Testing versus estimation of graph properties, revisited

A graph (Formula presented.) on (Formula presented.) vertices is (Formula presented.) -far from property (Formula presented.) if one should add/delete at least (Formula presented.) edges to turn (Formula presented.) into a graph satisfying (Formula presented.). A distance estimator for (Formula presented.) is an algorithm that given (Formula presented.) and (Formula presented.) distinguishes between the case that (Formula presented.) is (Formula presented.) -close to (Formula presented.) and the case that (Formula presented.) is (Formula presented.) -far from (Formula presented.). If (Formula presented.) has a distance estimator whose query complexity depends only on (Formula presented.), then (Formula presented.) is said to be estimable. Every estimable property is clearly also testable, since testing corresponds to estimating with (Formula presented.). A central result in the area of property testing is the Fischer–Newman theorem, stating that an inverse statement also holds, that is, that every testable graph 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 (Formula presented.) 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.We show that if P is hereditary, then one can turn a tester for 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.We show that for every P, one can turn a testing algorithm for 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.