Radiocoloring in planar graphs: Complexity and approximations

https://doi.org/10.1016/j.tcs.2005.03.013Get rights and content
Under an Elsevier user license
open archive

Abstract

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 G(V,E) is an assignment function Φ:VN such that |Φ(u)-Φ(v)|2, when u,v are neighbors in G, and |Φ(u)-Φ(v)|1 when the distance of u,v 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 NP=ZPP) within a factor of n1/2-ε (for any ε>0), 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 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 where λ4Δ+50.

Keywords

Mobile computing
Radio communication
Coloring
Computational complexity
Approximations
Planar graphs
Rapid mixing
Coupling

Cited by (0)

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.