Abstract
Let G = (V, E) be an undirected multi-graph where V and E are a set of nodes and a set of edges, respectively. Let k and l be fixed nonnegative integers. This paper considers location problems of finding a minimum size of node-subset S ⫅ V such that node-connectivity between S and x is greater than or equal to k and edge-connectivity between S and x is greater than or equal to l for every x ∈ V . This problem has important applications for multi-media network control and design. For a problem of considering only edge-connectivity, i.e., k = 0, an O(L(|V |, |E|, l)) = O( |E|+ |V |2 + |V |min {|E |; l |V |} min {l, |V |mponents for h = 1, 2,..., l. This paper presents an O(L(|V|, |E|, l)) time algorithm for 0≤ k ≤ 2 and l ≥ 0. It also shows that if k ≥ 3, the problem is NP-hard even for l = 0. Moreover, it shows that if the size of S is regarded as a parameter, the parameterized problem for k = 3 and l ≤ 1 is FPT (fixed parameter tractable).
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
Aho, A. V., Hopcroft, J. E., and Ullman, J. D., The Design and Analysis of Computer Algorithms, Addison-Wesley (1974).
Arata, K., Iwata, S., Makino, K., and Fujishige, S., Locating sources to meet ow demands in undirected networks, manuscript (2000).
Downey, R. G. and Fellows, M. R., Parameterized Complexity, Springer (1997).
Garey, M. R. and Johnson, D. S., Computers and Intractability: a Guide to the Theory of NP-Completeness, Freeman (1979).
Honami, S., Ito, H., Uehara, H., and Yokoyama, M., An algorithm for finding a node-subset having high connectivity from other nodes, IPSJ SIG Notes, AL-66, 99, 8, pp. 9–16 (1999). (in Japanese)
Hopcroft, J. E. and Tarjan, R. E., Dividing a graph into triconnected components, SIAM J. Comput., 2, pp. 135–158 (1973).
Ito, H. and Yokoyama, M., Edge connectivity between nodes and node-subsets, Networks, 31, pp. 157–164 (1998).
Ito, H., Uehara H., and Yokoyama, M., A faster and exible algorithm for a location problem on undirected ow networks, IEICE Trans., E83-A, 4, pp. 704–712 (2000).
Labbe, M., Peeters, D., and Thisse, J.-F., Location on networks, In M. O. Ball et al. (eds.), Handbooks in OR & MS, 8, North-Holland, pp. 551–624 (1995).
Nagamochi, H. and Watanabe, T., Computing k-edge-connected components in multigraphs, IEICE Trans., E76-A, pp. 513–517 (1993).
Tamura, H., Sengoku, M., Shinoda, S., and Abe, T., Location problems on undirected ow networks, IEICE Trans., E73, pp. 1989–1993 (1990).
Tamura, H., Sugawara, H., Sengoku, M., and Shinoda, S., Plural cover problem on undirected ow networks, IEICE Trans. A, J81-A, pp. 863–869 (1998). (in Japanese)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ito, H., Ito, M., Itatsu, Y., Uehara, H., Yokoyama, M. (2000). Location Problems Based on Node-Connectivity and Edge-Connectivity between Nodes and Node-Subsets. In: Goos, G., Hartmanis, J., van Leeuwen, J., Lee, D.T., Teng, SH. (eds) Algorithms and Computation. ISAAC 2000. Lecture Notes in Computer Science, vol 1969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-40996-3_29
Download citation
DOI: https://doi.org/10.1007/3-540-40996-3_29
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41255-7
Online ISBN: 978-3-540-40996-0
eBook Packages: Springer Book Archive