Finding forbidden minors in sublinear time: a n^1/2+o(1)-query one-sided tester for minor closed properties on bounded degree graphs
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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).
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