Let G be an undirected, bounded degree graph with n vertices. Fix a finite graph H, and suppose one must remove εn 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 εn 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 nontrivial 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).
Recently, Kumar-Seshadhri-Stolman (henceforth KSS) nearly resolved the BSS conjecture with an n 1/2+o(1) -query onesided tester for H -minor-freeness REF .
Finding Forbidden Minors in Sublinear Time: A n^1/2+o(1)-Query One-Sided Tester for Minor Closed Properties on Bounded Degree Graphs
2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
Mathematics Computer Science