Property testing algorithms are highly efficient algorithms, that come with probabilistic accuracy guarantees. For a property P, the goal is to distinguish inputs that have P from those that are far from having P with high probability correctly, by querying only a small number of local parts of the input. In property testing on graphs, the distance is measured by the number of edge modifications (additions or deletions), that are necessary to transform a graph into one with property P. Much research has focussed on the query complexity of such algorithms, i. e. the number of queries the algorithm makes to the input, but in view of applications, the running time of the algorithm is equally relevant. In (Adler, Harwath STACS 2018), a natural extension of the bounded degree graph model of property testing to relational databases of bounded degree was introduced, and it was shown that on databases of bounded degree and bounded tree-width, every property that is expressible in monadic second-order logic with counting (CMSO) is testable with constant query complexity and sublinear running time. It remains open whether this can be improved to constant running time. In this paper we introduce a new model, which is based on the bounded degree model, but the distance measure allows both edge (tuple) modifications and vertex (element) modifications. Our main theorem shows that on databases of bounded degree and bounded tree-width, every property that is expressible in CMSO is testable with constant query complexity and constant running time in the new model. We also show that every property that is testable in the classical model is testable in our model with the same query complexity and running time, but the converse is not true. We argue that our model is natural and our meta-theorem showing constant-time CMSO testability supports this.

中文翻译：

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在有界度模型的变体中更快的属性测试器

属性测试算法是高效的算法，具有概率精度保证。对于属性 P，目标是通过仅查询输入的少量局部部分，以高概率正确区分具有 P 的输入和远离 P 的输入。在图的属性测试中，距离是通过边修改（添加或删除）的数量来衡量的，这是将图转换为具有属性 P 所必需的。 许多研究都集中在此类算法的查询复杂度上，即数量算法对输入进行的查询的数量，但就应用而言，算法的运行时间同样重要。在（阿德勒，Harwath STACS 2018），引入了属性测试的有界度图模型对有界度关系数据库的自然扩展，表明在有界度和有界树宽数据库上，每个属性都可以用计数的一元二阶逻辑表达(CMSO) 是可测试的，具有恒定的查询复杂性和次线性运行时间。这是否可以改进为恒定的运行时间仍然是开放的。在本文中，我们介绍了一种新模型，该模型基于有界度模型，但距离度量允许边（元组）修改和顶点（元素）修改。我们的主要定理表明，在有界度和有界树宽的数据库上，在 CMSO 中可表达的每个属性都可以在新模型中以恒定的查询复杂度和恒定的运行时间进行测试。我们还表明，在经典模型中可测试的每个属性都可以在我们的模型中以相同的查询复杂度和运行时间进行测试，但反之则不然。我们认为我们的模型是自然的，我们的元定理显示恒定时间 CMSO 可测试性支持这一点。