Abstract
Suppose that each vertex of a graph G is either a supply vertex or a demand vertex and is assigned a supply or a demand. All demands and supplies are nonnegative constant numbers in a steady network, while they are functions of a variable λ in a parametric network. Each demand vertex can receive “power” from exactly one supply vertex through edges in G. One thus wishes to partition G to connected components by deleting edges from G so that each component has exactly one supply vertex whose supply is at least the sum of demands in the component. The “partition problem” asks whether G has such a partition. If G has no such partition, one wishes to find the maximum number r *, \(0\le r^* \textless 1\), such that G has such a partition when every demand is reduced to r * times the original demand. The “maximum supply rate problem” asks to compute r *. In this paper, we deal with a network in which G is a tree, and first give a polynomial-time algorithm for the maximum supply rate problem for a steady tree network, and then give an algorithm for the partition problem on a parametric tree network, which takes pseudo-polynomial time if all the supplies and demands are piecewise linear functions of λ.
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. 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 and RANDOM 2009. LNCS, vol. 5687, pp. 42–55. Springer, Heidelberg (2009)
Chekuri, C., Mydlarz, M., Shepherd, F.B.: Multicommodity demand flow in a tree. ACM Trans. on Algorithms 3, Article 3 (2007)
Gallo, G., Grigoriadis, M.D., Tarjan, R.E.: A fast parametric maximum flow algorithm and applications. SIAM J. Comput. 18(1), 30–55 (1989)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)
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., Hara, T., Zhou, X., Nishizeki, T.: Minimum cost partitions of trees with supply and demand. Algorithmica 64, 400–415 (2012)
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. Int. J. Found. Comput. Sci. 16, 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)
Minieka, E.: Parametric network flows. Operation Research 20(6), 1162–1170 (1972)
Morton, A.B., Mareels, I.M.Y.: An efficient brute-force solution to the network reconfiguration problem. IEEE Trans. Power Delivery 15, 996–1000 (2000)
Teng, J.-H., Lu, C.-N.: Feeder-switch relocation for customer interruption cost minimization. IEEE Trans. Power Delivery 17, 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
Morishita, S., Nishizeki, T. (2013). Parametric Power Supply Networks. In: Du, DZ., Zhang, G. (eds) Computing and Combinatorics. COCOON 2013. Lecture Notes in Computer Science, vol 7936. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38768-5_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-38768-5_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38767-8
Online ISBN: 978-3-642-38768-5
eBook Packages: Computer ScienceComputer Science (R0)