## Abstract

A graph G on n vertices is ε-far from property P if one should add/delete at least εn^{2} edges to turn G into a graph satisfying P. A distance estimator for P is an algorithm that given G and α, ε > 0 distinguishes between the case that G is (α − ε)-close to P and the case that G is α-far from P. If P has a distance estimator whose query complexity depends only on ε, then P is said to be estimable. Every estimable property is clearly also testable, since testing corresponds to estimating with α = ε. A central result in the area of property testing is the Fischer–Newman theorem, stating that an inverse statement also holds, that is, that every testable property is in fact estimable. The proof of Fischer and Newmann was highly ineffective, since it incurred a tower-type loss when transforming a testing algorithm for P into a distance estimator. This raised the natural problem, studied recently by Fiat–Ron and by Hoppen–Kohayakawa–Lang–Lefmann–Stagni, whether one can find a transformation with a polynomial loss. We obtain the following results. We show that if P is hereditary, then one can turn a tester for P into a distance estimator with an exponential loss. This is an exponential improvement over the result of Hoppen et. al., who obtained a transformation with a double exponential loss. We show that for every P, one can turn a testing algorithm for P into a distance estimator with a double exponential loss. This improves over the transformation of Fischer–Newman that incurred a tower-type loss. Our main conceptual contribution in this work is that we manage to turn the approach of Fischer–Newman, which was inherently ineffective, into an efficient one. On the technical level, our main contribution is in establishing certain properties of Frieze–Kannan Weak Regular partitions that are of independent interest.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023 |

Editors | Nicole Megow, Adam Smith |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772969 |

DOIs | |

State | Published - Sep 2023 |

Event | 26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023 - Atlanta, United States Duration: 11 Sep 2023 → 13 Sep 2023 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 275 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023 |
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Country/Territory | United States |

City | Atlanta |

Period | 11/09/23 → 13/09/23 |

### Funding

Funders | Funder number |
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NSF-BSF | 20196 |

European Research Council | 863438 |

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung | 200021_196965 |

## Keywords

- Frieze-Kannan Regularity
- Testing
- estimation
- graph theory
- randomized algorithms
- weak regularity