A tight Erd\H{o}s-P\'osa function for planar minors

TL;DR

This paper proves a tight bound on the Erdős-Pósa function for planar minors, showing it is linearithmic in the number of disjoint minors, and provides a polynomial-time approximation algorithm.

Contribution

It establishes the optimal $c k \, \log k$ bound for the Erdős-Pósa function for planar minors, improving previous super-logarithmic bounds.

Findings

The Erdős-Pósa function for planar minors is $O(k \log k)$, which is tight.

The proof is constructive and leads to a polynomial-time approximation algorithm.

The result generalizes and tightens classical bounds for planar graph minors.

Abstract

Let $H$ be a planar graph. By a classical result of Robertson and Seymour, there is a function $f:\mathbb{N} \to \mathbb{R}$ such that for all $k \in \mathbb{N}$ and all graphs $G$, either $G$ contains $k$ vertex-disjoint subgraphs each containing $H$ as a minor, or there is a subset $X$ of at most $f(k)$ vertices such that $G-X$ has no $H$-minor. We prove that this remains true with $f(k) = c k \log k$ for some constant $c=c(H)$. This bound is best possible, up to the value of $c$, and improves upon a recent result of Chekuri and Chuzhoy [STOC 2013], who established this with $f(k) = c k \log^d k$ for some universal constant $d$. The proof is constructive and yields a polynomial-time $O(\log \mathsf{OPT})$-approximation algorithm for packing subgraphs containing an $H$-minor.