Abstract
Let G =(V,E) be a requirements graph. Let d =(d ij n i,j be a length metric. For a tree T denote by d T(i,j) the distance between i and j in T (the length according to d of the unique i – j path in T).The restricted diameter of T,D T, is the maximum distance in T between pair of vertices with requirement between them. The minimum restricted diameter spanning tree problem is to find a spanning tree T such that the minimum restricted diameter is minimized. We prove that the minimum restricted diameter spanning tree problem is NP - hard and that unless P = NP there is no polynomial time algorithm with performance guarantee of less than 2.In the case that G contains isolated vertices and the length matrix is defined by distances over a tree we prove that there exist a tree over the non-isolated vertices such that its restricted diameter is at most 4 times the minimum restricted diameter and that this constant is at least 3 1/2. We use this last result to present an O(log (n))-approximation algorithm.
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
N. Alon, R. Karp, D. Peleg and D. West,“A graph theoretic game and its application to the k-server problem”, SIAM J.Comput.,24,78–100,1995.
Y. Bartal,“Probabilistic approximation of metric spaces and its algorithmic applications”, Proceedings of FOCS1996, pages 184–193.
Y. Bartal,“On approximating arbitrary metrics by tree metrics”,Proceedings of STOC1998, pages 161–168.
M. Charikar, C. Chekuri, A. Goel and S. Guha,“Rounding via trees:deterministic approximation algorithms for group steiner trees and k-median”, Proceedings of STOC1998,pages 114–123.
A. Gupta,“Steiner points in tree metrics don’t (really)help”, Proceeding of SODA2001, pages 220–227.
M.R. Garey and D.S. Johnson,“Computers and intractability:a guide to the theory of NP-completeness”, W.H.Freeman and Company,1979.
R. Hassin and A. Tamir,“On the minimum diameter spanning tree problem”, IPL,53,109–111,1995.
F. Harary,“Graph theory”, Addison-Wesley,1969.
J.-M. Ho, D.T. Lee, C.-H. Chang, and C.K. Wong,“Minimum diameter spanning trees and related problems”, SIAM J.Comput.,20(5),987–997,1991.
S. Khuller, B. Raghavachari and N. Young,“Designing multi-commodity flow trees”, IPL,50, 49–55,1994.
D. Peleg and E. Reshef, “Deterministic polylog approximation for minimum communication spanning trees”, Proceedings of ICALP1998,pages 670–681.
P. Seymour and R. Thomas,“Call routing and the rat catcher”,Combinatorica,14,217–241,1994.
A. Tamir, Private communication,2001.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hassin, R., Levin, A. (2002). Minimum Restricted Diameter Spanning Trees. In: Jansen, K., Leonardi, S., Vazirani, V. (eds) Approximation Algorithms for Combinatorial Optimization. APPROX 2002. Lecture Notes in Computer Science, vol 2462. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45753-4_16
Download citation
DOI: https://doi.org/10.1007/3-540-45753-4_16
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44186-1
Online ISBN: 978-3-540-45753-4
eBook Packages: Springer Book Archive