Abstract: 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. A Radiocoloring (RC) of a graph G(V,E) is an assignment function View the MathML source such that |{\"E}(u)−{\"E}(v)|greater-or-equal, slanted2, when u,v are neighbors in G, and |{\"E}(u)−{\"E}(v)|greater-or-equal, slanted1 when the distance of u,v in G is two. The discrete number of frequencies used is called order and the range of frequencies used, span. The optimization versions of the Radiocoloring Problem (RCP) are to minimize the span (min span RCP) or the order (min order RCP).
In this paper, we deal with an interesting, yet not examined until now, variation of the radiocoloring problem: that of satisfying frequency assignment requests which exhibit some periodic behavior. In this case, the interference graph (modelling interference between transmitters) is some (infinite) periodic graph. Infinite periodicgraphs usually model finite networks that accept periodic (in time, e.g. daily) requests for frequency assignment. Alternatively, they can 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. We focus on planar periodicgraphs, 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({\"A}(Gi)+{\'o})) time algorithm (where|Vi|=n, {\"A}(Gi) is the maximum degree of the graph Gi and {\'o} 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 View the MathML source as {\"A}(Gi)+{\'o} tends to infinity.
We remark that, any approximation algorithm for the min span RCP of a finite planar graph G, that achieves a span of at most {\'a}{\"A}(G)+constant, for any {\'a} and where {\"A}(G) is the maximum degree of G, can be used as a subroutine in our algorithm to produce an approximation for min span RCP of asymptotic ratio {\'a} for periodic planar graphs.

Abstract: 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 {\"O}: V → IN such that ∣{\"O}(u) - {\"O}(v)∣ ≥2, when u, v are neighbors in G, and ∣{\"O}(u) - {\"O}(v)∣ ≥1 when the distance of u, v 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 periodicgraphs 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 G i (V i ,E i ). The edge set of G is derived by connecting the vertices of each iteration G i to some of the vertices of the next iteration G i +1, the same for all G i . The model of periodicgraphs considered here is similar to that of periodicgraphs in Orlin [13], Marathe et al [10]. We focus on planar periodicgraphs, 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({\"A}(G i ) + {\'o})) time algorithm, (where ∣V i ∣ = n, {\"A}(G i ) is the maximum degree of the graph G i and {\'o} is the number of edges connecting each G i to G i +1), which obtains a radiocoloring of a periodic planar graph G that approximates the minimum span within a ratio which tends to 2 as {\"A}(Gi) + {\'o} tends to infinity.