Testing linear inequalities of subgraph statistics

Lior Gishboliner, Asaf Shapira, Henrique Stagni

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


Property testers are fast randomized algorithms whose task is to distinguish between inputs satisfying some predetermined property P and those that are far from satisfying it. Since these algorithms operate by inspecting a small randomly selected portion of the input, the most natural property one would like to be able to test is whether the input does not contain certain forbidden small substructures. In the setting of graphs, such a result was obtained by Alon et al., who proved that for any finite family of graphs F, the property of being induced F-free (i.e. not containing an induced copy of any F ∈ F) is testable. It is natural to ask if one can go one step further and prove that more elaborate properties involving induced subgraphs are also testable. One such generalization of the result of Alon et al. was formulated by Goldreich and Shinkar who conjectured that for any finite family of graphs F, and any linear inequality involving the densities of the graphs F ∈ F in the input graph, the property of satisfying this inequality can be tested in a certain restricted model of graph property testing. Our main result in this paper disproves this conjecture in the following strong form: some properties of this type are not testable even in the classical (i.e. unrestricted) model of graph property testing. The proof deviates significantly from prior non-testability results in this area. The main idea is to use a linear inequality relating induced subgraph densities in order to encode the property of being a pseudo-random graph.

Original languageEnglish
Title of host publication11th Innovations in Theoretical Computer Science Conference, ITCS 2020
EditorsThomas Vidick
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771344
StatePublished - Jan 2020
Event11th Innovations in Theoretical Computer Science Conference, ITCS 2020 - Seattle, United States
Duration: 12 Jan 202014 Jan 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference11th Innovations in Theoretical Computer Science Conference, ITCS 2020
Country/TerritoryUnited States


  • Graph property testing
  • Subgraph statistics


Dive into the research topics of 'Testing linear inequalities of subgraph statistics'. Together they form a unique fingerprint.

Cite this