Fast Algorithms for Graph Arboricity and Related Problems

TL;DR

This paper introduces faster algorithms for computing graph arboricity and cut hierarchies, significantly improving the time complexity over previous methods, with potential further improvements if related subproblems are optimized.

Contribution

The paper presents new algorithms for arboricity and cut hierarchy problems with improved time bounds, connecting these problems to max-entropy solutions in spanning tree packings.

Findings

Arboricity algorithm runs in 0^{1/2} m^{1+o(1)} time.

Cut hierarchy algorithm runs in m n^{1+o(1)} time.

Results improve previous bounds for weighted and unweighted graphs.

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 $\tilde{O}(nm)$ for weighted graphs and $\tilde{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 $\tilde{O}(n^2 m)$ for weighted graphs and $\tilde{O}(n m^{3/2})$ for unweighted graphs.