Abstract
Let G be a graph with a single source w, assigned a positive integer called the supply. Every vertex other than w is a sink, assigned a nonnegative integer called the demand. Every edge is assigned a positive integer called the capacity. Then a spanning tree T of G is called a spanning distribution tree if the capacity constraint holds when, for every sink v, an amount of flow, equal to the demand of v, is sent from w to v along the path in T between them. The spanning distribution tree problem asks whether a given graph has a spanning distribution tree or not. In the paper, we first observe that the problem is NP-complete even for series-parallel graphs, and then give a pseudo-polynomial time algorithm to solve the problem for a given series-parallel graph G.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Boulaxis, N.G., Papadopoulos, M.P.: Optimal feeder routing in distribution system planning using dynamic programming technique and GIS facilities. IEEE Trans. on Power Delivery 17(1), 242–247 (2002)
Chekuri, C., Ene, A., Korula, N.: Unsplittable flow in paths and trees and column-restricted packing integer programs. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX-RANDOM 2009. LNCS, vol. 5687, pp. 42–55. Springer, Heidelberg (2009)
Dinitz, Y., Garg, N., Goemans, M.X.: On the single-source unsplittable flow problem. Combinatorica 19(1), 17–41 (1999)
Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP-Completeness (Twenty-second printing), pp. 90–91. W.H. Freeman and Company (2000)
Ito, T., Demaine, E.D., Zhou, X., Nishizeki, T.: Approximability of partitioning graphs with supply and demand. Journal of Discrete Algorithms 6, 627–650 (2008)
Ito, T., Zhou, X., Nishizeki, T.: Partitioning graphs of supply and demand. Discrete Applied Math. 157, 2620–2633 (2009)
Ito, T., Zhou, X., Nishizeki, T.: Partitioning trees of supply and demand. IJFCS 16(4), 803–827 (2005)
Kawabata, M., Nishizeki, T.: Partitioning trees with supply, demand and edge-capacity. In: Proc. of ISORA 2011. Lecture Notes in Operation Research, vol. 14, pp. 51–58 (2011); Also IEICE Trans. on Fundamentals of Electronics, Communications and Computer Science (to appear)
Kim, M.S., Lam, S.S., Lee, D.-Y.: Optimal distribution tree for internet streaming media. In: Proc. 23rd Int. Conf. on Distributed Computing System (ICDCS 2003), pp. 116–125 (2003)
Kleinberg, J.M.: Single-source unsplittable flow. In: Proc. of 37th FOCS, pp. 68–77 (1996)
Morton, A.B., Mareels, I.M.Y.: An efficient brute-force solution to the network reconfiguration problem. IEEE Trans. on Power Delivery 15(3), 996–1000 (2000)
Nishizeki, T., Vygen, J., Zhou, X.: The edge-disjoint paths problem is NP-complete for series-parallel graphs. Discrete Applied Math. 115, 177–186 (2001)
Takamizawa, K., Nishizeki, T., Saito, N.: Linear-time computability of combinatorial problems on series-parallel graphs. J. Assoc. Comput. Mach. 29, 623–641 (1982)
Teng, J.-H., Lu, C.-N.: Feeder-switch relocation for customer interruption cost minimization. IEEE Trans. on Power Delivery 17(1), 254–259 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kawabata, M., Nishizeki, T. (2013). Spanning Distribution Trees of Graphs. In: Fellows, M., Tan, X., Zhu, B. (eds) Frontiers in Algorithmics and Algorithmic Aspects in Information and Management. Lecture Notes in Computer Science, vol 7924. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38756-2_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-38756-2_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38755-5
Online ISBN: 978-3-642-38756-2
eBook Packages: Computer ScienceComputer Science (R0)