Testing versus estimation of graph properties, revisited

A distance estimator for a graph property $\mathcal{P}$ is an algorithm that given $G$ and $\alpha ,\epsilon >0$ distinguishes between the case that $G$ is $(\alpha -\epsilon )$-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 $\epsilon$. Every estimable property is also testable, since testing corresponds to estimating with $\alpha =\epsilon$. 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.