Efficient testing of large graphs

Alon, Noga; Krivelevich, Michael; Fischer, Eldar; Szegedy, Márió

doi:10.1109/sffcs.1999.814642

Cited by 177 publications

(680 citation statements)

References 10 publications

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“…Again, Alon and Krivelevich [3] improved the analysis of the algorithm and obtained a bound ofÕ( 2 / 4 ) on the query complexity and 2Õ ( / 2 ) on the running time. Alon et al [1] presented another "constant-time" (i.e., independent of the size of the input graph) property testing algorithm; their algorithm uses the Szemerédi Regularity Lemma, and therefore the bounds for the query complexity and the running time, though independent of the size of the graph, have huge dependency of and . Fischer [15] extended the methods from [1] and investigated more general graph colorability properties.…”

Section: Context and Related Workmentioning

confidence: 99%

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Testing Hypergraph Coloring

Czumaj

Sohler

2001

Automata, Languages and Programming

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Abstract. In this paper we initiate the study of testing properties of hypergraphs. The goal of property testing is to distinguish between the case whether a given object has a certain property or is "far away" from the property. We prove that the fundamental problem of -colorability of k-uniform hypergraphs can be tested in time independent of the size of the hypergraph. We present a testing algorithm that examines only (k / ) O(k) entries of the adjacency matrix of the input hypergraph, where is a distance parameter independent of the size of the hypergraph. Notice that this algorithm tests only a constant number of entries in the adjacency matrix provided that , k, and are constant.

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Section: Context and Related Workmentioning

confidence: 99%

“…Recently, many researchers started studying a relaxation of the "exact decision task" and considered various forms of approximation algorithms for decision problems. In property testing (see, e.g., [1,11,13,17,16,18,19,26,29]), one considers the following class of problems:…”

Section: Introductionmentioning

confidence: 99%

Testing Hypergraph Coloring

Czumaj

Sohler

2001

Automata, Languages and Programming

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“…That is, with prefixes of these patterns and when equality is allowed, four firstorder quantifiers suffice to express an untestable graph property. The proof is a modification of the proof by Alon et al [1], which is made possible by the presence of directed edges and loops.…”

Section: Introductionmentioning

confidence: 90%

“…However, the (possible) loops seem to affect the notion of indistinguishability defined by Alon et al [1]. We use the following modification of Definition 2.…”

Section: Preliminariesmentioning

confidence: 99%

“…The study of combinatorial property testing was initiated by Goldreich et al [8], who focused on graphs. Alon et al [1] first considered the classification problem for testability, although they restricted their attention to undirected, loop-free graphs. They showed that all such first-order sentences 1 with quantifier prefixes of the form ∃ * ∀ * express testable properties.…”

Section: Introductionmentioning

confidence: 99%

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Untestable Properties Expressible with Four First-Order Quantifiers

Jordan

Zeugmann

2010

Language and Automata Theory and Applications

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Abstract. In property testing, the goal is to distinguish between structures that have some desired property and those that are far from having the property, after examining only a small, random sample of the structure. We focus on the classification of first-order sentences based on their quantifier prefixes and vocabulary into testable and untestable classes. This classification was initiated by Alon et al.[1], who showed that graph properties expressible with quantifier patterns ∃ * ∀ * are testable but that there is an untestable graph property expressible with quantifier pattern ∀ * ∃ * . In the present paper, their untestable example is simplified. In particular, it is shown that there is an untestable graph property expressible with each of the following quantifier patterns: ∀∃∀∃, ∀∃∀ 2 , ∀ 2 ∃∀ and ∀ 3 ∃.

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Stability‐type results for hereditary properties

Alon

Stav

2009

Journal of Graph Theory

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The classical Stability Theorem of Erdős and Simonovits can be stated as follows. For a monotone graph property P, let t ≥ 2 be such that t + 1 = min{χ(H) : H / ∈ P}. Then any graph G * ∈ P on n vertices, which was obtained by removing at most (n 2 edges from the complete graph G = K n , has edit distance o(n 2 ) to T n (t), the Turán graph on n vertices with t parts.In this paper we extend the above notion of stability to hereditary graph properties. It turns out that to do so the complete graph K n has to be replaced by a random graph. For a hereditary graph property P, consider modifying the edges of a random graph G = G(n, 1/2) to obtain a graph G * that satisfies P in (essentially) the most economical way. We obtain necessary and sufficient conditions on P which guarantee that G * has a unique structure. In such cases, for a pair of integers (r, s) which depends on P, G * has distance o(n 2 ) to a graph T n (r, s, 1 2 ) almost surely. Here T n (r, s, 1 2 ) denotes a graph which consists of almost equal sized r + s parts, r of them induce an independent set, s induce a clique and all the bipartite graphs between parts are quasi-random (with edge density 1 2 ). In addition, several strengthened versions of this result are shown.

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Efficient testing of large graphs

Cited by 177 publications

References 10 publications

Testing Hypergraph Coloring

Testing Hypergraph Coloring

Untestable Properties Expressible with Four First-Order Quantifiers

Stability‐type results for hereditary properties

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