O.R. ApplicationsAn improved algorithm to determine lower bounds for the fixed spectrum frequency assignment problem
Introduction
The last decade has witnessed a dramatic increase in the importance of large scale radio networks, particularly in cellular mobile telephone applications. The corresponding growth in demand for frequencies has highlighted the importance of good network planning. The available radio spectrum is a limited resource, and frequency assignment algorithms aim to balance the economies achieved by frequency reuse with any consequent loss of quality in the network.
The form of the frequency assignment problem known as the fixed spectrum frequency assignment problem (FS-FAP), is of major importance to network operators. The available frequencies are given and some measure of interference must be minimised. The constraints considered here are binary constraints, specifying the necessary frequency separation between given pairs of transmitters. There may be penalties (or weights) associated with the violation of each constraint. The objective can be to minimise the number of constraints violated or, as is increasingly done by operators, to minimise the sum of the weights associated with violated constraints.
The ability to generate good lower bounds for the cost function in specific instances of the FS-FAP makes it possible to determine the quality of assignments generated for each instance. The general application of the techniques may establish the overall effectiveness of the algorithms used for frequency assignment. For these reasons, lower bounding techniques for fixed spectrum problems have generated particular interest among operators of mobile telephone networks and many other types of radio network. Only recently has any significant progress been made.
The main results presented in the literature in the field of lower bounds for the FS-FAP are due to Hurkens and Tiourine [9], Tiourine et al. [19], Smith et al. [17], Montemanni [13], Maniezzo and Montemanni [12], Koster [10] and Koster et al. [11]. All these lower bounds are studied for problems with particular features and none are general purpose.
More recently Helmberg has presented in [8] (see also [7]) a new lower bound based on semidefinite programming. The method provides a lower bound for the Min k-partition problem, that is seen as a simplification of FS-FAP. The main limitations of this approach are that it can model only co-channel constraints (requiring separations of at least 1 channel) and the intrinsic high computational complexity.
A different lower bounding approach, based on linear programming, has been presented in Montemanni et al. [15]. The technique gives good results for many problems of a moderate size, although there are some problems for which the quality of the bound is less good.
The aim of this paper is to improve the method of [15], in terms of the quality of the bounds, the speed of the algorithms and the memory requirement for variables and constraints. The potential for improvement of the quality of the results is variable. Indeed in some cases the lower bound is equal to a known upper bound. However, in other cases improvements will be demonstrated, which can be of over 26%. The improvement in memory efficiency, in combination with the use of more computer memory, allows much larger problems to be handled.
The paper is organised as follows. In Section 2 the FS-FAP is formally described through a graph theoretical model and then through its integer programming formulation. In Section 3 the method we propose is described and the differences from the technique presented in Montemanni et al. [15] are highlighted. In Section 4 some computational experiments, in which the new method and the one presented in [15] are compared, are presented. In Section 5 an attempt is made to apply the technique to some larger and more recent benchmarks. Finally, in Section 6 conclusions are given.
Section snippets
Graph theoretical model
The FS-FAP may aim to minimise interference, maximise service or minimise blocking. Here we concentrate on the most common variation, sometimes also referred to as the minimum interference frequency assignment problem. This variation can be represented through a weighted undirected graph. Formally, each instance is a 5-tuple FS-FAP={V,E,D,P,F} with:
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V: vertex set of an undirected graph G. Every vertex represents a transmitter of the original frequency assignment problem;
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E: set of edges of the
Lower bounding technique
The lower bounding technique we propose is described initially in terms of the linear programming relaxation (LR) of IP, as used by Montemanni et al. [15]. This is obtained by replacing constraints (5) byAs LR provides very poor lower bounds, some reinforcing inequalities for LR have to be defined, aiming to improve the quality of the estimates. They are presented in the following sections, together with a description of a simplification of formulation LR.
Computational results I: comparison with previous results
In this section we compare the lower bounds obtained by the technique described in this paper with those of the method presented in Montemanni et al. [15]. Section 5 will consider the possible application of the method to larger benchmarks. Three different sets of benchmarks are used in this section. The problems from the first two sets were used in Montemanni et al. [15], while those of the third set have been generated in Montemanni [14].
Problems of the first set are obtained from existing
Computational results II: feasibility of attempting larger benchmarks
As well as the improvements in solution quality and run time, the simplified formulations should allow larger problems to be handled. In this section an attempt is made to apply the method to two benchmarks from the COST 259 collection [21].
Firstly, it should be noted that the method presented here has no special provision for locally or globally blocked channels (i.e. channels within F which are unavailable to some or all transmitters). These tend to occur near national boundaries to avoid
Conclusions
In this paper the evolution of a lower bounding technique recently presented in the literature [15] has been proposed.
The improvements presented in this paper, after having been described in detail from a theoretical point of view, have confirmed their effectiveness in practice. The new lower bounding technique works better than the old one (either in terms of run times or in terms of quality) on most of the benchmarks considered.
The requirement for good lower bounding techniques able to
References (21)
- et al.
The partial constraint satisfaction problem: Facets and lifting theorems
Operations Research Letters
(1998) - et al.
Improving heuristics for the frequency assignment problem
European Journal of Operational Research
(1998) - S.M. Allen, D.H. Smith, S. Hurley, Confidential report, University of Glamorgan, May...
A simulation study of some dynamic channel assignment algorithms in a high capacity mobile telecommunications system
IEEE Transactions on Communications
(1973)- R. Borndörfer, A. Eisenblätter, M. Grötschel, A. Martin, The orientation model for frequency assignment problems,...
- et al.
Algorithm 457: Finding all cliques of an undirected graph [H]
Communications of the ACM
(1973) - et al.
A tabu search algorithm for frequency assignment
Annals of Operations Research
(1996) - A. Eisenblätter, Frequency assignment in GSM networks: Models, heuristics and lower bounds, Ph.D. Thesis, Technische...
- A. Eisenblätter, M. Grötschel, A.M.C.A. Koster, Frequency assignment and ramifications of coloring, Technical Report...
- C. Helmberg, Semidefinite programming for combinatorial optimization, Technical Report 00-34, Konrad-Zuse-Zentrum für...
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2022, EURO Journal on Computational OptimizationCitation Excerpt :In fact we can also formulate the problem using the standard representation of interference as given in Section 2.1 above. In order to investigate the computational performance of our formulation to minimise the number of frequency changes subject to a constraint upon interference we used the standard set of interference test problems that have been used by other authors in the literature (e.g. most recently by Ceschia et al. [3], but also (in whole or in part) by [8–10,12–15,20–22]). Table 3 shows the results obtained for the same test problems as in Table 2, but using the formulation based upon the standard representation of interference as given in Section 3.4 above.
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Path relinking for the fixed spectrum frequency assignment problem
2015, Expert Systems with ApplicationsCitation Excerpt :Population-based approaches are also very popular: ANTS algorithm (Maniezzo & Carbonaro, 2000), genetic algorithms (Dorne & Hao, 1996; Park et al., 2002; San José-Revuelta, 2007), hybrid evolutionary algorithms (Dorne & Hao, 1995; Jin, Wu, Horng, & Tsai, 2001; Kim et al., 2007; Lai & Coghill, 1996), multiple strategy method (Hurley et al., 1997). In addition to heuristic methods, approaches using integer programming were also proposed in the literature to determine lower bounds for the FS-FAP problem, such as those in Montemanni, Smith, and Allen (2001, 2004). Recently, the population-based path relinking approach (Glover, 1997; Glover, Laguna, & Martí, 2000) has attracted special attention in combinatorial optimization, and shows outstanding performances in solving a number of difficult problems, such as antibandwidth problem (Duarte, Martí, Resende, & Silva, 2011), bandwidth coloring problem (Lai, Lü, Hao, Glover, & Xu, 2014), clustering (Martins de Oliveira, Nogueira Lorena, Chaves, & Mauri, 2014), flow shop sequencing and scheduling (Costa, Goldbarg, & Goldbarg, 2012; Reeves & Yamada, 1998), unconstrained binary quadratic optimization (Wang, Lü, Glover, & Hao, 2012), and web services composition (Parejo, Segura, Fernandez, & Ruiz-Cortés, 2014).
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Present address: Istituto Dalle Molle di Studi sull' Intelligenza Artificiale (IDSIA), Galleria 2, CH-6928 Manno, Switzerland.