GSF-locality is not sufficient for proximity-oblivious testing

TL;DR

This paper demonstrates that GSF-locality alone does not guarantee the existence of proximity-oblivious testers by providing a counterexample of a GSF-local property that propagates.

Contribution

It shows that the non-propagation condition is not necessary for GSF-local properties to admit proximity-oblivious testing, answering an open question.

Findings

Identified a GSF-local property that propagates and is not testable by POTs.

Connected FO properties with GSF-local properties through neighborhood profiles.

Provided a counterexample to the necessity of the non-propagation condition.

Abstract

In Property Testing, proximity-oblivious testers (POTs) form a class of particularly simple testing algorithms, where a basic test is performed a number of times that may depend on the proximity parameter, but the basic test itself is independent of the proximity parameter. In their seminal work, Goldreich and Ron [STOC 2009; SICOMP 2011] show that the graph properties that allow constant-query proximity-oblivious testing in the bounded-degree model are precisely the properties that can be expressed as a generalised subgraph freeness (GSF) property that satisfies the non-propagation condition. It is left open whether the non-propagation condition is necessary. Indeed, calling properties expressible as a generalised subgraph freeness property GSF-local properties, they ask whether all GSF-local properties are non-propagating. We give a negative answer by exhibiting a property of graphs that is GSF-local and propagating. Hence in particular, our property does not admit a POT, despite being GSF-local. We prove our result by exploiting a recent work of the authors which constructed a first-order (FO) property that is not testable [SODA 2021], and a new connection between FO properties and GSF-local properties via neighbourhood profiles.