Years and Authors of Summarized Original Work
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1998; Goldreich, Ron
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2004; Kaufman, Krivelevich, Ron
Problem Definition
A graph is bipartite (or 2-colorable) if its vertices can be partitioned into two sets such that there are no edges between pairs of vertices that reside in the same set.
Given a (simple) graph, the task is to determine whether it is bipartite or is “far” from being bipartite. Thus, the standard decision problem is relaxed by allowing any answer when the graph is not bipartite but is “close” to some bipartite graph. We wish to solve this “approximate decision” problem in sublinear time, given access to a data structure that answers adjacency and incidence queries in unit time.
To complete the formulation of the problem, we need to define the distance between graphs and describe how the graph is accessed. The distance between two graphs G1 = (V, E1) and \(G_{2} = (V,E_{2})\) is determined by the symmetric difference between their edge sets (i.e., \(E_{1}\varDelta E_{2}\)...
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Recommended Reading
Ben-Eliezer I, Kaufman T, Krivelevich M, Ron D (2012) Comparing the strength of query types in property testing: the case of testing k-colorability. Comput Complex 22(1):89–135
Bogdanov A, Obata K, Trevisan L (2002) A lower bound for testing 3-colorability in bounded-degree graphs. In: Proceedings of the forty-third annual symposium on foundations of computer science (FOCS), Los Alamitos, pp 93–102
Czumaj A, Goldreich O, Ron D, Seshadhri C, Shapira A, Sohler C (2012) Finding cycles and trees in sublinear time. Technical Report TR12-035, Electronic Colloquium on Computational Complexity (ECCC), to appear in Random Structures and Algorithms
Goldreich O (2010) Introduction to testing graph properties, in [5]
Goldreich O (ed) (2010) Property testing: current research and surveys. LNCS, vol 6390. Springer, Heidelberg
Goldreich O, Ron D (1999) A sublinear bipartite tester for bounded degree graphs. Combinatorica 19(3):335–373
Goldreich O, Ron D (2002) Property testing in bounded degree graphs. Algorithmica 32(2):302–343
Goldreich O, Goldwasser S, Ron D (1998) Property testing and its connections to learning and approximation. J ACM 45:653–750
Kaufman T, Krivelevich M, Ron D (2004) Tight bounds for testing bipartiteness in general graphs. SIAM J Comput 33(6):1441–1483
Parnas M, Ron D (2002) Testing the diameter of graphs. Random Struct Algorithms 20(2):165–183
Ron D (2008) Property testing: a learning theory perspective. Found Trends Mach Learn 1(3):307–402
Ron D (2010) Algorithmic and analysis techniques in property testing. Found Trends Theor Comput Sci 5:73–205
Rubinfeld R, Sudan M (1996) Robust characterization of polynomials with applications to program testing. SIAM J Comput 25(2):252–271
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Goldreich, O., Ron, D. (2016). Testing Bipartiteness of Graphs in Sublinear Time. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_701
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