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arXiv · 2409.14881

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

Pavel Arkhipov·Vladimir Kolmogorov
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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 $O(k δm \log n)$ time, where $(n,m)$ is the size of $G$ and $δ$ 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 $O(k δ(m+δn)\log n)$ [Gabow, STOC 1994] to $O(k δm \log n)$, by exploiting our solution for the first problem. A similar approach with the same complexity also works for the undirected version of the problem.

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Pavel Arkhipov, Vladimir Kolmogorov. 2025-10-15. Bounded indegree $k$-forests problem and a faster algorithm for directed graph augmentation. https://arxiv.org/abs/2409.14881

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