On the Extension Theorem for Packing Steiner Forests

We consider the problem of packing edge-disjoint Steiner forests in a graph. The input consists of a multi-graph $G=(V,E)$ and a collection of $h$ vertex subsets $S = \{S_1,S_2,\ldots,S_h\}$. A Steiner forest for $S$, also called an $S$-forest, is a forest of $G$ in which each $S_i$ is connected. In the case where $h=1$, this is the Steiner Tree packing problem. Kriesell's conjecture postulates that $2k$-edge-connectivity of $S_1$ is sufficient to find $k$ edge-disjoint $S_1$-trees. Lau showed that $24k$-edge-connectivity suffices for the Steiner Tree packing problem, which was improved to $6.5k$ by West and Wu and $5k+4$ by Devos, McDonald and Pivotto. In his thesis, Lau asserts that for the Steiner Forest problem, if each $S_i$ is $30k$-edge-connected in $G$, then there exist $k$ edge-disjoint $S$-forests. However, Lau's proof relies on an intermediate theorem called the Extension Theorem, which in this paper we will demonstrate has a gap by providing a counterexample to Lau's Extension Theorem. Furthermore, we will resolve this gap by correcting Lau's proof to show that $36k$-edge-connectivity of each $S_i$ suffices to pack $k$ $S$-forests. More careful analysis yields that $35k$-edge-connectivity of each $S_i$ is sufficient when $k \geq 8$.