Tight Bounds for Testing Bipartiteness in General Graphs

Kaufman, Tali; Krivelevich, Michael; Ron, Dana

doi:10.1137/s0097539703436424

Cited by 86 publications

(134 citation statements)

References 19 publications

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“…Informally, the notion of an oblivious tester means that the size of the input is not an important resource when studying property testing of "natural" graph properties in the dense graph model, such as hereditary properties, whose definition is independent of the input size. We stress that as opposed to the dense graph model, property-testing in the bounded degree model [13] and the general density model [20,15], usually requires query complexity, which depends on the size of the input graph. Therefore, the notion of oblivious testing is not adequate for those models.…”

Section: The Main Result: Uniform Vs Non-uniform Property Testingmentioning

confidence: 99%

A separation theorem in property testing

Alon

Shapira

2008

Combinatorica

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Consider the following seemingly rhetorical question: Is it crucial for a property-tester to know the error parameter in advance? Previous papers dealing with various testing problems, suggest that the answer may be no, as in these papers there was no loss of generality in assuming that is given as part of the input, and is not known in advance. Our main result in this paper, however, is that it is possible to separate a natural model of property testing in which is given as part of the input from the model in which is known in advance (without making any computational hardness assumptions). To this end, we construct a graph property P, which has the following two properties:(i) There is no tester for P accepting as part of the input, whose number of queries depends only on .(ii) For any fixed , there is a tester for P (that works only for that specific ), which makes a constant number of queries.Interestingly, we manage to construct a separating property P, which is combinatorially natural as it can be expressed in terms of forbidden subgraphs and also computationally natural as it can be shown to belong to coN P . The main tools in this paper are efficiently constructible graphs of high girth and high chromatic number, a result about testing monotone graph properties, as well as basic ideas from the theory of recursive functions. Of independent interest is a precise characterization of the monotone graph properties that can be tested with being part of the input, which we obtain as one of the main steps of the paper.

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Section: The Main Result: Uniform Vs Non-uniform Property Testingmentioning

confidence: 99%

A separation theorem in property testing

Alon

Shapira

2008

Combinatorica

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“…Focusing on properties that have efficient testing algorithms for bounded-degree and general sparse graphs, we ask which of these also have efficient distance approximation algorithms. We establish that all properties shown to have efficient property testing algorithms in [GR02] also have efficient distance 1 A third model, appropriate for testing properties of graphs that are neither dense nor sparse [KKR04], also allows vertex-pair queries.…”

Section: Introductionmentioning

confidence: 90%

Distance Approximation in Bounded-Degree and General Sparse Graphs

Marko

Ron

2006

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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We address the problem of approximating the distance of bounded degree and general sparse graphs from having some predetermined graph property P. Namely, we are interested in sublinear algorithms for estimating the fraction of edges that should be added to / removed from a graph so that it obtains P. This fraction is taken with respect to a given upper bound m on the number of edges. In particular, for graphs with degree bound d over n vertices, m = dn. To perform such an approximation the algorithm may ask for the degree of any vertex of its choice, and may ask for the neighbors of any vertex.The problem of estimating the distance to having a property was first explicitly addressed by Parnas et. al. (ECCC 2004). In the context of graphs this problem was studied by Fischer and Newman (FOCS 2005) in the dense-graphs model. In this model the fraction of edge modifications is taken with respect to n 2 , and the algorithm may ask for the existence of an edge between any pair of vertices of its choice. Fischer and Newman showed that every graph property that has a testing algorithm in this model with query complexity that is independent of the size of the graph, also has a distance-approximation algorithm with query complexity that is independent of the size of the graph.In this work we focus on bounded-degree and general sparse graphs, and give algorithms for all properties that were shown to have efficient testing algorithms by Goldreich and Ron (Algorithmica, 2002). Specifically, these properties are k-edge connectivity, subgraph-freeness (for constant size subgraphs), being a Eulerian graph, and cycle-freeness. A variant of our subgraph-freeness algorithm approximates the size of a minimum vertex cover of a graph in sublinear time. This approximation improves on a recent result of Parnas and Ron (ECCC 2005).

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“…More recently, [PR02] and [KKR04] have proposed a general-graph model in which the tester is given query access to both the adjacency-matrix and the incidence-list representations of the input graph. Distance, in this model, is measured with respect to |E|, which means that the absolute number of allowable incorrect edges cannot be a priori bounded.…”

Section: Sublinear Algorithms and Property Testingmentioning

confidence: 99%

A Nearly-Quadratic Gap between Adaptive and Non-adaptive Property Testers

Hurwitz

2011

Algorithms and Computation

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We show that for all integers t ≥ 8 and arbitrarily small > 0, there exists a graph property Π (which depends on ) such that -testing Π has non-adaptive query complexity Q = Θ(q 2−2/t ), where q = Θ( −1 ) is the adaptive query complexity. This resolves the question of how beneficial adaptivity is, in the context of proximity-dependent properties ([9]). This also gives evidence that the canonical transformation of Goldreich and Trevisan ([8]) is essentially optimal when converting an adaptive property tester to a non-adaptive property tester. To do so, we provide optimal adaptive and non-adaptive testers for the combined property of having maximum degree O( N ) and being a blowup collection of an arbitrary base graph H.

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Tight Bounds for Testing Bipartiteness in General Graphs

Cited by 86 publications

References 19 publications

A separation theorem in property testing

A separation theorem in property testing

Distance Approximation in Bounded-Degree and General Sparse Graphs

A Nearly-Quadratic Gap between Adaptive and Non-adaptive Property Testers

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