Fast Property Testing and Metrics for Permutations

Abstract

The goal of property testing is to quickly distinguish between objects which satisfy a property and objects that are -far from satisfying the property. There are now several general results in this area which show that natural properties of combinatorial objects can be tested with "constant" query complexity, depending only on and the property, and not on the size of the object being tested. The upper bound on the query complexity coming from the proof techniques is often enormous and impractical. It remains a major open problem if better bounds hold.Maybe surprisingly, for testing with respect to the rectangular distance, we prove there is a universal (not depending on the property), polynomial in 1/ query complexity bound for two-sided testing hereditary properties of sufficiently large permutations. We further give a nearly linear bound with respect to a closely related metric which also depends on the smallest forbidden subpermutation for the property. Finally, we show that several different permutation metrics of interest are related to the rectangular distance, yielding similar results for testing with respect to these metrics.

Finding and Counting Permutations via CSPs

“…Newman et al [47] study pattern matching in a property-testing framework (aiming to distinguish pattern-avoiding sequences from those that contain many copies of the pattern). In this setting, the focus is on the query complexity of different approaches, and sampling techniques are often used; see also [9,32].…”

Property testing and parameter estimation

“…a k-blow-up of a sample Q of size q. An analogous statement is true when considering other combinatorial objects, like graphs [ES83], permutation [FW18,Lemma 3.2], etc.…”

“…More than that, this sample complexity is universal, that is, it does not depend on the property P being tested. However, the testers given by this result need to assume that the input configuration is larger than some n 0 , which depends on the property P. More precisely, n 0 depends on a parameter that we call the blow-up parameter k * P of P, which is directly inspired in [FW18] (see also the function ψ F in [AS08b,AFNS09]). The blow-up parameter will be formally defined in Section 11.2.…”

“…We define the concept of blow-ups of configurations, which is analogous to its counterpart in [FW18] for permutations. In Fig.…”

“…The main argument used to prove Theorem 11.2 resembles the one used by Fox and Wei [FW18] to show that, under a certain notion of distance, hereditary permutation properties are testable with polynomial (on the error parameter) sample complexity. It also applies the Fox-Pach-Suk Regularity Lemma for uniform semi-algebraic hypergraphs [FPS16] to get a regular partition of configurations of points.…”