Bounded indegree $k$-forests problem and a faster algorithm for directed graph augmentation

TL;DR

This paper introduces a new approach to find maximum edge-disjoint forests with bounded indegree in directed graphs and improves the algorithmic complexity for the directed edge-connectivity augmentation problem.

Contribution

It provides a min-max characterization and a faster algorithm for the bounded indegree forest problem, which in turn enhances the complexity of the directed augmentation problem.

Findings

Achieved an $O(k \delta m \log n)$ time algorithm for the forest problem.

Improved the augmentation problem complexity from $O(k \delta (m+\delta n)\log n)$ to $O(k ext ilde{} ext ilde{} m \log n)$.

Applicable similar approach to undirected graph problems.

Abstract

We consider two problems for a directed graph $G$, which we show to be closely related. The first one is to find $k$ edge-disjoint forests in $G$ 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 $O(k \delta m \log n)$ time, where $(n,m)$ is the size of $G$ and $\delta$ is the difference between $k$ and the edge connectivity of the graph. The second problem is the directed edge-connectivity augmentation problem, which has been extensively studied before: find a smallest set of directed edges whose addition to the graph makes it strongly $k$-connected. We improve the complexity for this problem from $O(k \delta (m+\delta n)\log n)$ [Gabow, STOC 1994] to $O(k \delta m \log n)$, by exploiting our solution for the first problem. A similar approach with the same complexity also works for the undirected version of the problem.