Radiocolorings in periodic planar graphs: PSPACE-completeness and efficient approximations for the optimal range of frequencies

The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. The Radiocoloring (RC) of a graph G(V,E) is an assignment function Φ: V → IN such that |Φ(u) - Φ(ν)| ≥ 2, when u, ν are neighbors in G, and |Φ(u) - Φ(ν)| ≥ 1 when the distance of u, ν in G is two. The range of frequencies used is called span. Here, we consider the optimization version of the Radiocoloring Problem (RCP) of finding a radiocoloring assignment of minimum span, called min span RCP. In this paper, we deal with a variation of RCP: that of satisfying frequency assignment requests with some periodic behavior. In this case, the interference graph is an (infinite) periodic graph. Infinite periodic graphs model finite networks that accept periodic (in time, e.g. daily) requests for frequency assignment. Alternatively, they may model very large networks produced by the repetition of a small graph. A periodic graph G is defined by an infinite two-way sequence of repetitions of the same finite graph Gi(Vi,Ei). The edge set of G is derived by connecting the vertices of each iteration Gi to some of the vertices of the next iteration Gi+1, the same for all Gi. The model of periodic graphs considered here is similar to that of periodic graphs in Orlin [13], Marathe et al [10]. We focus on planar periodic graphs, because in many cases real networks are planar and also because of their independent mathematical interest. We give two basic results: -We prove that the min span RCP is PSPACE-complete for periodic planar graphs.-We provide an O(n(Δ(Gi) + σ)) time algorithm, (where |Vi| = n, Δ(Gi) is the maximum degree of the graph Gi and σ is the number of edges connecting each Gi to Gi+1), which obtains a radiocoloring of a periodic planar graph G that approximates the minimum span within a ratio which tends to 2 as Δ(Gi) + σ tends to infinity.