Real-valued Frequency Assignment

We consider the binary constraints formulation of the frequency assignment problem in its most general form: for an arbitrary metric space, with frequencies taking arbitrary real values, and with possibly innnitely many constraints. We obtain some necessary and suucient conditions for the problem to have a solution with a nite span. When the metric space is the set of integers, we give an exact criterion. Also we demonstrate a connection of this problem in one-dimensional case with one combinatorial question about nite permutations; and pose some unsolved problems. The practical problem of assigning frequencies to transmitters in a cellular network has been approximated by various combinatorial and/or geometrical formulations. Most often it is treated as a problem of nding a colouring of a metric space (e.g. a graph). The required colouring should satisfy a certain set of restrictions; usually each restriction involves two vertices, and the restrictions therefore are called binary constraints. In this note we consider a variant of the standard binary constraints problem in which frequencies (or colours) can take on arbitrary real values. Apart from its theoretical signiicance, this problem can also have some practical importance; especially for the question to what extent one can improve the performance by using more channels within the same frequency interval.