Finding Forbidden Minors in Sublinear Time: A n^1/2+o(1)-Query One-Sided Tester for Minor Closed Properties on Bounded Degree Graphs

Let G be an undirected, bounded degree graph with n vertices. Fix a finite graph H, and suppose one must remove ε n edges from G to make it H-minor free (for some small constant ε > 0). We give an n
^1/2+o(1)
-time randomized procedure that, with high probability, finds an H-minor in such a graph. As an application, suppose one must remove ε 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 prior sublinear time bound was known for this problem. 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 Ω(√n) lower bound of Czumaj et al (RSA 2014). Prior to this work, the only graphs H for which non-trivial one-sided 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× 2)-grid, and the k-circus (Fichtenberger et al, Arxiv 2017).