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.

Parts of this work are supported by a stipend of the “Gemeinsames Hochschulson- derprogramm III von Bund und Länder” through the DAAD.