solved by querying only a small number of entries of the given permutations, we say that E is testable. For example, when d=2 and E consists of the single relation\mathrm{XY}=\mathrm{YX}, this corresponds to testing whether σ1σ2=σ2σ1, where σ1σ2 and σ2σ1 denote composition of permutations. We define a collection of graphs, naturally associated with the system E, that encodes all the information relevant to the testability of E. We then prove two theorems that provide criteria for testability and non-testability in terms of expansion properties of these graphs. By virtue of a deep connection with group theory, both theorems are applicable to wide classes of systems of relations. In addition, we formulate the well-studied group-theoretic notion of stability in permutations as a special case of the testa-bility notion above, interpret all previous works on stability as testability results, survey previous results on stability from a computational perspective, and describe many directions for future research on stability and testability. This is an extended abstract. The full version is available at https://arxiv.org/abs/2011.05234. All references beyond Sections I and II refer to the full version.

Original language | English |
---|---|

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

Publisher | Institute of Electrical and Electronics Engineers |

Pages | 286-297 |

Number of pages | 12 |

ISBN (Electronic) | 9781665420556 |

DOIs | |

State | Published - 1 Jan 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 |
---|---|

Volume | 2022-February |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021 |
---|---|

Country/Territory | United States |

City | Virtual, Online |

Period | 7/02/22 → 10/02/22 |

## Keywords

- Graph expansion via group theory
- Property testing
- Stability in permutations
- Testability of relations between permutations

## ASJC Scopus subject areas

- General Computer Science