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

2024-09-23

arXiv.org

AI Summary

This paper unifies the study of two classical problems on directed graphs: (1) constructing $k$ edge-disjoint forests such that each vertex has in-degree at most $k$, while maximizing the total number of edges; and (2) augmenting the graph with a minimum number of directed edges to make it strongly $k$-connected. We establish, for the first time, an intrinsic combinatorial equivalence between these problems and derive a unified min-max characterization. Leveraging network flow modeling and a greedy augmentation strategy, we design an efficient $O(kdelta m log n)$-time algorithm—improving the best-known time bound for strong $k$-connectivity augmentation from $O(kdelta(m + delta n)log n)$ to the current optimum. Moreover, we provide the first exact combinatorial characterization and polynomial-time algorithm for bounded-in-degree $k$-forests. These results significantly advance both the theory and algorithms for $k$-connectivity augmentation in directed and undirected graphs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Social Networks and Social Media: Social mining and social search on the Web

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.

Problem

• Finding maximum k edge-disjoint forests with bounded indegree
• Improving directed graph edge-connectivity augmentation algorithms
• Developing faster strongly k-connected graph construction methods

Innovation

• Min-max characterization for bounded indegree forests
• Improved algorithm for directed graph augmentation
• Exploiting forest solution to reduce complexity