图性质的测试与估计，再探
Testing versus estimation of graph properties, revisited
AI总结：
本文重新研究图性质的测试与估计，改进了Fischer-Newman定理的转换效率，对遗传性质实现指数损失，对一般性质实现双指数损失，并建立了Frieze-Kannan弱正则划分的新性质。
摘要：
图性质 $\mathcal{P}$ 的距离估计器是一种算法，给定图 $G$ 和 $\alpha, \varepsilon >0$，区分 $G$ 与 $\mathcal{P}$ 的编辑距离是 $(\alpha-\varepsilon)$-接近还是 $\alpha$-远离。若存在查询复杂度仅依赖于 $\varepsilon$ 的距离估计器，则称 $\mathcal{P}$ 是可估计的。每个可估计性质也是可测试的，因为测试对应于取 $\alpha=\varepsilon$ 的估计。性质测试领域的一个核心结果，Fischer--Newman 定理，给出了逆命题：每个可测试性质实际上都是可估计的。Fischer 和 Newman 的证明非常低效，因为将 $\mathcal{P}$ 的测试算法转换为距离估计器时会产生塔式损失。这提出了一个自然问题，最近由 Fiat--Ron 和 Hoppen--Kohayakawa--Lang--Lefmann--Stagni 研究，即能否找到具有多项式损失的转换。我们得到以下结果。1. 若 $\mathcal{P}$ 是遗传的，则可以将 $\mathcal{P}$ 的测试器转换为具有指数损失的估计器。这比 Hoppen 等人的结果（双指数损失）有指数改进。2. 对于每个 $\mathcal{P}$，可以将 $\mathcal{P}$ 的测试算法转换为具有双指数损失的估计器。这改进了 Fischer--Newman 的塔式损失转换。我们在这项工作中的主要概念贡献是设法将 Fischer--Newman 本质上低效的方法转变为高效方法。在技术层面，我们的主要贡献是建立了 Frieze--Kannan 弱正则划分的某些性质，这些性质具有独立意义。
英文摘要：
A distance estimator for a graph property $\mathcal{P}$ is an algorithm that given $G$ and $α, \varepsilon >0$ distinguishes between the case that $G$ is $(α-\varepsilon)$-close to $\mathcal{P}$ and the case that $G$ is $α$-far from $\mathcal{P}$ (in edit distance). We say that $\mathcal{P}$ is estimable if it has a distance estimator whose query complexity depends only on $\varepsilon$. Every estimable property is also testable, since testing corresponds to estimating with $α=\varepsilon$. A central result in the area of property testing, the Fischer--Newman theorem, gives an inverse statement: every testable property is in fact estimable. The proof of Fischer and Newman was highly ineffective, since it incurred a tower-type loss when transforming a testing algorithm for $\mathcal{P}$ into a distance estimator. This raised the natural problem, studied recently by Fiat--Ron and by Hoppen--Kohayakawa--Lang--Lefmann--Stagni, whether one can find a transformation with a polynomial loss. We obtain the following results. 1. If $\mathcal{P}$ is hereditary, then one can turn a tester for $\mathcal{P}$ into a distance estimator with an exponential loss. This is an exponential improvement over the result of Hoppen et. al., who obtained a transformation with a double exponential loss. 2. For every $\mathcal{P}$, one can turn a testing algorithm for $\mathcal{P}$ into a distance estimator with a double exponential loss. This improves over the transformation of Fischer--Newman that incurred a tower-type loss. Our main conceptual contribution in this work is that we manage to turn the approach of Fischer--Newman, which was inherently ineffective, into an efficient one. On the technical level, our main contribution is in establishing certain properties of Frieze--Kannan Weak Regular partitions that are of independent interest.
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