Graph limits and parameter testing

Borgs, Christian; Chayes, Jennifer; Lovász, László; Sós, Vera T.; Szegedy, Balázs; Vesztergombi, K.

doi:10.1145/1132516.1132556

Cited by 126 publications

(93 citation statements)

References 18 publications

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“…The technique used in [3] lays in the core of the proof of Theorem 1.1. Similar results on testing of hereditary properties were obtained by Lovász and Szegedy in [18] using convergent graph sequences (see also [13]). It should also be possible to apply this approach, initiated in [19], to give an alternative proof of Theorem 1.2.…”

Section: Related Worksupporting

confidence: 84%

What is the furthest graph from a hereditary property?

Alon

Stav

2008

Random Struct Algorithms

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For a graph property P, the edit distance of a graph G from P, denoted E P (G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G in order to turn it into a graph satisfying P. What is the furthest graph on n vertices from P and what is the largest possible edit distance from P? Denote this maximal distance by ed(n, P). This question is motivated by algorithmic edge-modification problems, in which one wishes to find or approximate the value of E P (G) given an input graph G.A monotone graph property is closed under removal of edges and vertices. Trivially, for any monotone property, the largest edit distance is attained by a complete graph. We show that this is a simple instance of a much broader phenomenon. A hereditary graph property is closed under removal of vertices. We prove that for any hereditary graph property P, a random graph with an edge density that depends on P essentially achieves the maximal distance from P, that is: ed(n, P) = E P (G(n, p(P))) + o(n 2 ) with high probability. The proofs combine several tools, including strengthened versions of the Szemerédi Regularity Lemma, properties of random graphs and probabilistic arguments.

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Section: Related Worksupporting

confidence: 84%

What is the furthest graph from a hereditary property?

Alon

Stav

2008

Random Struct Algorithms

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“…Subsequently the study of combinatorial property testing, and in particular, graph property testing has developed into a rich study and by now we have almost complete understanding (at least in the dense-graph model) of which graph properties are locally testable [Alon et al, 2006, Borgs et al, 2006.…”

Section: Algebraic Property Testingmentioning

confidence: 99%

“…Section 3.1),case when m = 1 and the functions of interest map K to a prime subfield F p , there was no interesting relationships known between the degrees of the functions in the family and the locality of the test. And such understanding is essential to get a characterization of affine-invariant locally testable codes that would be analogous to the characterizations of graph properties of [Alon et al, 2006, Borgs et al, 2006.…”

Section: Algebraic Property Testingmentioning

confidence: 99%

Limits on the Rate of Locally Testable Affine-Invariant Codes

Ben–Sasson¹,

Sudan²

2011

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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Despite its many applications, to program checking, probabilistically checkable proofs, locally testable and locally decodable codes, and cryptography, "algebraic property testing" is not wellunderstood. A significant obstacle to a better understanding, was a lack of a concrete definition that abstracted known testable algebraic properties and reflected their testability. This obstacle was removed by [Kaufman and Sudan, STOC 2008] who considered (linear) "affine-invariant properties", i.e., properties that are closed under summation, and under affine transformations of the domain. Kaufman and Sudan showed that these two features (linearity of the property and its affine-invariance) play a central role in the testability of many known algebraic properties. However their work does not give a complete characterization of the testability of affine-invariant properties, and several technical obstacles need to be overcome to obtain such a characterization. Indeed, their work left open the tantalizing possibility that locally testable codes of rate dramatically better than that of the family of Reed-Muller codes (the most popular form of locally testable codes, which also happen to be affine-invariant) could be found by systematically exploring the space of affine-invariant properties.In this work we rule out this possibility and show that general (linear) affine-invariant properties are contained in Reed-Muller codes that are testable with a slightly larger query complexity. A central impediment to proving such results was the limited understanding of the structural restrictions on affine-invariant properties imposed by the existence of local tests. We manage to overcome this limitation and present a clean restriction satisfied by affine-invariant properties that exhibit local tests. We do so by relating the problem to that of studying the set of solutions of a certain nice class of (uniform, homogenous, diagonal) systems of multivariate polynomial equations. Our main technical result completely characterizes (combinatorially) the set of zeroes, or algebraic set, of such systems.

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“…This line of research was sparked by limits of dense graphs [7][8][9]32], which we focus on here, followed by limits of other structures, e.g. permutations [22,23,28], sparse graphs [5,14] and partial orders [25].…”

Section: Introductionmentioning

confidence: 99%

Weak regularity and finitely forcible graph limits

Cooper

Kaiser

Kráľ

et al. 2018

Trans. Amer. Math. Soc.

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Graphons are analytic objects representing limits of convergent sequences of graphs. Lovász and Szegedy conjectured that every finitely forcible graphon, i.e. any graphon determined by finitely many graph densities, has a simple structure. In particular, one of their conjectures would imply that every finitely forcible graphon has a weak ε-regular partition with the number of parts bounded by a polynomial in ε −1 . We construct a finitely forcible graphon W such that the number of parts in any weak ε-regular partition of W is at least exponential in ε −2 /2 5 log * ε −2 . This bound almost matches the known upper bound for graphs and, in a certain sense, is the best possible for graphons.

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Graph limits and parameter testing

Cited by 126 publications

References 18 publications

What is the furthest graph from a hereditary property?

What is the furthest graph from a hereditary property?

Limits on the Rate of Locally Testable Affine-Invariant Codes

Weak regularity and finitely forcible graph limits

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