Faster algorithms for packing forests in graphs and related problems

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 $O(k δm \log n)$, where $n, m$ are the number of vertices and edges in $G$ respectively, and $δ= \max\{1, k - k_G\}$, 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 $O(k δ(m+δn)\log n)$ [Gabow, STOC 1994] to $O(k δm \log n)$. A similar approach with the same complexity also works for the undirected version of the problem.