The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters, by exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. A Radiocoloring (RC) of a graph is an assignment function such that , when are neighbors in G, and when the distance of in G is two. The number of discrete frequencies and the range of frequencies used are called order and span, respectively. The optimization versions of the Radiocoloring Problem (RCP) are to minimize the span or the order. In this paper we prove that the radiocoloring problem for general graphs is hard to approximate (unless ) within a factor of (for any ), where n is the number of vertices of the graph. However, when restricted to some special cases of graphs, the problem becomes easier. We prove that the min span RCP is NP-complete for planar graphs. Next, we provide an time algorithm which obtains a radiocoloring of a planar graph G that approximates the minimum order within a ratio which tends to 2 (where the maximum degree of G). Finally, we provide a fully polynomial randomized approximation scheme (fpras) for the number of valid radiocolorings of a planar graph G with colors, in the case where .
A preliminary version of this work [8] has appeared in the Proceedings of the 25th International Symposium on Mathematical Foundations of Computer Science (MFCS 2000). This research is partially supported by the European Union Fifth Framework Programme Projects ALCOM-FT, ARACNE and the Greek GSRT PENED’99 Project ALKAD.