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Flexible Models for Testing Graph Properties

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Book cover Computational Complexity and Property Testing

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12050))

Abstract

The standard models of testing graph properties postulate that the vertex-set consists of \(\{1,2,...,n\}\), where n is a natural number that is given explicitly to the tester. Here we suggest more flexible models by postulating that the tester is given access to samples the arbitrary vertex-set; that is, the vertex-set is arbitrary, and the tester is given access to a device that provides uniformly and independently distributed vertices. In addition, the tester may be (explicitly) given partial information regarding the vertex-set (e.g., an approximation of its size). The flexible models are more adequate for actual applications, and also facilitates the presentation of some theoretical results (e.g., reductions among property testing problems).

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Notes

  1. 1.

    See Observations 1.2, 2.2 and 3.2.

  2. 2.

    That is, if a graph \(G=(V,E)\) has the property, then, for any bijection \(\pi :V\rightarrow V'\), the graph \(G'=(V',\{\{\pi (u),\pi (v)\}:\{u,v\}\!\in \!E\}\) has the property.

  3. 3.

    A more refined definition, following [3], may consider the number of queries to each of the oracles. In such a case, it makes sense to refer to the number of queries to the adjacency predicate (resp., the sampling device) as the query (resp., sample) complexity of the tester.

  4. 4.

    For simplicity, we adopt the standard convention by which the neighbors of v appear in arbitrary order in the sequence \((g(v,1),...,g(v,\mathrm{deg}(v)))\), where \(\mathrm{deg}(v){\mathop {=}\limits ^\mathrm{def}}|\{j\in [d]:g(v,j)\ne \bot \}|\).

  5. 5.

    In some cases (e.g. [4, Sec. 9.2.3]), these testers use an estimate of |V| in order to avoided pathological problems that arise when \(|V|<B{\mathop {=}\limits ^\mathrm{def}}O(1/\epsilon )\). But determining whether not \(|V|<B\) holds can be done by using \({\widetilde{O}}(1/\epsilon )\) samples of V, let alone that it actually suffices to distinguish between \(|V|<B\) and \(|V|\ge 2B\).

  6. 6.

    The obvious procedure is to keep sampling till seeing, say, 100 pairwise collisions, and then outputting the square of the number of trials (divided by 200).

  7. 7.

    In a previous version of this paper, we considered \(g_1:V\times [|V|-1]\rightarrow V\cup \{\bot \}\), where \(|V|-1\) served as a trivial degree bound. In retrospect, we feel that using such an upper bound is problematic, because it may allow the tester to determine the number of vertices in the graph (assuming that querying \(g_1\) on an input that is not in its domain results in a suitable indication). On the other hand, allowing an infinite representation of finite graphs is not problematic, because the representation is not used as a basis for the definition of the relative distance between graphs.

  8. 8.

    Here we follow [4, Def. 10.2], rather than [4, Def. 10.1]. See discussion in [4, Sec. 10.1.2].

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel

    Oded Goldreich

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  1. Oded Goldreich
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Correspondence to Oded Goldreich .

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  1. Weizmann Institute of Science, Rehovot, Israel

    Oded Goldreich

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Goldreich, O. (2020). Flexible Models for Testing Graph Properties. In: Goldreich, O. (eds) Computational Complexity and Property Testing. Lecture Notes in Computer Science(), vol 12050. Springer, Cham. https://doi.org/10.1007/978-3-030-43662-9_19

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  • DOI: https://doi.org/10.1007/978-3-030-43662-9_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-43661-2

  • Online ISBN: 978-3-030-43662-9

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