Graph-Theoretical Models for Frequency Assignment Problems

In the present dissertation we investigate structural and algorithmic aspects of the frequency assignment problem in mobile telephone networks. This problem is of particular interest for the graph theory because of its close relationship to graph coloring. The frequency assignment problem includes the characteristic features of T-coloring, list coloring, and set coloring, and belongs thereby to NP-hard combinatorial problems. The main part of this thesis is dedicated to the so-called hybrid networks, with some stations operating according to the static principle and other stations according to the dynamic principle. We develop a graph-theoretical model and introduce two optimization criteria for the frequency assignment problem in hybrid networks. The computational complexity of evaluating these criteria is first examined for graphs with bounded treewidth and for complete graphs. In the further investigations we concentrate on the criterion called channel stability number, which generalizes the k-partial coloring problem. We develop efficient algorithms for some special cases of the channel stability number in complete graphs and study its approximability. Theoretical complexity results are used in the development of heuristic algorithms for the computation of the channel stability number in general graphs. For our test cases based on real world data, the comparison of lower and upper bounds shows that the algorithms provide results with a relative error between 1 and 6 percent. Mobile telephone networks can be in a simplified way modeled using disk graphs. In the last part of the thesis the properties of four classes of disk graphs are examined. We show that the chromatic number of the members of any of these classes is bounded by a linear function of the clique number. A linear relation is also shown for some subclasses of intersection graphs of rectangles. Finally, we compare disk graphs and planar graphs and present improved algorithms for the maximum clique problem in unit disk graphs and for the vertex cover problem in intersection disk graphs.