Abstract
Let G=(V,E) denote a weighted graph of n nodes and m edges, and let G[ V ′ ] denote the subgraph of G induced by a subset of nodes V′ ⊆ V. The radius of G[ V ′ ] is the maximum length of a shortest path in G[ V ′ ] emanating from its center (i.e., a node of G[ V ′ ] of minimum eccentricity). In this paper, we focus on the problem of partitioning the nodes of G into exactly p non-empty subsets, so as to minimize the sum of the induced subgraph radii. We show that this problem – which is of significance in facility location applications – is NP-hard when p is part of the input, but for a fixed constant p > 2 it can be solved in O(n 2p/p!) time. Moreover, for the notable case p=2, we present an efficient O(mn 2+n 3 logn) time algorithm.
Work partially supported by the Research Project GRID.IT, funded by the Italian Ministry of Education, University and Research, by the European Union under COST 295 (DYNAMO), and by the Swiss SBF under grant no. C05.0047. Part of this work has been developed while the first author was visiting ETH Zürich.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alt, H., Arkin, E.M., Brönnimann, H., Erickson, J., Fekete, S.P., Knauer, C., Lenchner, J., Mitchell, J.S.B., Whittlesey, K.: Minimum-cost coverage of point sets by disks. In: Proc. 22nd ACM Symp. on Computat. Geometry (SoCG 2006), pp. 449–458 (2006)
Agarwal, P.K., Procopiuc, C.M.: Exact and approximation algorithms for clustering. Algorithmica 33(2), 201–226 (2002)
Biló, V., Caragiannis, I., Kaklamanis, C., Kanellopoulos, P.: Geometric clustering to minimize the sum of cluster sizes. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 460–471. Springer, Heidelberg (2005)
Charikar, M., Panigrahy, R.: Clustering to minimize the sum of cluster diameters. J. of Computer and Systems Sciences 68(2), 417–441 (2004)
Cormack, R.M.: A review of classification. J. of the Royal Statistical Society 134, 321–367 (1971)
Doddi, S.R., Marathe, M.V., Ravi, S.S., Taylor, D.S., Widmayer, P.: Approximation algorithms for clustering to minimize the sum of diameters. Nordic Journal of Computing 7(3), 185–203 (2000)
Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theor. Comp. Science 38(23), 293–306 (1985)
Hochbaum, D.S., Shmoys, D.B.: A best possible heuristic for the k-center problem. Mathematics of Operations Research 10, 180–184 (1985)
Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems. I: The p-centers. SIAM J. Applied Mathematics 37(3), 519–538 (1979)
Labbe, M., Peeters, D., Thisse, J.F.: Location on networks. In: Ball, M., Magnanti, T., Francis, R.L. (eds.) Handbooks in Operations Research and Management Science: Network Routing, Elsevier, Amsterdam (1995)
Lev-Tov, N., Peleg, D.: Polynomial time approximation schemes for base station coverage with minimum total radii. Computer Networks 47(4), 489–501 (2005)
Plesník, J.: On the computational complexity of centers locating in a graph. Aplikace Matematiky 25(6), 445–452 (1980)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Proietti, G., Widmayer, P. (2006). Partitioning the Nodes of a Graph to Minimize the Sum of Subgraph Radii. In: Asano, T. (eds) Algorithms and Computation. ISAAC 2006. Lecture Notes in Computer Science, vol 4288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11940128_58
Download citation
DOI: https://doi.org/10.1007/11940128_58
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-49694-6
Online ISBN: 978-3-540-49696-0
eBook Packages: Computer ScienceComputer Science (R0)