Adler, I., Koehler, N. orcid.org/0000-0002-1023-6530 and Peng, P. (2024) On Testability of First-Order Properties in Bounded-Degree Graphs and Connections to Proximity-Oblivious Testing. SIAM Journal on Computing (SICOMP), 53 (4). 825- 883. ISSN 0097-5397
Abstract
We study property testing of properties that are definable in first-order logic (FO) in the bounded-degree graph and relational structure models. We show that any FO property that is defined by a formula with quantifier prefix ∃∗∀∗ is testable (i.e., testable with constant query complexity), while there exists an FO property that is expressible by a formula with quantifier prefix ∀∗∃∗ that is not testable. In the dense graph model, a similar picture has long been known [N. Alon, E. Fischer, M. Krivelevich, and M. Szegedy, Combinatorica, 20 (2000), pp. 451–476] despite the very different nature of the two models. In particular, we obtain our lower bound by an FO formula that defines a class of bounded-degree expanders, based on zig-zag products of graphs. We expect this to be of independent interest. We then use our class of FO definable bounded-degree expanders to answer a long-standing open problem for proximity-oblivious testers (POTs). POTs are 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; SIAM J. Comput., 40 (2011), pp. 534–566] show that the graph properties that are constant-query proximity-oblivious testable in the bounded-degree model are precisely the properties that can be expressed as a generalized 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 generalized subgraph freeness property GSF-local properties, they ask whether all GSF-local properties are non-propagating. We give a negative answer by showing that our FO definable property is GSF-local and propagating. Hence, in particular, our property does not admit a POT, despite being GSF-local. For this result we establish a new connection between FO properties and GSF-local properties via neighborhood profiles.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2024 Society for Industrial and Applied Mathematics. This is an author produced version of an article published in SIAM Journal on Computing. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | graph property testing, first-order logic, proximity-oblivious testing, locality, lower bound |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) > Algorithms & Complexity |
Depositing User: | Symplectic Publications |
Date Deposited: | 10 May 2024 14:54 |
Last Modified: | 26 Jul 2024 13:15 |
Status: | Published |
Publisher: | Society for Industrial and Applied Mathematics |
Identification Number: | 10.1137/23M1556253 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:212370 |