# A Characterization of Testable Hypergraph Properties

@article{Joos2017ACO,
title={A Characterization of Testable Hypergraph Properties},
author={Felix Joos and Jaehoon Kim and Daniela K{\"u}hn and Deryk Osthus},
journal={2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)},
year={2017},
pages={859-867}
}
• Published 11 July 2017
• Mathematics
• 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
We provide a combinatorial characterization of all testable properties of k-graphs (i.e. k-uniform hypergraphs). Here, a k-graph property P is testable if there is a randomized algorithm which makes a bounded number of edge queries and distinguishes with probability 2/3 between k-graphs that satisfy P and those that are far from satisfying P. For the 2-graph case, such a combinatorial characterization was obtained by Alon, Fischer, Newman and Shapira. Our results for the k-graph setting are in…

## Figures from this paper

Earthmover Resilience and Testing in Ordered Structures
• Mathematics, Computer Science
Electron. Colloquium Comput. Complex.
• 2018
A wide class of properties of ordered structures - the earthmover resilient (ER) properties - are identified and it is shown that the "good behavior" of such properties allows us to obtain general testability results that are similar to (and more general than) those of unordered graphs.
Edge Correlations in Random Regular Hypergraphs and Applications to Subgraph Testing
• Mathematics
SIAM J. Discret. Math.
• 2019
An edge-switching technique for hypergraphs is developed which allows us to show that correlations are limited for a large range of densities, and several corollaries on subgraph counts in random $d$-regularhypergraphs are deduced.
Testability of Homomorphism Inadmissibility: Property Testing Meets Database Theory
• Mathematics, Computer Science
PODS
• 2019
The characterization shows that homomorphism inadmissibility from A is constant-query testable with one-sided error if and only if the core of A is alpha-acyclic; this result generalizes existing results for testing subgraph-freeness in the general graph model.
On the query complexity of estimating the distance to hereditary graph properties
• Mathematics, Computer Science
SIAM J. Discret. Math.
• 2021
It is proved that the normalized edit distance of any given graph to be induced or induced is estimable with a query complexity that depends only on the bounds of the Frieze--Kannan Regularity Lemma and on a Removal Lemma for $\mathcal{F}$.

## References

SHOWING 1-10 OF 66 REFERENCES
Efficient testing of large graphs
• Mathematics
40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)
• 1999
This theorem is used to prove that first order graph properties not containing a quantifier alternation of type "/spl forall//spl exist/" are always testable, while it is shown that some properties containing this alternation are not.
Property testing in hypergraphs and the removal lemma
• Mathematics
STOC '07
• 2007
This paper proves thathereditary graph properties are testable (with one-sided error) for hypergraphs, an immediate consequence of a (hyper)graph theoretic statement, which is an extension of the so-called removal lemma.
A combinatorial characterization of the testable graph properties: it's all about regularity
• Mathematics, Computer Science
STOC '06
• 2006
One of the main open problems in the area of property-testing, which was raised in the 1996 paper of Goldreich, Goldwasser and Ron, is resolved by a purely combinatorial characterization of the graph properties that are testable with a constant number of queries.
Efficient Testing of Hypergraphs
• Mathematics
ICALP
• 2002
It is proved that if more than ?n3 (? > 0) triples must be added or deleted from a 3-graph H on n vertices to destroy all induced copies of F, then H must contain ? cn |V(F)| induced copiesof F, as long as n ? n0(?,F).
Efficient testing of hypergraphs: (Extended abstract)
• Mathematics
• 2002
We investigate a basic problem in combinatorial property testing, in the sense of Goldreich, Goldwasser, and Ron [9,10], in the context of 3-uniform hypergraphs, or 3-graphs for short. As customary,
Testing properties of graphs and functions
• Mathematics
• 2008
We define an analytic version of the graph property testing problem, which can be formulated as studying an unknown 2-variable symmetric function through sampling from its domain and studying the
Every Monotone 3-Graph Property is Testable
• Mathematics
Electron. Notes Discret. Math.
• 2005
H is η-far from P if no hypergraph G with |E(G)4E(H)| ≤ ηn satisfies P .
Every minor-closed property of sparse graphs is testable
• Mathematics
Electron. Colloquium Comput. Complex.
• 2008
The proof combines results from the theory of graph minors with results on convergent sequences of sparse graphs, which rely on martingale arguments, to infer that many well studied graph properties, like being planar, outer-planar, series-parallel, bounded genus, bounded tree-width and several others, are testable with a constant number of queries.
Every property of hyperfinite graphs is testable
• Mathematics
STOC '11
• 2011
It is shown that the structure of a planar graph on large enough number of vertices, n, and with constant maximum degree d, is determined, up to the modification (insertion or deletion) of at most ε d n edges, by the frequency of k-discs for certain k=k(ε,d) that is independent of the size of the graph.
Testability and repair of hereditary hypergraph properties
• Mathematics
Random Struct. Algorithms
• 2010
This paper extends the result to continuous graphs on probability spaces, and shows that the repair algorithm is "local" in the sense that it only depends on a bounded amount of data; in particular, the graph can be repaired in a time linear in the number of edges.