Random walks and forbidden minors II: A $\text{poly}(d\varepsilon^{-1})$-query tester for minor-closed properties of bounded-degree graphs

Abstract: Let $G$ be a graph with $n$ vertices and maximum degree $d$. Fix some minor-closed property $\mathcal{P}$ (such as planarity). We say that $G$ is $\varepsilon$-far from $\mathcal{P}$ if one has to remove $\varepsilon dn$ edges to make it have $\mathcal{P}$. The problem of property testing $\mathcal{P}$ was introduced in the seminal work of Benjamini-Schramm-Shapira (STOC 2008) that gave a tester with query complexity triply exponential in $\varepsilon<sup>{-1}$. Levi-Ron (TALG 2015) have given the best tester to date, with a quasipolynomial (in $\varepsilon<sup>{-1}$) query complexity. It is an open problem to get property testers whose query complexity is $\text{poly}(d\varepsilon<sup>{-1})$, even for planarity. In this paper, we resolve this open question. For any minor-closed property, we give a tester with query complexity $d\cdot \text{poly}(\varepsilon<sup>{-1})$. The previous line of work on (independent of $n$, two-sided) testers is primarily combinatorial. Our work, on the other hand, employs techniques from spectral graph theory. This paper is a continuation of recent work of the authors (FOCS 2018) analyzing random walk algorithms that find forbidden minors.