Every Property of Hyperfinite Graphs Is Testable

Newman, Ilan; Sohler, Christian

doi:10.1137/120890946

Cited by 76 publications

(110 citation statements)

References 15 publications

(39 reference statements)

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“…Newman and Sohler [21] showed that every graph property of a hyperfinite undirected graph is testable with constant query complexity in the bounded degree (undirected) graph model, where the testing algorithm in the bounded degree graph model is allowed to query the neighbors of any given vertex. This together with Theorem 1.1 implies the following theorem.…”

Section: Testing Hyperfinite Propertiesmentioning

confidence: 99%

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Relating two property testing models for bounded degree directed graphs

Czumaj

Pan

Sohler

2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

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We study property testing algorithms in directed graphs (digraphs) with maximum indegree and maximum outdegree upper bounded by d. For directed graphs with bounded degree, there are two different models in property testing introduced by Bender and Ron (2002). In the bidirectional model, one can access both incoming and outgoing edges while in the unidirectional model one can only access outgoing edges. In our paper we provide a new relation between the two models: we prove that if a property can be tested with constant query complexity in the bidirectional model, then it can be tested with sublinear query complexity in the unidirectional model.A corollary of this result is that in the unidirectional model (the model allowing only queries to the outgoing neighbors), every property in hyperfinite digraphs is testable with sublinear query complexity.

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Section: Testing Hyperfinite Propertiesmentioning

confidence: 99%

“…It is known that every graph property of hyperfinite graphs is testable with query complexity O ε,d (1) [21] and the result here follows by an easy modification of the proofs, which we only sketch here.…”

Section: Testing Hyperfinite Propertiesmentioning

confidence: 99%

Relating two property testing models for bounded degree directed graphs

Czumaj

Pan

Sohler

2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

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“…(of Lemma 11.2) Our lemma holds for simple planar graphs with a vertex degree bound by the main result of [22] (see also Theorem 5. in [15]). Our strategy is to reduce our lemma to the Newman-Sohler result, by enconding our Q 1 -edge colored, directed graph with a simple planar graph.…”

Section: Benjamini-schramm Convergence Implies the Convergence Of Strmentioning

confidence: 85%

Infinite dimensional representations of finite dimensional algebras and amenability

Elek

2017

Math. Ann.

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We present a novel quantitative approach to the representation theory of finite dimensional algebras motivated by the emerging theory of graph limits.We introduce the rank spectrum of a finite dimensional algebra R over a finite field. The elements of the rank spectrum are representations of the algebra into von Neumann regular rank algebras, and two representations are considered to be equivalent if they induce the same Sylvester rank functions on R-matrices.Based on this approach, we can divide the finite dimensional algebras into three types: finite, amenable and non-amenable representation types. We prove that string algebras are of amenable representation type, but the wild Kronecker algebras are not. Here, the amenability of the rank algebras associated to the limit points in the rank spectrum plays a very important part.We also show that the limit points of finite dimensional representations of algebras of amenable representation type can always be viewed as representations of the algebra in the continuous ring invented by John von Neumann in the 1930's.As an application in algorithm theory, we introduce and study the notion of parameter testing of modules over finite dimensional algebras, that is analogous to the testing of bounded degree graphs introduced by Goldreich and Ron. We shall see that for string algebras all the reasonable (stable) parameters are testable.

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“…A notable example of hyperfinite graphs is planar graphs. Exploiting this result, [23] showed that every hyperfinite property is constant-query testable.…”

Section: Introductionmentioning

confidence: 95%

“…In the bounded-degree model with a degree bound d, we only deal with graphs with maximum degree at most d. We have access to the incidence list and a graph G is called -far from a property P if we must add or remove at least dn edges to make G satisfy P . [23] showed that we can test H-isomorphism in constant time when H is hyperfinite, that is, for any > 0 we can decompose H into connected components of size s = s( ) by deleting n edges. A notable example of hyperfinite graphs is planar graphs.…”

Section: Introductionmentioning

confidence: 99%

Testing Linear-Invariant Function Isomorphism

Wimmer

Yoshida

2013

Automata, Languages, and Programming

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Abstract. A function f : F n 2 → {−1, 1} is called linear-isomorphic to g if f = g • A for some non-singular matrix A. In the g-isomorphism problem, we want a randomized algorithm that distinguishes whether an input function f is linear-isomorphic to g or far from being so. We show that the query complexity to test g-isomorphism is essentially determined by the spectral norm of g. That is, if g is close to having spectral norm s, then we can test g-isomorphism with poly(s) queries, and if g is far from having spectral norm s, then we cannot test g-isomorphism with o(log s) queries. The upper bound is almost tight since there is indeed a function g close to having spectral norm s whereas testing gisomorphism requires Ω(s) queries. As far as we know, our result is the first characterization of this type for functions. Our upper bound is essentially the Kushilevitz-Mansour learning algorithm, modified for use in the implicit setting. Exploiting our upper bound, we show that any property is testable if it can be well-approximated by functions with small spectral norm. We also extend our algorithm to the setting where A is allowed to be singular.

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Every Property of Hyperfinite Graphs Is Testable

Cited by 76 publications

References 15 publications

Relating two property testing models for bounded degree directed graphs

Relating two property testing models for bounded degree directed graphs

Infinite dimensional representations of finite dimensional algebras and amenability

Testing Linear-Invariant Function Isomorphism

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