一个扩展的有界度图模型VDF的二部性检验
Testing Bipartitness in an Augmented VDF Bounded-Degree Graph Model
摘要：
在最近的一项工作（ECCC，TR 18 -171，2018）中，我们介绍了测试图形属性的模型，其中，除了对通常的图形查询的答案外，测试人员还获得{根据任意分布$D$绘制的\em随机顶点}。需要这样的测试器来区分具有该属性的图和远离该属性的图，{ \em，其中图之间的距离基于未知顶点分布$D$来定义}。这些模型（“顶点分布自由”（VDF））推广了$D$假设在顶点集上是均匀分布的标准模型，并在稠密图模型和有界度图模型下进行了研究。上述工作的重点是测试人员，称为{ \sf strong}，其查询复杂度仅取决于邻近参数$ε$。不幸的是，在标准的有界度图模型中，一些自然属性（如二分性）没有强的测试者，而其他属性（如无环性）没有强的单侧错误测试者（而单侧错误是VDF模型固有的）。因此，建议研究一般性问题（即，非强）“次线性”复杂性的测试者。在这项工作中，我们追求上述建议，但这样做的模型，增强了上述工作中提出的模型。具体来说，我们为测试人员提供了一个未知分布$D$的评估oracle，以及$D$的样本和对测试图的oracle访问。我们的主要结果是测试的二分性和循环自由，在这个增强模型中，具有复杂性，几乎是线性的平方根的“有效支持大小”的$D$。
英文摘要：
In a recent work (ECCC, TR18-171, 2018), we introduced models of testing graph properties in which, in addition to answers to the usual graph-queries, the tester obtains {\em random vertices drawn according to an arbitrary distribution $D$}. Such a tester is required to distinguish between graphs that have the property and graphs that are far from having the property, {\em where the distance between graphs is defined based on the unknown vertex distribution $D$}. These ("vertex-distribution free" (VDF)) models generalize the standard models in which $D$ is postulated to be uniform on the vertex-set, and they were studies both in the dense graph model and in the bounded-degree graph model. The focus of the aforementioned work was on testers, called {\sf strong}, whose query complexity depends only on the proximity parameter $ε$. Unfortunately, in the standard bounded-degree graph model, some natural properties such as Bipartiteness do not have strong testers, and others (like cycle-freeness) do not have strong testers of one-sided error (whereas one-sided error was shown inherent to the VDF model). Hence, it was suggested to study general (i.e., non-strong) testers of "sub-linear" complexity. In this work, we pursue the foregoing suggestion, but do so in a model that augments the model presented in the aforementioned work. Specifically, we provide the tester with an evaluation oracle to the unknown distribution $D$, in addition to samples of $D$ and oracle access to the tested graph. Our main results are testers for Bipartitness and cycle-freeness, in this augmented model, having complexity that is almost-linear in the square root of the "effective support size" of $D$.
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