Fast Algorithms for Graph Arboricity and Related Problems

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