Abstract
We investigate the problem of broadcasting information in a given undirected network. At the beginning information is given at some processors, called sources. Within each time unit step every informed processor can inform only one neighboring processor. The broadcasting problem is to determine the length of the shortest broadcasting schedule for a network, called the broadcasting time of the network. We show that there is no efficient approximation algorithm for the broadcasting time of a network with a single source unless P = NP. More formally, it is NP-hard to distinguish between graphs G = (V,E) with broadcasting time smaller than \( b \in \theta \left( {\sqrt {|V|} } \right) \) and larger than \( (\tfrac{{57}} {{56}} - \varepsilon )b \) for any ∈ > 0. For ternary graphs it is NP-hard to decide whether the broadcasting time is b ε Θ(log |V|) or b + Θ(\( \sqrt b \)) in the case of multiples sources. For ternary networks with single sources, it is NP-hard to distinguish between graphs with broadcasting time smaller than \( b \in \theta \left( {\sqrt {|V|} } \right) \) and larger than \( b + c\sqrt {\log {\mathbf{ }}b} \). We prove these statements by polynomial time reductions from E3-SAT. Classification: Computational complexity, inapproximability, network communication.
Parts of this work are supported by a stipend of the “Gemeinsames Hochschulson- derprogramm III von Bund und Länder” through the DAAD.
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
BGNS98. A. Bar-Noy, S. Guha, J. Naor, B. Schieber, Multicasting in heterogeneous networks, In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, 1998, 448–453.
BHLP92. J.-C. Bermond, P. Hell, A. Liestman, and J. Peters, Broadcasting in Bounded Degree Graphs, SIAM J. Disc. Math. 5, 1992, 10–24.
Feig98. U. Feige, A threshold of ln n for approximating set cover, Journal of the ACM, Vol. 45, 1998 (4):634–652.
GaJo79. M. Garey and D. Johnson, Computers and Intractability, A Guide To the Theory of NP-Completeness, Freeman 1979.
Håast97. J. Håastad, Some optimal inapproximability results, In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, 1997, 1–10.
HHL88. S. Hedetniemi, S. Hedetniemi, and A. Liestman, A Survey of Gossiping and Broadcasting in Communication Networks, Networks 18, 1988, 319–349.
JRS98. A. Jakoby, R. Reischuk, C. Schindelhauer, The Complexity of Broadcasting in Planar and Decomposable Graphs, Discrete Applied Mathematics 83, 1998, 179–206.
LP88. A. Liestman and J. Peters, Broadcast Networks of Bounded Degree, SIAM J. Disc. Math. 4, 1988, 531–540.
Midd93. M. Middendorf, Minimum Broadcast Time is NP-complete for 3-regular planar graphs and deadline 2, Information Processing Letters, 46, 1993, 281–287.
MRSR95. M. V. Marathe, R. Ravi, R. Sundaram, S.S. Ravi, D.J. Rosenkrantz, H.B. Hunt III, Bicriteria network design problems, Proc. 22nd Int. Colloquium on Automata, Languages and Programming, Lecture Notes in Comput. Sci. 944, Springer-Verlag, 1995, 487–498.
SCH81. P. Slater, E. Cockayne, and S. Hedetniemi, Information Dissemination in Trees, SIAM J. Comput. 10, 1981, 692–701.
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
Schindelhauer, C. (2000). On the Inapproximability of Broadcasting Time. In: Jansen, K., Khuller, S. (eds) Approximation Algorithms for Combinatorial Optimization. APPROX 2000. Lecture Notes in Computer Science, vol 1913. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44436-X_23
Download citation
DOI: https://doi.org/10.1007/3-540-44436-X_23
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67996-7
Online ISBN: 978-3-540-44436-7
eBook Packages: Springer Book Archive