# On the Inapproximability of Broadcasting Time

## 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.

## Keywords

Tree Decomposition Reduction Graph Busy Schedule Binomial Tree Complete Binary Tree## Preview

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## 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.Google Scholar - BHLP92. J.-C. Bermond, P. Hell, A. Liestman, and J. Peters,
*Broadcasting in Bounded Degree Graphs*, SIAM J. Disc. Math. 5, 1992, 10–24.MATHCrossRefMathSciNetGoogle Scholar - Feig98. U. Feige,
*A threshold of ln n for approximating set cover*, Journal of the ACM, Vol. 45, 1998 (4):634–652.MATHCrossRefMathSciNetGoogle Scholar - GaJo79. M. Garey and D. Johnson,
*Computers and Intractability, A Guide To the Theory of NP-Completeness*, Freeman 1979.Google Scholar - Håast97. J. Håastad,
*Some optimal inapproximability results*, In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, 1997, 1–10.Google Scholar - HHL88. S. Hedetniemi, S. Hedetniemi, and A. Liestman,
*A Survey of Gossiping and Broadcasting in Communication Networks*, Networks 18, 1988, 319–349.MATHCrossRefMathSciNetGoogle Scholar - JRS98. A. Jakoby, R. Reischuk, C. Schindelhauer,
*The Complexity of Broadcasting in Planar and Decomposable Graphs*, Discrete Applied Mathematics 83, 1998, 179–206.MATHCrossRefMathSciNetGoogle Scholar - LP88. A. Liestman and J. Peters,
*Broadcast Networks of Bounded Degree*, SIAM J. Disc. Math. 4, 1988, 531–540.CrossRefMathSciNetGoogle Scholar - Midd93. M. Middendorf,
*Minimum Broadcast Time is NP-complete for 3-regular planar graphs and deadline 2*, Information Processing Letters, 46, 1993, 281–287.MATHCrossRefMathSciNetGoogle Scholar - 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.Google Scholar - SCH81. P. Slater, E. Cockayne, and S. Hedetniemi,
*Information Dissemination in Trees*, SIAM J. Comput. 10, 1981, 692–701.MATHCrossRefMathSciNetGoogle Scholar