Constructive Arboricity in Almost-Linear Time for Dense Graphs

The arboricity of a graph is the minimum number of forests needed to partition its edges. We present a randomized algorithm that computes the exact arboricity of a simple graph and constructs an optimal forest partition in $\min \{n^{2+o(1)},m^{4\mathrm{/}3+o(1)}\}$ time both in expectation and with high probability, where $n$ and $m$ denote the numbers of vertices and edges. In particular, it runs in almost-linear time, $O(m^{1+o(1)})$, on dense graphs with $m=\mathrm{\Theta }(n^2)$. We also give the first known almost-linear-time algorithm that constructs a partition into at most one more forest than optimal.