Hereditary properties of permutations are strongly testable

TL;DR

This paper proves that hereditary permutation properties are strongly testable using Kendall's tau distance, and establishes a link between rectangular and Kendall's tau distances for such properties.

Contribution

It resolves an open problem by demonstrating strong testability of hereditary permutation properties and confirms a conjecture relating two permutation distance measures.

Findings

Hereditary permutation properties are strongly testable.

A bound exists for subpermutation size to test properties.

Rectangular and Kendall's tau distances are closely related for hereditary properties.

Abstract

We show that for every hereditary permutation property P and every eps>0, there exists an integer M such that if a permutation p is eps-far from P in the Kendall's tau distance, then a random subpermutation of p of order M has the property P with probability at most eps. This settles an open problem whether hereditary permutation properties are strongly testable, i.e., testable with respect to the Kendall's tau distance. In addition, our method also yields a proof of a conjecture of Hoppen, Kohayakawa, Moreira and Sampaio on the relation of the rectangular distance and the Kendall's tau distance of a permutation from a hereditary property.