### TCS+ talk: Wednesday, October 17 — C. Seshadhri, UC Santa Cruz

The next TCS+ talk will take place this coming Wednesday, October 17th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). C. Seshadhri from UC Santa Cruz will speak about “Finding forbidden minors through random walks: (almost) $n^{1/2}$-query one-sided testers for minor closed properties” (abstract below).

Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: 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 a nearly $n^{1/2}$ time algorithm that, with high probability, finds an $H$-minor in such a graph.

As an application, consider a graph $G$ that requires $\varepsilon n$ edge removals 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 result 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 lower bounds of Czumaj et al (RSA 2014).

Joint work with Akash Kumar and Andrew Stolman