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GSF-Locality Is Not Sufficient For Proximity-Oblivious Testing

Authors Isolde Adler, Noleen Köhler, Pan Peng

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Isolde Adler
  • School of Computing, University of Leeds, UK
Noleen Köhler
  • School of Computing, University of Leeds, UK
Pan Peng
  • Department of Computer Science, University of Sheffield, UK

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Isolde Adler, Noleen Köhler, and Pan Peng. GSF-Locality Is Not Sufficient For Proximity-Oblivious Testing. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 34:1-34:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.CCC.2021.34

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Finite Model Theory
Keywords
  • Property testing
  • proximity-oblivous testing
  • locality
  • first-order logic
  • lower bound

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References

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