Abstract
In a digraph with real-valued edge capacities, we pack the greatest number of arborescences in time O(n 3 m log (n 2/m)); the packing uses at most m distinct arborescences. Here n and m denote the number of vertices and edges respectively. Similar results hold for integral packing: we pack the greatest number of arborescences in time O(minn, log (nN)n 2 m log (n 2/m)), using at most m+nā2 distinct arborescences; here N denotes the largest capacity. These results improve all previous strong- and weak-polynomial bounds for capacitated digraphs. The algorithm extends to related problems, including packing spanning trees in an undirected capacitated graph, and covering such graphs by forests. The algorithm provides a new proof of Edmonds' theorem for arborescence packing, for both integral and real capacities. The algorithm works by maintaining a certain laminar family of sets.
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Gabow, H.N., Manu, K.S. (1995). Packing algorithms for arborescences (and spanning trees) in capacitated graphs. In: Balas, E., Clausen, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 1995. Lecture Notes in Computer Science, vol 920. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59408-6_67
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DOI: https://doi.org/10.1007/3-540-59408-6_67
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