NP-Completeness Results and Efficient Approximations for Radiocoloring in 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 Φ : V → IN such that |Φ(u) - Φ(v)| ≥ 2, when u, v are neighbors in G, and |Φ(u) - Φ(v)| ≥ 1 when the minimum distance of u, v in G is two. The discrete number 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 min span RCP is NP-complete for planar graphs. Next, we provide an O(nΔ) time algorithm (|V| = n) which obtains a radiocoloring of a planar graph G that approximates the minimum order within a ratio which tends to 2 (where Δ 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 λ colors, in the case λ ≥ 4Δ + 50.