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

TL;DR

This paper introduces a sublinear time randomized algorithm to detect forbidden minors in bounded degree graphs, significantly advancing property testing by nearly matching the lower bounds and resolving longstanding conjectures.

Contribution

It presents the first sublinear time one-sided tester for minor-closed properties, applicable to any minor-closed property, with near-optimal running time.

Findings

Achieves an $n^{1/2+o(1)}$-time algorithm for finding minors

Resolves a conjecture on one-sided property testers for minor-closed properties

Nearly optimal due to matching lower bounds

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 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 $\varepsilon n$ 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 $\Omega(\sqrt{n})$ 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), $K_{2,k}$, $(k\times 2)$-grid, and the $k$-circus (Fichtenberger et al, Arxiv 2017).