Counting Minimum Weight Arborescences

Abstract

In a directed graph $D = (V, A)$ with a specified vertex $r \in V$, an arc subset $B \subseteq A$ is called an r-arborescence if B has no arc entering r and there is a unique path from r to v in (V, B) for each $v \in V \backslash \{ r \}$. The problem of finding a minimum weight r-arborescence in a weighted digraph has been studied for decades starting with Chu and Liu (Sci Sin 14:1396–1400, 1965), Edmonds (J Res Natl Bur Stand 71B:233–240, 1967) and Bock (Developments in operations research, Gordon and Breach, New York, pp 29–44, 1971). In this paper, we focus on the number of minimum weight arborescences. We present an algorithm for counting minimum weight r-arborescences in $O(n^{\omega })$ time, where n is the number of vertices of an input digraph and $\omega$ is the matrix multiplication exponent.