A lower bound for testing 3-colorability in bounded-degree graphs

Bogdanov, Andrej; Obata, K.; Trevisan, Luca

doi:10.1109/sfcs.2002.1181886

Cited by 65 publications

(96 citation statements)

References 14 publications

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“…3 Testing k-colorability has previously been studied in the neighbor query model for the case that k = 3 and the graph has constant maximum degree (that is, d = O(1), and furthermore, the maximum degree d max is O(1) as well). In this case, Bogdanov et al [6] proved that is necessary to perform Ω(n) queries (that is, there is no algorithm with sublinear query complexity).…”

Section: Related Work Onmentioning

confidence: 99%

“…Next we build on a reduction from [6] to get a lower bound of Ω n d for two-sided error algorithms in the group query model. The lower bound in [6] is for the case k = 3, and therefore we start with a reduction for general k. by an induced copy of F (while keeping all edges between S and V (G ′ ) \ S), and we denote the resulting graph by G ′′ , then we have:…”

Section: Corollary 47 Every One-sided Error Algorithm For Testingmentioning

confidence: 99%

“…The lower bound in [6] is for the case k = 3, and therefore we start with a reduction for general k. by an induced copy of F (while keeping all edges between S and V (G ′ ) \ S), and we denote the resulting graph by G ′′ , then we have:…”

Section: Corollary 47 Every One-sided Error Algorithm For Testingmentioning

confidence: 99%

“…We first prove the bound for k = 3, and in the end of the proof we explain how to adjust it to other values of k. The authors of [6] show that there exist two distributions of b-regular graphs (for a constant b) overñ vertices with the following properties. All graphs in the support of the first distribution, denoted D 3col , are 3-colorable, and with high probability over the choice of a graph according to the second distribution, denoted D 3col-far , the graph is Θ(1)-far from being 3-colorable.…”

Section: Corollary 47 Every One-sided Error Algorithm For Testingmentioning

confidence: 99%

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Comparing the Strength of Query Types in Property Testing: The Case of Testing k-Colorability

et al. 2010

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Abstract. We study the power of four query models in the context of property testing in general graphs, where our main case study is the problem of testing k-colorability. Two query types, which have been studied extensively in the past, are pair queries and neighbor queries. The former corresponds to asking whether there is an edge between any particular pair of vertices, and the latter to asking for the i th neighbor of a particular vertex. We show that while for pair queries, testing kcolorability requires a number of queries that is a monotone decreasing function in the average degree d, the query complexity in the case of neighbor queries remains roughly the same for every density and for large values of k. We also consider a combined model that allows both types of queries, and we propose a new, stronger, query model, related to the field of Group Testing. We give upper and lower bounds on the query complexity for one-sided error in all the models, where the bounds are nearly tight for three of the models. In some of the cases our lower bounds extend to two-sided error algorithms. The problem of testing k-colorability was previously studied in the contexts of dense graphs and of sparse graphs, and in our proofs we unify approaches from those cases, and also provide some new tools and techniques that may be of independent interest.

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Section: Related Work Onmentioning

confidence: 99%

Section: Corollary 47 Every One-sided Error Algorithm For Testingmentioning

confidence: 99%

Section: Corollary 47 Every One-sided Error Algorithm For Testingmentioning

confidence: 99%

Section: Corollary 47 Every One-sided Error Algorithm For Testingmentioning

confidence: 99%

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Comparing the Strength of Query Types in Property Testing: The Case of Testing k-Colorability

et al. 2010

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“…One of our reductions is strong gap-preserving local reduction, which maps a decision problem to another decision problem. This is derived from the gap-preserving lo-cal reduction introduced by [1]. The other one is strong Lreduction, which maps an optimization problem to another optimization problem.…”

Section: Introductionmentioning

confidence: 99%

Query-Number Preserving Reductions and Linear Lower Bounds for Testing

Yoshida

Ito

2010

IEICE Trans. Inf. & Syst.

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SUMMARYIn this paper, we study lower bounds on the query complexity of testing algorithms for various problems. Given an oracle that returns information of an input object, a testing algorithm distinguishes the case that the object has a given property P from the case that it has a large distance to having P with probability at least 2 3 . The query complexity of an algorithm is measured by the number of accesses to the oracle. We introduce two reductions that preserve the query complexity. One is derived from the gap-preserving local reduction and the other is from the L-reduction. By using the former reduction, we show linear lower bounds on the query complexity for testing basic NP-complete properties, i.e., 3-edge-colorability, directed Hamiltonian path/cycle, undirected Hamiltonian path/cycle, 3-dimensional matching and NP-complete generalized satisfiability problems. Also, using the second reduction, we show a linear lower bound on the query complexity of approximating the size of the maximum 3-dimensional matching.

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Testing monotonicity over graph products

Halevy

Kushilevitz

2008

Random Struct Algorithms

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ABSTRACT:We consider the problem of monotonicity testing over graph products. Monotonicity testing is one of the central problems studied in the field of property testing. We present a testing approach that enables us to use known monotonicity testers for given graphs G 1 , G 2 , to test monotonicity over their product G 1 × G 2 . Such an approach of reducing monotonicity testing over a graph product to monotonicity testing over the original graphs, has been previously used in the special case of monotonicity testing over [n] d for a limited type of testers; in this article, we show that this approach can be applied to allow modular design of testers in many interesting cases: this approach works whenever the functions are boolean, and also in certain cases for functions with a general range. We demonstrate the usefulness of our results by showing how a careful use of this approach improves the query complexity of known testers. Specifically, based on our results, we provide a new analysis for the known tester for [n] d which significantly improves its query complexity analysis in the low-dimensional case. For example, when d = O(1), we reduce the best known query complexity from O(log 2 n/ ) to O(log n/ ).

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A lower bound for testing 3-colorability in bounded-degree graphs

Cited by 65 publications

References 14 publications

Comparing the Strength of Query Types in Property Testing: The Case of Testing k-Colorability

Comparing the Strength of Query Types in Property Testing: The Case of Testing k-Colorability

Query-Number Preserving Reductions and Linear Lower Bounds for Testing

Testing monotonicity over graph products

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