Minimum-Cost $k$-Arborescence Packing in $\tilde{O}(km+k^{4}n)$ Time

Abstract

We give a deterministic, strongly polynomial algorithm running in $\tilde{O}(km+k^{4}n)$ time that finds $k$ arc-disjoint arborescences of minimum total cost in a directed multigraph with $n$ vertices and $m$ arcs. This significantly improves upon the state of the art for $k\ge 2$ with $k=n^{o(1)}$: the prior fastest weakly polynomial and strongly polynomial algorithms take $\tilde{O}(km\sqrt{n}\log C)$ and $\tilde{O}(kmn)$ time (Gabow and Xu, 1989; Gabow, 1995), respectively. Roughly, our technique can be viewed as a way to apply a linear number of \emph{shortest} augmenting paths in near-linear total time for the special case of the weighted matroid intersection problem between the $k$-fold graphic matroid and the indegree partition matroid. This was previously done in the unweighted case by Gabow (1995), but our approach is different. As an extension, we also obtain a deterministic $\tilde{O}(k{m}^{1+o(1)}+k^{5}n)$-time algorithm for minimum-cost maximum-cardinality packing of $k$ forests in undirected graphs. In particular, this finds $k$ edge-disjoint spanning trees of minimum total cost, if they exist. The previous fastest algorithm takes $\tilde{O}(k{n}^2+m)$ time (Gabow and Westermann, 1992).