Let $G$ be an $n$-vertex $m$-edge graph with weighted vertices. A pair of vertex sets $A,B \subseteq V(G)$ is a $\frac{2}{3} - \textit{separation}$ of $\textit{order}$ $|A \cap B|$ if $A \cup B = V(G)$, there is no edge between $A \backslash B$ and $B \backslash A$, and both $A \backslash B$ and $B \backslash A$ have weight at most $\frac{2}{3}$ the total weight of $G$. Let $\ell \in \mathbb{Z}^+$ be fixed. Alon, Seymour and Thomas [$\textit{J. Amer. Math. Soc.}$ 1990] presented an algorithm that in $\mathcal{O}(n^{1/2}m)$ time, either outputs a $K_\ell$-minor of $G$, or a separation of $G$ of order $\mathcal{O}(n^{1/2})$. Whether there is a $\mathcal{O}(n+m)$ time algorithm for this theorem was left as open problem. In this paper, we obtain a $\mathcal{O}(n+m)$ time algorithm at the expense of $\mathcal{O}(n^{2/3})$ separator. Moreover, our algorithm exhibits a tradeoff between running time and the order of the separator. In particular, for any given $\epsilon \in [0,\frac{1}{2}]$, our algorithm either outputs a $K_\ell$-minor of $G$, or a separation of $G$ with order $\mathcal{O}(n^{(2-\epsilon )/3})$ in $\mathcal{O}(n^{1+\epsilon} +m)$ time.
Discrete Mathematics & Theoretical Computer Science
