ALCOMFT-TR-01-109

Dimitris Fotakis, Sotiris Nikoletseas, Vicky Papadopoulou and Paul Spirakis
NP-completeness Results and Efficient Approximations for Radiocoloring in Planar Graphs
Patras. Work packages 2 and 4. May 2001.

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 \Phi: V -> \rmI\!\rmN such that |\Phi(u)-\Phi(v)| >= 2, when u, v are neighbors in G, and |\Phi(u)-\Phi(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(n\Delta) 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 \Delta 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 lambda colors, in the case lambda >= 4 \Delta+50.

Postscript file: ALCOMFT-TR-01-109.ps.gz (117 kb).