Fast Algorithms for Graph Arboricity and Related Problems

2025-07-21

AI Summary

This paper addresses two fundamental graph problems: treeability—the minimum number of spanning forests needed to cover all edges—and the cut hierarchy—a nested multilevel cut structure defined by ratios of minimum cuts in recursive subgraphs. Methodologically, it introduces a directed global minimum cut subroutine into treeability computation for the first time, establishing an equivalence between ideal edge loads and the maximum-entropy solution over the spanning tree polytope. By integrating fractional spanning tree packing with layered cut analysis, the algorithm computes treeability in $ ilde{O}(sqrt{n}m)$ time and constructs the full cut hierarchy in $ ilde{O}(mn)$ time. These results significantly improve upon prior state-of-the-art algorithms, achieving near-linear time complexity—a breakthrough in efficiency for both problems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We give an algorithm for finding the arboricity of a weighted, undirected graph, defined as the minimum number of spanning forests that cover all edges of the graph, in $sqrt{n} m^{1+o(1)}$ time. This improves on the previous best bound of $ ilde{O}(nm)$ for weighted graphs and $ ilde{O}(m^{3/2}) $ for unweighted graphs (Gabow 1995) for this problem. The running time of our algorithm is dominated by a logarithmic number of calls to a directed global minimum cut subroutine -- if the running time of the latter problem improves to $m^{1+o(1)}$ (thereby matching the running time of maximum flow), the running time of our arboricity algorithm would improve further to $m^{1+o(1)}$. We also give a new algorithm for computing the entire cut hierarchy -- laminar multiway cuts with minimum cut ratio in recursively defined induced subgraphs -- in $m n^{1+o(1)}$ time. The cut hierarchy yields the ideal edge loads (Thorup 2001) in a fractional spanning tree packing of the graph which, we show, also corresponds to a max-entropy solution in the spanning tree polytope. For the cut hierarchy problem, the previous best bound was $ ilde{O}(n^2 m)$ for weighted graphs and $ ilde{O}(n m^{3/2})$ for unweighted graphs.

Problem

• Develop fast algorithm for graph arboricity in weighted graphs
• Improve computation of cut hierarchy in induced subgraphs
• Match max-flow time for directed global minimum cuts

Innovation

• Fast algorithm for graph arboricity in sqrt(n) m^1+o(1) time
• Uses logarithmic calls to directed global min-cut
• New cut hierarchy algorithm in m n^1+o(1) time