IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 1, JANUARY 2007 331 Measuring the Effectiveness of Frequency Assignment Algorithms Derek H. Smith, Lesley A. Hughes, Jim N. J. Moon, and Roberto Montemanni Abstract—Lower bounds are used to assess the quality of fre- quency assignments and the effectiveness of the algorithms used to obtain them. This paper makes three contributions. First, a technique is described for modifying an effective existing lower bound for the span to take account of multiple interference. Multiple interference may increase the span and the modification captures some or all of this increase. Second, new results are given for a lower bound for some of the two level penalty-based COST259 problems. It remains true that for these problems, the gap between upper and lower bounds is large, by comparison with other benchmarks. Third, some evidence is presented to suggest that the assignments available today for problems of the COST259 type are still capable of very significant improvement. Index Terms—Lower bounds, multiple interference, radio frequency assignment. I. I NTRODUCTION T HE TASK of assigning frequencies to the transmitters of a radio network is usually known as the frequency assign- ment problem (FAP), or channel assignment problem. The aim is to use the separation of assigned frequencies to minimize interference in the network, while at the same time using spec- trum efficiently. The task is applied directly to real radio net- works and has to be addressed by network operators regularly as the network grows and evolves. There has been an enormous volume of literature devoted to the FAP over the past 30 years; see [1]–[3] for surveys and [4] for an extensive bibliography. Although third-generation mobile telephone networks are code- division-multiple-access systems and do not use frequency assignment, the problem remains important in the continuing operation of second-generation mobile telephone systems and in many other radio systems, both civil and military. These include satellite systems where a number of different services of different types and bandwidths may need to be assigned [5]. Two versions of the FAP are considered here: the minimum span FAP and the fixed spectrum FAP. The minimum span FAP aims to find assignments using the minimum number of Manuscript received February 16, 2005; revised November 22, 2005 and February 20, 2006. The work of R. Montemanni was supported in part by the Swiss National Science Foundation through project 200020-109854/1. The review of this paper was coordinated by Prof. T. Hou. D. H. Smith and L. A. Hughes are with the Division of Mathematics and Statistics, University of Glamorgan, CF37 1DL Pontypridd, U.K. (e-mail: dhsmith@glam.ac.uk; lahughe1@glam.ac.uk). J. N. J. Moon is with the School of Computing, University of Glamorgan, CF37 1DL Pontypridd, U.K. (e-mail: jnjmoon@glam.ac.uk). R. Montemanni is with the Istituto Dalle Molle di Studi sull’Intelligenza Artificiale (IDSIA), 6928 Manno, Switzerland (e-mail: roberto@idsia.ch). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2006.883770 consecutive equally spaced channels for which all specified frequency separation constraints are satisfied. These are usually constraints of the form |f (T i ) - f (T j )|≥ c ij , where c ij is an integer giving the required frequency separation, and f (T i ) is the frequency assigned to a transmitter T i . Generally, these binary constraints are derived by a computation of interference involving pairs of transmitters and their receivers. This version of the FAP is of most interest to spectrum authorities and in the initial planning of systems. In the fixed spectrum FAP, the spectrum available is defined initially as the spectrum available to the operator for the service. It normally consists of a set of consecutive equally spaced channels, possibly with some gaps of one or more channels that are unavailable for some or all transmitters. Some measure of interference must then be minimized. This may consist of the number of frequency separation constraints violated, or a sum of penalties associated with the violated constraints, or a more detailed signal-to- interference measure. This version of the FAP is of importance to operators of a wide range of terrestrial and satellite-based radio systems, both commercial and military. Lower bounds on the span have always been important in frequency assignment work (see [6]–[8] for early references). These are used to assess the quality of frequency assignments and, therefore, to assess the effectiveness of algorithms used to find them. Generally, lower bounds for the span are successful for realistic FAPs, as first observed in [6]. This can be seen from the results for the Philadelphia and minimum span CELAR benchmarks at [4], for example. Usually, the actual span equals the lower bound, and it is rare to find a realistic FAP for which the best lower bound is less than 90% of the achievable span. The same is not true for randomly generated FAPs [9], where the lower bound can be much too small to be useful. Theoretical reasons for this situation are known [10], [11]. Lower bounds for fixed spectrum problems also exist [12]–[15]. Good results have been achieved for many of the fixed spectrum (minimum interference) CELAR instances [4] and for many of the unweighted and weighted problems con- sidered by Montemanni et al. [14]. However, for the COST259 problems [4], the bounds are poor. The gap between the up- per and lower bounds is too large for the lower bound to be useful. Recently, there has been increasing interest in the possibility of including multiple interference in FAPs [16]. The most sat- isfactory approaches appear to be those that use a cost function based on signal-to-interference ratios (SIRs) at many reception points [17], [18]. If multiple interference is considered, the gap between the achievable span and existing lower bounds may well increase to the point where the bounds are no longer useful. 0018-9545/$25.00 © 2007 IEEE