Efficient Testing of Large Graphs

  title={Efficient Testing of Large Graphs},
  author={Noga Alon and Eldar Fischer and Michael Krivelevich and Mario Szegedy},
P be a property of graphs. An -test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than edges to make it satisfy P. The property P is called testable, if for every there exists an -test for P whose total number of queries is independent of… 

Testing graphs for colorability properties *

  • E. Fischer
  • Mathematics
    Random Struct. Algorithms
  • 2005
It is proven here that other classes of graph properties, describable by various generalizations of the coloring notion used in Alon et al. are testable, showing that this approach can broaden the understanding of the nature of the testable graph properties.

A combinatorial characterization of the testable graph properties: it's all about regularity

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.

Testing Property of graphs: Introduction and a few examples

A property tester determines whether a graph G = (V,E) has a given property or is far from having the property (‘far’ from having a property will be defined later on depending on the graph

A Characterization of Graph Properties Testable for General Planar Graphs with one-Sided Error (It's all About Forbidden Subgraphs)

  • A. CzumajC. Sohler
  • Mathematics, Computer Science
    2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2019
The sufficient condition in the characterization reduces the problem to the task of testing H-freeness in planar graphs, and is the main and most challenging technical contribution of the paper.

A characterization of testable hypergraph properties [ Extended Abstract ]

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

Testing Hereditary Properties of Nonexpanding Bounded-Degree Graphs

It is shown that every hereditary graph property is testable with a constant number of queries provided that every sufficiently large induced subgraph of the input graph has poor expansion.

Testing Hereditary Properties of Ordered Graphs and Matrices

The proof bridges the gap between techniques related to the regularity lemma, used in the long chain of papers investigating graph testing, and string testing techniques and develops a Ramsey-type lemma for multipartite graphs with undesirable edges.

Efficient Testing of Hypergraphs

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).

Testing graphs against an unknown distribution

This paper completely solve Goldreich’s problem by giving a precise characterization of the graph properties that are testable in the Vertex-Distribution-Free model.

A Characterization of Testable Hypergraph Properties

This work provides a combinatorial characterization of all testable properties of k-graphs (i.e. k-uniform hypergraphs) and shows that for the somewhat stronger concept of local repairability, the testability results for graphs do not extend to the 3-graph setting.



The algorithmic aspects of the regularity lemma

The authors first demonstrate the computational difficulty of finding a regular partition; they show that deciding if a given partition of an input graph satisfies the properties guaranteed by the lemma is co-NP-complete, and prove that despite this difficulty theLemma can be made constructive.

Regular languages are testable with a constant number of queries

This paper discusses testability of more complex languages and shows that the query complexity required for testing context free languages cannot be bounded by any function of /spl epsiv/.

Property testing and its connection to learning and approximation

The authors study the question of determining whether an unknown function has a particular property or is /spl epsiv/-far from any function with that property, and devise algorithms to test whether a graph has properties such as being k-colorable or having a /spl rho/-clique.

On graphs with small subgraphs of large chromatic number

This paper establishes the following generalization which was suggested by Erdös: for each positive constantc and positive integerk there exist positive integersfk(c) andno such that ifG is any graph with more thanno vertices having the property that the chromatic number ofG cannot be made less thank by the omission of at mostcn2 edges, thenG contains ak-chromatic subgraph with at mostfk (c) vertices.

The regularity lemma and approximation schemes for dense problems

  • A. FriezeR. Kannan
  • Mathematics
    Proceedings of 37th Conference on Foundations of Computer Science
  • 1996
The central point here is that the Regularity Lemma provides an explanation of why these Max-SNP hard problems turn out to be easy in dense graphs.

Regular Partitions of Graphs

Abstract : A crucial lemma in recent work of the author (showing that k-term arithmetic progression-free sets of integers must have density zero) stated (approximately) that any large bipartite graph

On Sets of Integers Which Contain No Three Terms in Arithmetical Progression.

  • R. SalemD. Spencer
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1942
example there does not exist a sequence of domains Dl, D2, D3, . .. closing down on the point 0 and such that, for each n, the boundary of D. is compact. ,'Jones, F. B., "Concerning Certain

Robust Characterizations of Polynomials with Applications to Program Testing

The characterizations provide results in the area of coding theory by giving extremely fast and efficient error-detecting schemes for some well-known codes and play a crucial role in subsequent results on the hardness of approximating some NP-optimization problems.

On Sets of Integers Which Contain No Three Terms in Arithmetical Progression.

  • F. Behrend
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1946
By a modification of Salem and Spencer' method, the better estimate 1-_2/2log2 + e v(N) > N VloggN is shown.