We consider several problems related to packing forests in graphs. The first one is to find $k$ edge-disjoint forests in a directed graph $G$ of maximal size such that the indegree of each vertex in these forests is at most $k$. We describe a min-max characterization for this problem and show that it can be solved in almost linear time for fixed $k$, extending the algorithm of [Gabow, 1995]. Specifically, the complexity is $O(k \delta m \log n)$, where $n, m$ are the number of vertices and edges in $G$ respectively, and $\delta = \max\{1, k - k_G\}$, where $k_G$ is the edge connectivity of the graph. Using our solution to this problem, we improve complexities for two existing applications: (1) $k$-forest problem: find $k$ forests in an undirected graph $G$ maximizing the number of edges in their union. We show how to solve this problem in $O(k^3 \min\{kn, m\} \log^2 n + k \cdot{\rm MAXFLOW}(m, m) \log n)$ time, breaking the $O_k(n^{3/2})$ complexity barrier of previously known approaches. (2) Directed edge-connectivity augmentation problem: find a smallest set of directed edges whose addition to the given directed graph makes it strongly $k$-connected. We improve the deterministic complexity for this problem from $O(k \delta (m+\delta n)\log n)$ [Gabow, STOC 1994] to $O(k \delta m \log n)$. A similar approach with the same complexity also works for the undirected version of the problem.

翻译：我们研究了图中森林打包相关的若干问题。第一个问题是在有向图$G$中寻找$k$个边不相交的森林，使得这些森林中每个顶点的入度至多为$k$，并最大化森林总规模。我们给出了该问题的极小-极大刻画，并证明对于固定$k$可在近似线性时间内求解，这推广了[Gabow, 1995]的算法。具体复杂度为$O(k \delta m \log n)$，其中$n, m$分别为图$G$的顶点数和边数，$\delta = \max\{1, k - k_G\}$，$k_G$为图的边连通度。利用该问题的解决方案，我们改进了两个现有应用的复杂度：(1) $k$-森林问题：在无向图$G$中寻找$k$个森林，最大化其并集的边数。我们证明了该问题可在$O(k^3 \min\{kn, m\} \log^2 n + k \cdot{\rm MAXFLOW}(m, m) \log n)$时间内求解，突破了现有方法$O_k(n^{3/2})$的复杂度障碍。(2) 有向边连通度增强问题：寻找最小规模的有向边集合，将其添加到给定有向图中使其成为强$k$-连通图。我们将该问题的确定性复杂度从$O(k \delta (m+\delta n)\log n)$ [Gabow, STOC 1994]改进至$O(k \delta m \log n)$。类似方法以相同复杂度也适用于该问题的无向图版本。