The complexity of broadcasting in planar and decomposable graphs

Abstract

Broadcasting in processor networks means disseminating a single piece of information, which is originally known only at some nodes, to all members of the network. The goal is to inform everybody using as few rounds as possible, that is to minimize the broadcasting time.

Given a graph and a subset of nodes, the sources, the problem to determine its specific broadcast time, or more general to find a broadcast schedule of minimal length has been shown to be NP-complete. In contrast to other optimization problems for graphs, like vertex cover or traveling salesman, little was known about restricted graph classes for which polynomial time algorithms exist, for example for graphs of bounded treewidth. The broadcasting problem is harder in this respect because it does not have the finite state property. Here, we will investigate this problem in detail and prove that it remains hard even if one restricts to planar graphs of bounded degree or constant broadcasting time. A simple consequence is that the minimal broadcasting time cannot even be approximated with an error less than 1/8, unless P=NP.

On the other hand, we will investigate for which classes of graphs this problem can be solved efficiently and show that broadcasting and even a more general version of this problem becomes easy for graphs with good decomposition properties. The solution strategy can efficiently be parallelized, too. Combining the negative and the positive results reveals the parameters that make broadcasting difficult. Depending on simple graph properties the complexity jumps from NC or P to NP.

Supported by Deutsche Forschungsgemeinschaft, Grant DFG Re672/2, and German-Israeli Foundation for Scientific Research & Development, Grant GIF I-2007-199.06/91