A distance estimator for a graph property $\mathcal{P}$ is an algorithm that given $G$ and $\alpha, \varepsilon >0$ distinguishes between the case that $G$ is $(\alpha-\varepsilon)$-close to $\mathcal{P}$ and the case that $G$ is $\alpha$-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 $\alpha=\varepsilon$. 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.

翻译：一个图性质$\mathcal{P}$的距离估计器是一种算法，给定图$G$、参数$\alpha$和$\varepsilon >0$，能区分$G$在编辑距离上$(\alpha-\varepsilon)$-接近$\mathcal{P}$与$G$ $\alpha$-远离$\mathcal{P}$这两种情形。若$\mathcal{P}$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。由于测试对应于取$\alpha=\varepsilon$时的估计，每个可估计性质也都是可测试的。性质测试领域的核心结果——Fischer–Newman定理——给出了逆命题：每个可测试性质实际上也是可估计的。Fischer和Newman的证明高度非有效，因为将$\mathcal{P}$的测试算法转化为距离估计器时，其损失呈指数塔形式。这自然引出一个问题，即是否存在多项式损失下的转化，该问题近期由Fiat–Ron以及Hoppen–Kohayakawa–Lang–Lefmann–Stagni研究。我们获得以下结果：1. 若$\mathcal{P}$是遗传的，则可将$\mathcal{P}$的测试器转化为指数损失的距离估计器。这比Hoppen等人的结果（双指数损失）实现了指数级改进。2. 对任意$\mathcal{P}$，可将$\mathcal{P}$的测试算法转化为双指数损失的距离估计器。这改进了Fischer–Newman的转化（指数塔损失）。本文的主要概念贡献在于：我们将原本内在非有效的Fischer–Newman方法转化为有效方法。技术层面，我们的主要贡献是建立了Frieze–Kannan弱正则划分的若干性质，这些性质具有独立意义。