## Counting minimum weight arborescences.(English)Zbl 1401.05280

Algorithmica 80, No. 12, 3908-3919 (2018); erratum ibid. 81, No. 1, 418 (2019).
Summary: 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 $$v \in V \setminus \{ r \}$$. The problem of finding a minimum weight $$r$$-arborescence in a weighted digraph has been studied for decades starting with Y.-j. Chu and T.-h. Liu [Sci. Sin. 14, 1396–1400 (1965; Zbl 0178.27401)], J. Edmonds [J. Res. Natl. Bur. Stand., Sect. B 71, 233–240 (1967; Zbl 0155.51204)] and F. Bock [in: Developments Operations Res. 1, Proc. 3rd annual Israel Conf. Operations Res. 1969, 29–44 (1971; Zbl 0235.90056)]. 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.

### MSC:

 05C85 Graph algorithms (graph-theoretic aspects) 05C20 Directed graphs (digraphs), tournaments

### Keywords:

minimum weight arborescence; matrix tree theorem; counting

### Citations:

Zbl 0178.27401; Zbl 0155.51204; Zbl 0235.90056
Full Text:

### References:

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