Let G be an undirected, bounded degree graph with n vertices. Fix a finite graph H, and suppose one must remove \varepsilon n edges from G to make it H-minor free (for some small constant \varepsilon > 0).
We give an n^{1/2+o(1)}-time randomized procedure that, with high probability, finds an H-minor in such a graph. For an example application, suppose one must remove \varepsilon n edges from a bounded degree graph G to make it planar. This result implies an algorithm, with the same running time, that produces a K_{3,3} or K_5 minor in G.
No sublinear time bound was known for this problem, prior to this result.

By the graph minor theorem, we get an analogous result for any minor-closed property. Up to n^{o(1)} factors, this resolves a conjecture of Benjamini-Schramm-Shapira (STOC 2008) on the existence of one-sided property testers for minor-closed properties. Furthermore, our algorithm is nearly optimal, by an \Omega(\sqrt{n}) lower bound of Czumaj et al (RSA 2014).

Prior to this work, the only graphs H for which non-trivial property testers were known for H-minor freeness are the following: H being a forest or a cycle (Czumaj et al, RSA 2014), K_{2,k}, (k\times 2)-grid, and the k-circus (Fichtenberger et al, Arxiv 2017).