Intelligent Search for the Radio Links Frequency Assignment Problem

Intelligent Search for the Radio Links Frequency Assignment Problem Dr. A. Bouju, Dr. J. F. Boyce, C. H. D. Dimitropoulos, G. vom Scheidt, Prof. J. G. Taylor Wheatstone Laboratory, King's College London, Strand, London WC2R 2LS, U.K. Dr. A. Likas, G. Papageorgiou, Prof. A. Stafylopatis Department of Electrical & Computer Engineering National Technical University of Athens, 157 73 Zographou, Athens, Greece Abstract. In this paper we present results on solving a dicult constraint satisfaction problem, namely the Radio Links Frequency Assignment Problem (RLFAP). Three methods have been compared through application to a set of realistic test problems, the CELAR problems. The methods considered are Tabu Search, GENET and Boltzmann Machine variations.

1 Introduction The radio links frequency assignment problem occurs in many civil and military applications [1, 2]. The main objective is to assign radio frequencies to a number of transmitters, subject to a number of constraints, so that no interference occurs. However, it may be impossible to satisfy all constraints, in which case trying to minimise the number of violated constraints (i.e. the interference) is a more realistic goal. The problem is NP-complete, and is a variant of the general T ?graph colouring problem, as introduced by Hale [3]: we want to colour the vertices of a graph in such a way that the two colours assigned to adjacent vertices i and j dier by at least Cij , where Cij , is a xed coecient associated to the edge [i; j ]. In this paper we assess the performance of three techniques for the Radio Links Frequency Assignment Problem (RLFAP), by application to a set of realistic test problems, the CELAR problems 1 . The methods investigated are: the Tabu Search heuristic, a GENeral NETwork algorithm (GENET) for constraint satisfaction problems, and variations of the Boltzmann Machine based on double-update operation. Tabu Search seems to be a promising method for these types of problems, as shown in the literature [4, 5]. Part of the work presented in this paper is a result of the Combinatorial Algorithms for Military Applications (CALMA) project, where funding for King's College London was provided by the DRA. 2 GvS would like to thank the German People's Scholarship Foundation for their nancial support.

2 The CELAR data set of problems In the CELAR problems, a set of possible frequencies is given to operate the radio links. Each of the radio links has to be assigned one of the frequencies, while satisfying a usually large number of constraints of the following type: for any two "neighbouring" links i and j , if fi (resp: fj ) is the frequency assigned to link i (resp: to link j ) then the absolute dierence jfi ? fj j should not belong to some speci ed set S of forbidden values (of course, other more complicated constraints often have to be taken into account). The concept of "neighbouring" links, here, not only depends on the fact that the links lie geographically close to each other, but also on various electromagnetic characteristics (such as propagation, transmission power, etc : : : ) which may result in electromagnetic incompatibility (at certain frequencies) between the links. As the number of pairs of neighboring links may be quite large (10; 000 pairs for a 1000 link network would be typical), the number of constraints which have to be taken into account to obtain a feasible frequency 1 2

The data was provided by the French "Centre d'Electronique de l'Armement"(CELAR). The CALMA project belongs to the EUCLID program (European Cooperation for the Long Term in Defence) [6].

1

assignment (assuming that some exists) is usually very large. This class of problems therefore appears to be extremely dicult to solve, not only from the theoretical point of view. Indeed, the practical complexity of such problems is essentially related to the total potential number of candidate assignments (among which one feasible or one close to feasibility has to be selected): even for a moderate size example of, say, 50 frequencies and 500 links (Celar data set), this number would be as large as 50500. In addition to this, in practical situations, the set of constraints de ning the problem is subject to frequent changes over time. Therefore, it is not just a matter of solving one extremely complex problem every now and then. Instead, it appears that such problems have to be solved repetitively, at time intervals which may be quite short. From the CELAR data set that contains eleven problems (scen01-scen11) we will present results on six of the problems (scen01-scen05,scen11), which all have feasible solutions (all constraints can be satis ed). For problems scen06-scen10 there exists no feasible solution, so these are constraint optimisation problems and are not considered in this present paper. We concentrate on the constraint satisfaction problems of the CELAR data set. In these problems we have the following kind of constraints:  jfi ? fj j > ci;j `Inequality' constraints.  jfi ? fj j = ci;j `Equality' constraints. Each variable(link) can be assigned a frequency from its respective allowed domain of frequencies, i.e. fi 2 Di , where (i = 1; : : :; 7) as there are seven available frequency domains for these problems. We have the following data set : scen01: 916 variables (links), 5548 constraints, no variable pre-assigned. scen02: 200 variables, 1235 constraints, no variable pre-assigned. scen03: 400 variables, 2760 constraints, no variable pre-assigned. scen04: 680 variables, 3967 constraints, with 280 variables pre-assigned. scen05: 400 variables, 2598 constraints, no variable pre-assigned. scen11: 680 variables, 4103 constraints, no variable pre-assigned. For the above problems we have to nd solutions that satisfy all the constraints; there exist more than one feasible solutions. In addition: we have to minimise the number of dierent frequencies used in the assignment (for scen01-scen04) and we have to minimise the largest frequency value used (for scen05).

3 Tabu Search

3.1 The Algorithm 3.1.1 Rationale

Tabu Search is a modern heuristic method which was introduced by Glover [7, 8] as an ecient way of nding high quality solutions to hard combinatorial optimisation problems, such as the Radio Links Frequency Assignment Problem (RLFAP). The underlying premise is that intelligent search should be based on more systematic forms of guidance than purely random. It is hypothesised that any class of problems exhibits general characteristics which may be discovered and subsequently utilised in the solution of any speci c case of a problem within the class. The meta problem is then to determine how such characteristics may be codi ed and incorporated within a search algorithm. One objective of past research in the eld has been to derive and exploit a collection of principles of intelligent problem solving. It is assumed that any approximate solution to a search problem has an associated neighbourhood of possible approximate solutions which can be regarded as adjacent to the present solution and which are accessible from it by given transformations or \moves". The search proceeds by following a sequence of moves. A memory is maintained of the recent moves and the properties of the recent neighbourhood. The name \Tabu" derives from the implementation of search guidance by the imposition of restrictions, either by direct exclusion of some moves or by modi ed probabilities for possible moves. The rules which govern the restrictions codify the a priori knowledge of the problem class. They depend both on the current structure of the neighbourhood of the solution and its history. The rules themselves may vary dynamically

during the course of a search both in response to recent search behaviour and to the number of search moves since initiation. The rules are based on the interaction between restriction and aspiration, where the former seeks to limit the moves available at any search state whilst the latter permits the overriding of restrictions if the search appears to be trapped in a local minimum.

3.1.2 The RLFAP Implementation The RLFAP implementation of Tabu Search outlined below 3 requires three parameters to run: the size of the tabu list (jT j), the patience parameter (p), and the ratio parameter (N ). 1. Begin with a random assignment of frequencies to links. 2. Consider all links within N % of the maximum number of violations, (a) Check all frequencies for each link, (b) Check for TABU and ASPIRATION. 3. Select the best move for the iteration & GOTO (2). 4. Store the best move so far. 5. Update tabu list 4 and aspiration criteria. 5 6. Repeat from step (2), until a solution is found or until there is no improvement in the cost for more than p iterations.

3.2 Results with Tabu Search We have obtained feasible solutions for all the problems of the CELAR data that have feasible solutions. Detailed results on feasibility with Tabu Search have been reported in [9]. Satisfying, all the constraints (feasibility) is not the only objective (apart from scen11); we need to minimise the number of dierent assigned frequencies as well. We use the cost function: Cost = (V iolations) + m  (Frequencies), where m is the weighting term. This gives poor results even with a large m value, unless the search begins from an advantageous initial assignment that uses a minimal number of distinct frequencies (Starting Point Selection Strategy for Frequency Minimisation in [9]). Tabu Search results are tabulated with the GENET results in x4:4.

4 GENET GENET is a connectionist approach to constraint satisfaction problems developed by Dr E.P.K. Tsang et al. [10]. An implementation of the algorithm was used to solve the Radio Link Frequency Assignment Problem presented in the CALMA data set [6]. It was extended to also handle constraint optimisation problems and modi cations of the escape heuristic and optimisations of the search mechanism that signi cantly improve GENET's performance were studied. For a detailed description of this implementation see [9]. The new move is added to the recency based tabu list, which means that for the next jT j iterations this move will be tabu. Recency based tabu was all that was required to nd feasible assignments in most cases. However, on some occasions the same link would be repeatedly selected for assignment on each iteration and tabu search would cycle with dierent frequency assignments for the same link. To prevent this cycling and to diversify the search a frequency based tabu is imposed on that link for a xed number of iterations. 5 A simple type of aspiration criteria is employed: a candidate move is licit (non-tabu) if the assignment yields a solution with a cost better than the best found so far. Hence, the updated aspiration value is A := min(A; cost ? 1). 3 4

4.1 Problem representation

GENET can be viewed as a network of nodes with weighted inhibitory connections, that are modi ed during the optimisation process. Each variable is represented as a cluster of such nodes, one node for every value in the variable's domain. One node per cluster may be active at any time, representing the currently assigned value. If a node is active, it is de ned to have an output of 1, if it is inactive, an output of 0. If for example a variable had the domain (10; 20; 30; 40; 50) and its current assignment was to a frequency of 40, the ve corresponding nodes would have values (0; 0; 0; 1; 0). Constraints are represented by inhibitory connections linking incompatible nodes. All connections receive an initial weight of '-1'. The weights can be stored in a m  n matrix for each constrained variable-pair, m and n are the respective domain sizes.

4.2 Minimisation technique

After activating a randomly chosen node in each cluster, the input to all nodes is computed by summing over the weighted outputs of all connected nodes:

inputx =

X y

weightxy  outputx;

(1)

In each cluster, one of the nodes with the least negative input is activated and all others are inactivated. The newly active nodes represent the new assignment. This process is repeated until the network settles to a stable state and no further updates occur. If the network has converged to a global minimum (no constraints are violated) or its resources are exhausted (e.g. a time limit exceeded), GENET stops. Otherwise the following rule is applied to try to escape from the local minimum:

weightxy = weightxy ? outputx  outputy (2) If a constraint was violated, i.e. if both nodes of a connection were active and had output 1, the inhibition between these nodes is strengthened further by adding '-1' to its weight. The algorithm then loops back and recomputes the inputs. 0

4.3 Interpretation of the algorithm

The escape technique employed by GENET systematically deforms the cost surface by adding a localised cost at the nodes that caused the violation of a constraint. Taking these inhibitions into account enables the algorithm to escape from local minima by increasingly penalising violations. The weight factor at each link can be separated into a constant value of '-1' and monotonically growing additional inhibitions. Hence the sum over all inputs of a node is the negative equivalent of the number of constraints that would be violated by activating it, plus additional inhibitions. By choosing the node with the least negative input, GENET essentially implements a modi ed min-con icts hill-climbing technique (a "greedy" algorithm).

4.4 GENET Results

The following results for the Celar data set were obtained with a C++ implementation of a modi ed version of GENET and a C implementation of Tabu Search. For each scenario at least 20 runs were completed on a 130MHz DEC Alpha. Current optimisations for GENET include full cross-indexing of variables, constraints and domains and minimal-neighbourhood propagation of changes (Tabu was not optimised). The following table shows results for cost functions with additional terms. If all constraints can be satis ed (scenarios 0105,11) the number of distinct frequencies used (out of 48 available) (scenarios 01-04) or the highest assigned frequency (scenario 05) has to be minimised. Scenario 11 only requires nding a feasible assignment (no violations). Scenario names followed by an "arc" indicate preprocessing using arc-consistency 6 . 6 For scenarios 04 and 05 arc-consistency was used as a pre-processing technique. Arc-consistency reduces the available values in the domain for each variable. Before imposing the arc-consistency the average number of values available for a variable was

CELAR

best known best found found optimum average time cost TABU GENET TABU GENET TABU GENET scen01 16 18 16 n.a. 5% 3hrs 32s scen02 14 14 14 70% 50% 4min 13s scen03 14 14 14 20% 10% 34min 28s scen04 46 n.a. 46 n.a. 100% n.a. 12s scen04arc 46 46 46 100% 100% 10s 0.24s scen05 792 n.a. 792 n.a. 30% n.a. 460s scen05arc 792 792 792 40% 100% 7min 2s scen11 0 0 0 60% 60% 54min 25s

5 Double-update Boltzmann Machine

5.1 Network architecture

The representation of a constraint satisfaction problem in terms of a binary Hop eld-type network can be obtained by arranging the network units in a matrix structure with a matrix row corresponding to each of the variables. The units in each row correspond to the values in the variable's domain, so that, if a unit (k; l) is active, the value corresponding to the lth unit of the kth row is assigned to variable k (at most one unit should be active in each row). A binary constraint refers to a pair of incompatible units (variable assignments). The connection weights wkl;mn and the threshold values kl are chosen so that the network states corresponding to solutions of a given problem are equilibrium states of minimum energy:

8 ? < wkl;mn = : ? 0

kl =

(3)

if units (k; l) and (m; n) are incompatible if k = m and l 6= n (4) otherwise where ,  and  are positive constants and ;  > . The above speci cation of weight and threshold values ensures that a state corresponding to a problem solution is an equilibrium point of minimum energy ?N , where N is the number of variables. If the pure Hop eld network is used, the equilibria obtained are mostly local minima in which there exist rows with no active unit. The Boltzmann Machine Optimizer [11], which applies an annealing schedule to the operation of the Hop eld network, can be used for evading local minima and continuing the search for a global one. Its eciency, however, is reduced in the case of large problem instances.

5.2 Double-update operation

To solve the CELAR problems we have considered a modi cation to the operation of the Boltzmann Machine Optimizer based on the group update technique [12], which leads to a much more eective search of the problem state space. We start from a consistent network state (having the property that there is exactly one active unit in each row) and perform state transitions so that the network always passes through consistent states. To achieve this, we consider that at every time step a double update is performed. (The notion of group update constitutes a generalization of the notion of single update [12]. In analogy with the single unit case, rst a trial is performed by calculating the dierence in the energy that would result if the state of all units in the group were altered. Then a decision must be made as to whether a group update must take place, i.e., whether the states of all these units must change or not. Convergence and stability properties of Hop eld-type networks carry over to the case of group-update operation as well.) Consider that the network is in a consistent state and let (i; i? ) denote the index of the active unit in each row i (i = 1; : : :; N ). We randomly select a row k and a unit (k; l) in that row that is dierent from 39.5 for scen04 and 39.4 for scen05 and after this pre-processing the average number of values available for a variable was 3.2 for scen04 (making it an easy problem) and 20.2 for scen05. The value of arc-consistency has been demonstrated in [9].

(k; k? ). The selected unit along with unit (k; k?) constitute a group that will be tested for update. If this trial is successful, the network will move to a consistent state and we will set k? = l. Computation of the energy dierence caused by the double update yields:
E =

X

m6=k

wmm? ;kk? ?

X

m6=k

wmm? ;kl

(5)

It can be shown that the states of the network that correspond to valid problem solutions are global minimum equilibrium points under double-update operation. In addition to favouring an ecient search, the doubleupdate scheme has an immediate impact on the computational complexity of the approach. Indeed, the cost per update under single-update operation is of the order of the number of units in the network, whereas double-update operation reduces the cost to the order of N . The above formulation of the double-update approach is sucient for providing feasible solutions to constraint satisfaction problems. Some extensions are necessary to also satisfy extra criteria like the ones considered here. In particular, an additional row is included with units that correspond to the elements of the union of the variable domains. Depending upon whether we must minimise the number of dierent frequencies used in the assignment or the largest frequency used, we can choose appropriate (symmetric) connection weights and threshold values for these additional units, so that minimisation of the network energy leads to optimisation of the respective criterion. The units of the extra row follow the single-update mode of operation and are randomly selected for update. This extension preserves the stability properties of the network. Other types of extensions can also be considered, that improve search performance or allow inclusion of other types of requirements.

5.3 Results The results displayed below were obtained on a Silicon Graphics Power Challenge with R4000 processors using a C implementation of the double-update Boltzmann Machine. For each problem, 15 runs were carried out and the appropriate extension was applied following the respective optimisation criteria. Double-update Boltzmann Machine CELAR best known best found average time scen01 16 20 4hrs scen02 14 14 5min scen03 14 16 57min scen04 46 46 18min scen05 792 792 84min scen11 0 0 94min

6 Conclusions All three methods are eective and easy ways for solving the Radio Links Frequency Assignment problems that have feasible solutions. GENET appears to be much faster than the other two methods. However, since only the GENET implementation was optimised it is dicult to directly compare the processing times. No pre-processing technique has been applied to scenarios 04 and 05 when using the Boltzmann Machine approach. Tabu Search and GENET were also tested for stability under perturbations of the constraints and exhibited good behaviour 7 . 7 The requirement for perturbations is that we can handle situations in which less than 5% of the constraints are aected. The Cij values for these constraints can change by up to 30% of their values. For all problems with feasible solutions the perturbations described above can be handled very easily. For most trials, either the original solution could still satisfy the perturbed constraints, or a new solution could be found in a few seconds. For example, when 5% of the constraints of scen11 are randomly modi ed so that their Cij values increase by 30% of their initial values, a new solution can be found in under seven seconds.

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