RANDOM WALKS AND FORBIDDEN MINORS I: AN 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 epsilon n edges from G to make it H-minor-free (for some small constant epsilon > 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 epsilon 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 K3,3- or K5-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, and Shapira [Adv. Math., 223 (2010), pp. 2200-2218] on the existence of one-sided property testers for minor-closed properties. Furthermore, our algorithm is nearly optimal by an Omega (root n) lower bound of Czumaj et al. [Random Structures Algorithms, 45 (2014), pp. 139-184]. Prior to this work, the only graphs H for which nontrivial one-sided property testers were known for H-minor-freeness were the following: H being a forest or a cycle [Czumaj et al., Random Structures Algorithms, 45 (2014), pp. 139-184], K-2,K-k, (k x 2)-grid, and the k-circus [Fichtenberger et al., preprint, arXiv:1707.06126v1, 2017].