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 $k \geq 8$.

翻译：我们考虑在图中打包边不交斯坦纳森林的问题。输入包含一个多重图$G=(V,E)$和一组$h$个顶点子集$S = \{S_1,S_2,\ldots,S_h\}$。对于$S$的斯坦纳森林，也称为$S$-森林，是$G$的一个森林，其中每个$S_i$是连通的。当$h=1$时，这对应于斯坦纳树打包问题。Kriesell猜想提出，$S_1$的$2k$-边连通性足以找到$k$个边不交的$S_1$-树。Lau证明$24k$-边连通性足以解决斯坦纳树打包问题，该结果随后被West和Wu改进至$6.5k$，并被Devos、McDonald和Pivotto改进至$5k+4$。在其论文中，Lau断言，对于斯坦纳森林问题，如果每个$S_i$在$G$中是$30k$-边连通的，则存在$k$个边不交的$S$-森林。然而，Lau的证明依赖于一个称为扩展定理的中间定理，在本文中我们将通过提供一个对Lau扩展定理的反例来证明该定理存在漏洞。此外，我们将通过修正Lau的证明来解决这个漏洞，表明每个$S_i$的$36k$-边连通性足以打包$k$个$S$-森林。更仔细的分析表明，当$k \geq 8$时，每个$S_i$的$35k$-边连通性就足够了。