Solving Frequency Assignment Problems via Tree-Decomposition 1

In this extended abstract we describe a computational study to solve hard frequency assignment problems (FAPs) to optimality using a tree decomposition of the graph that models interference constraints. We present a dynamic programming algorithm which solves FAPs based on this tree decomposition. With the use of several dominance and bounding techniques it is possible to solve small and medium-size real-life instances of the frequency assignment problem to optimality. Moreover, with an iterative version of the algorithm we obtain good lower bounds for large-size instances within reasonable time and memory limits. The Frequency Assignment Problem (FAP) has two basic structural properties: the limited availability of frequencies to be assigned to wireless connections, and signal interference between connections for some combinations of frequencies. Practical applications range from military communication and television broadcasting to (the most popular example) mobile telephone communication. This diversity has not only resulted in many different models, but also in many different types of instances. The model we consider is fairly general in the sense that most variants of the FAP can be transformed to it. It consists of a set of antennas that are all to be assigned a frequency from an antenna-dependent set of available frequencies, the domain. In some applications certain frequencies are favored over others. We model this by introducing a penalty on each frequency from the domain of an antenna. For pairs of antennas specific combinations of frequencies may interfere, resulting in loss of quality of the reception of the signals. This loss of quality is measured and 1 Extended abstract of presentation at the 6th Twente Workshop on Graphs and Combinatorial Optimization 2 Corresponding author, e-mail: A.Koster@KE.UniMaas. NL; website: http://www.unimaas. nl/ ̃akoster/ Preprint submitted to Elsevier Preprint 26 April 1999 penalized with an amount related to the level of interference. For each pair of antennas, the penalties for all possible combinations of frequencies are stored in a penalty matrix. The penalty matrices have a structure that is often useful in solution methods, namely that frequencies within a given distance have a high penalty, and frequencies at larger distances have no penalty. The pairwise relationship between the antennas allows for the following graph model: the vertices represent the antennas, each antenna pair with a nonzero penalty matrix is connected by an edge. This graph is called the constraint or interference graph. The standard objectives are to find a frequency plan that minimizes the sum of the vertex and edge-penalties, or that minimizes the maximum penalty. We refer to Hale [5] for an overview of models for the FAP with other objectives. Summarized, the FAP is defined by the quadruple (G, D, p, q), where G = (V, E) is the constraint graph. The set D = {Dv : v ∈ V } is the collection of all domains. For each of the n vertices v ∈ V in the graph the domain Dv contains the frequencies available for that vertex. The last two components of the FAP are two penalty functions p and q. For each pair of adjacent vertices and corresponding choice of frequencies, the function p determines the interference penalty. The function q denotes the level of preference for all domain elements. The functions p and q are called the edge-penalty function, and the vertex-penalty function, respectively. The objective of the problem is to select from every domain Dv exactly one element in such a way that the total sum of the edgeand vertex-penalties is minimized. The FAP is, in most of its variants, hard to solve, due to its close relation to the vertex coloring problem. In Koster, van Hoesel and Kolen [8] it is proved with a reduction from Maximum Satisfiability that the FAP is NP-hard, even if all domains have size 2. Therefore, many heuristic approaches have been suggested using all of the known methods in operations research and artificial intelligence, like simulated annealing (cf. Hurkens and Tiourine [6]), tabu search (cf. Castelino, Hurley and Stephens [4]) and genetic algorithms (Kolen [7]). A comparison of these techniques on a specific set of data can be found in [10]. In this paper we concentrate on finding exact solutions, or (second best) on finding good lower bounds for the FAP. To obtain lower bounds for this problem, Hurkens and Tiourine [6] use nonlinear programming techniques. Good lower bounds are only obtained for very special cost structures and fairly simple constraint graphs. An exact solution technique has been studied by Koster, van Hoesel and Kolen [8]. They investigate the polyhedral approach which can also be used to obtain lower bounds. This approach only works within reasonable time for problems with a limited number of frequencies available for every antenna. Aardal et al. [1] use a branch and cut algorithm to solve the FAP in which a solution without penalty is to be find. Forced by the limited success of the exact solution methods so far, we tried to