Abstract
Let G=(VG,AG) be a digraph and let S ⊔ T be a bipartition of VG. A bibranching is a subset B⊆AG such that for each node s∈S there exists a directed s–T path in B and, vice versa, for each node t∈T there exists a directed S–t path in B.
Bibranchings generalize both branchings and bipartite edge covers. Keijsper and Pendavingh proposed a strongly polynomial primal-dual algorithm that finds a minimum weight bibranching in O(n′(m+nlog n)) time (where n:=|VG|, m:=|AG|, n′:=min (|S|,|T|)).
Assuming that arc weights are integers we develop a weight-scaling algorithm of time complexity \(O(m\sqrt{n}\;\log n\log(nW))\) for the minimum weight bibranching problem (where W denotes the maximum magnitude of arc weights).
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Babenko, M.A. An Efficient Scaling Algorithm for the Minimum Weight Bibranching Problem. Algorithmica 61, 898–922 (2011). https://doi.org/10.1007/s00453-009-9377-1
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DOI: https://doi.org/10.1007/s00453-009-9377-1