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.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
J. Edmonds: Minimum partition of a matroid into independent subsets. J. Res. National Bureau of Standards 69B, 1965, pp. 67ā72
J. Edmonds: Edge-disjoint branchings. In: Combinatorial Algorithms, R. Rustin, Ed., Algorithmics Press, New York, 1972, pp. 91ā96
A. Frank: On disjoint trees and arborescences. In: Algebraic Methods in Graph Theory, L. LovĆ”sz and V.T. SĆ³s, Eds., Colloq. Math. Soc. JĆ”nos Bolyai 18, North-Holland, New York, 1978, pp. 159ā169
A. Frank: Kernel systems of directed graphs. Acta Sci. Math. 41, 1979, pp. 63ā76
H.N. Gabow: A matroid approach to finding edge connectivity and packing arborescences. Proc. 23rd Annual ACM Symp. on Theory of Comp., 1991, pp. 112-122; J. CSS, to appear
H.N. Gabow: Algorithms for graphic polymatroids and parametric s-sets. Proc. 6th Annual ACM-SIAM Symp. on Disc. Algorithms, 1995, pp. 88-97
M. Grƶtschel, L. LovƔsz, A. Schrijver: Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, New York, 1988
H.N. Gabow, H.H. Westermann: Forests, frames and games: Algorithms for matroid sums and applications. Algorithmica 7, 1992, pp. 465ā497
J. Hao, J.B. Orlin: A faster algorithm for finding the minimum cut in a directed graph. J. Algorithms 17, 3, 1994, pp. 424ā446
H. Imai: Network-flow algorithms for lower-truncated transversal polymatroids. J. Op. Res. Soc. of Japan 26, 3, 1983, pp. 186ā210
L. LovĆ”sz: On two minimax theorems in graph theory. J. Comb. Theory, B, 21, 1976, pp. 96ā103
W. Mader: On n-edge-connected digraphs. Annals Discr. Math. 17, 1983, pp. 439ā441
C. Mao-cheng: Arc-disjoint arborescences of digraphs. J. Graph Theory 7, 1983, pp. 235ā240
M.W. Padberg, L.A. Wolsey: Fractional covers for forests and matchings. Math. Programming 29, 1984, pp. 1ā14
T. Radzik: Newton's method for fractional combinatorial optimization. Proc. 33rd Annual Symp. on Found. of Comp. Sci., 1992, pp. 659ā669
J. Roskind, R.E. Tarjan: A note on finding minimum-cost edge-disjoint spanning trees. Math. Op. Res. 10, 4, 1985, pp. 701ā708
Y. Shiloach: Edge-disjoint branchings in directed multigraphs. Inf. Proc. Letters 8, 2, 1979, pp. 24ā27
R.E. Tarjan: A good algorithm for edge-disjoint branching. Inf. Proc. Letters 3, 2, 1974, pp. 51ā53
L.E. Trotter, Jr.: Discrete packing and covering. In: Combinatorial Optimization: Annotated Bibliographies, M. O'hEigeartaigh, J.K. Lenstra and A.H.G. Rinnooy Kan, Eds., John Wiley, New York, 1985, pp. 21ā31
V.A. Trubin: Strength and reinforcement of a network and tree packing. Kibernetika 2, 1991, pp. 67ā75
V.A. Trubin: Strength of a graph and packing of trees and branchings. Kibernetika i Sistemnyi Analiz 3, 1993, pp. 94ā99
P. Tong, E.L. Lawler: A faster algorithm for finding edge-disjoint branchings. Inf. Proc. Letters 17, 2, 1983, pp. 73ā76
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
Ā© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/3-540-59408-6_67
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-59408-6
Online ISBN: 978-3-540-49245-0
eBook Packages: Springer Book Archive