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The clique number ␻G is the maximum number of vertices in a clique of G, and the clique problem is, given G and integer k ⱖ 0, to decide whether G has a clique of size at least k.. Each

CHAPTER 12

Ensemble Planning for Digital

Audio Broadcasting

ALBERT GRÄF and THOMAS McKENNEY

Department of Music Informatics, Johannes Gutenberg University, Mainz, Germany

It is expected that in many countries digital broadcasting systems will mostly replace rent FM radio and television technology in the course of the next one or two decades Thedigital media not only offer superior image and audio quality and interesting new types ofmultimedia data services “on the air,” but also have the potential to employ the scarce re-source of broadcast frequencies much more efficiently Thus, broadcast companies andnetwork providers have a demand for new planning methods that help to fully exploitthese capabilities in the large-scale digital broadcasting networks of the future

cur-In this chapter, we consider in particular the design of DAB (digital audio ing) networks Although channel assignment methods for analog networks, which areusually based on graph coloring techniques (see, e.g., [3, 9, 11, 19, 21]), are also appli-cable to DAB planning, they are not by themselves sufficient for the effective planning

broadcast-of large DAB networks This is due to the fact that, in contrast to classical radio works, the DAB system transmits whole “ensembles” consisting of multiple radio pro-grams and other (data) services, and allows an ensemble to be transmitted on a singlechannel even if the corresponding transmitters may interfere Hence, one can span largeareas with so-called single frequency networks, which makes it possible to utilize elec-tromagnetic spectrum much more efficiently To make the best use of this feature, how-ever, it is necessary to integrate the planning of the ensembles with the frequencyassignment step This is not possible with existing methods, which are all simply adap-tions of known graph coloring techniques that are applied to a prescribed ensemble col-lection

net-We first show how to formulate this generalized planning problem, which we call theensemble planning problem, as a combined bin packing/graph coloring problem We thendiscuss some basic solution techniques and algorithms to compute lower bounds in order

to assess the quality of computed solutions Finally, we develop, in some detail, a more vanced tabu search technique for the problem Experimental results are used to point outthe strengths and weaknesses of current solution approaches

ad-267

Handbook of Wireless Networks and Mobile Computing, Edited by Ivan Stojmenovic´

Copyright © 2002 John Wiley & Sons, Inc ISBNs: 0-471-41902-8 (Paper); 0-471-22456-1 (Electronic)

12.1.1 Mathematical Preliminaries

We assume familiarity with the basic notions of graph theory (see, e.g., [8] or [14]) andNP-completeness [7] All graphs are simple, undirected, and loopless The subgraph of a

graph G = (V, E) induced by a subset of vertices W
V is denoted G W; it consists of the

vertex set W and all edges of G between the vertices in W A coloring of a graph G = (V,

E) is a function f mapping vertices to “colors” in such a manner that f (v) ⫽ f (w) ᭙ vw 

E If | f (V)| ⱕ k then f is also called a k-coloring The chromatic number ␹(G) of G is fined to be the minimum k for which such a k-coloring exists The graph coloring problem

de-is, given G = (V, E) and integer k ⱖ 0, to decide whether G has a k-coloring A clique of a graph G = (V, E) is a subset W of V s.t G Wis complete, i.e., vw  E ᭙ v, w  W : v ⫽ w.

The clique number ␻(G) is the maximum number of vertices in a clique of G, and the clique problem is, given G and integer k ⱖ 0, to decide whether G has a clique of size at least k The clique number is an obvious lower bound on the chromatic number of a graph,

since all vertices in a clique must be colored differently (this bound is not tight as there are

graphs G for which the gap between ␻(G) and ␹(G) becomes arbitrarily large [8]) It is

well known that both the coloring and the clique problem are NP-complete Moreover,both problems are also difficult to approximate: the problem of determining a coloring

using at most |V|1/7–␧␹(G) colors, and the problem of finding a clique of size at least

|V| ␧–1/4␻(G), are NP-hard for each ␧> 0 [6] This implies, in particular, that it is not ble to approximate the optimization versions of these problems within constant perfor-mance ratio in polynomial time, unless P = NP

possi-Another type of NP-complete problem we deal with in this chapter is the bin packing

problem Given a set X with positive element sizes ␮x , x  X, and M > 0, an M-packing of

X is a set B of mutually disjoint subsets of X s.t X = B B B and the total size ␮B =

⌺x B␮x of each “bin” B  B is at most M The bin packing problem is, given X, ␮, M and

a nonnegative integer k, to decide whether X has an M-packing B of size at most k We also

let p M (X) denote the minimum size of an M-packing of X, called the M-packing number of

X In contrast to the chromatic and clique numbers, the packing number can be

approxi-mated with good performance quite easily, using a simple kind of “greedy” procedureknown as the first-fit packing algorithm [7]

Two special aspects of the DAB technology motivate the problem discussed in this chapter:First, a DAB frequency block usually carries several different radio programs and addition-

al data services Therefore we have to deal with service ensembles instead of single radioprograms Each service needs a certain bandwidth, and the total bandwidth of all servicesforming an ensemble must not exceed the bandwidth available in a DAB frequency block.Second, an ensemble may be transmitted on a single frequency block channel, even bypotentially interfering transmitters; collections of transmitters that all broadcast the sameensemble on the same channel are called single frequency networks (SFNs) By making ajudicious choice of SFNs, one can often save a considerable amount of frequency re-sources compared to conventional broadcast networks in which interfering transmitters

must always be given different channels, even if they transmit the same signal However,

to fully exploit the potential frequency savings, new planning algorithms are needed.Therefore, in our approach to DAB network planning we not only consider the channelassignment aspect, but also the search for ensemble collections that help to reduce the over-all frequency demand In our model, the network is described by a graph whose vertices cor-respond to the (geographical) areas for which a certain service supply is to be provided Twovertices are connected by an edge if transmitters in the corresponding areas may interfere.(In practice, the areas are usually polygons on the earth’s surface, and edges connect areasthat are within a given geometric distance, the so-called block repetition distance.)The goal is to find ensembles that realize the desired service supply, and a correspond-ing assignment of frequency blocks that prevents interference and minimizes frequencyrequirements We call this the ensemble planning problem Thus an instance of the ensem-ble planning problem consists of the following items:

앫 A set S of services, where each service s  S has a positive bandwidth ␮s

앫 The area graph G = (V, E), where V is the set of areas and the edge set E is

interpret-ed as the interference relationship between areas

앫 The requirements R v,  V, which denote, for each area v  V, the set of services

to be supplied in that area

앫 The maximum ensemble size M > 0

We generally assume (w.l.o.g.) that all bandwidths and the ensemble size are positive

integers R and ␮are considered as functions R : V哫 2Sand ␮: S哫 ⺞ In the following

we usually only specify the parameters G, R, and M and assume a corresponding service set S with bandwidths ␮without further notice

A solution to the ensemble planning problem consists of two items, an ensemble signment and a corresponding block assignment An ensemble assignment is a relation B

as-
V × 2 S, which assigns to each v  V a set B v = {B : ( v, B)  B} of ensembles (service

sets) that are to be transmitted in the corresponding area For an ensemble assignment B to

be admissible, it must satisfy the supply requirements, and the individual ensembles mustnot exceed the maximum ensemble size:

R v
BBv B ᭙ v  V (12.1)

␮B ⱕ M ᭙ B  B v,  V (12.2)where the total bandwidth ␮B of an ensemble B
S is defined by ␮B= ⌺s B␮s

The second part of the solution is the block assignment f, which maps each ( v, B)  B

to a corresponding frequency block or “color” f ( v, B) To be admissible, the block

assign-ment must not introduce any interferences, i.e., different ensembles in the same or fering areas must always be assigned different frequency blocks:

inter-f (v, B) ⫽ f (w, C) ᭙ (v, B), (w, C)  B : B ⫽ C ⵩ (v = w ⵪ vw  E) (12.3)

12.2 THE ENSEMBLE PLANNING PROBLEM 269

Finally, the target function to be minimized is the number of distinct frequency blocks,

i.e., | f ( B)| = |{ f (v, B) : (v, B)  B}| We can now formulate the ensemble planning

prob-lem in the usual decision format as follows:

Problem 1 (Ensemble Planning Problem) Given G = (V, E), R : V哫 2S , M  ⺞ and K

 ⺞, decide whether there is an admissible ensemble assignment B and corresponding missible block assignment f s.t | f ( B)| ⱕ K

ad-In the following, for given G, R, and M, by ␹M R (G) we denote the minimum number of

frequency blocks in any admissible solution It is not difficult to see that the admissible

block assignments f for a given ensemble assignment B actually are in one-to-one

corre-spondence to the valid colorings of an associated ensemble graph G Bwhich has B as its set

of vertices and whose edges are precisely those pairs (v, B)(w, C) for which B ⫽ C ⵩ (v =

w ⵪ vw  E); cf Equation (12.3) Consequently we have that

␹M R (G) = min{␹(G B) : B admissible} (12.4)Hence the ensemble planning problem is nothing but a graph coloring problem on top of akind of packing problem We further explore this in the following sections

Example 1 Consider the problem instance in Figure 12.1, which involves five areas I–Vand eight services A–H We assume that only adjacent areas (such as I and V, or II and IV,

but not I and IV) interfere Thus, the area graph in this case is W4, the wheel with 4

spokes, where area V forms the hub of the wheel The requirements R vand service widths ␮s are given in the figure The maximum ensemble size is M = 9

band-Our task is (1) to arrange the services into a collection of ensembles for each area (this

is what we call the ensemble assignment), and (2) to assign channels to the resulting

en-Figure 12.1 Ensemble planning problem

sembles (this is the block assignment) We want to do this in such a manner that the tions (12.1), (12.2), and (12.3) are satisfied And, of course, we want the number of re-quired frequency blocks to be as small as possible (This is the hard part.)

condi-Figure 12.2 shows two different solutions The first solution (on the left-hand side) isfairly straightforward: we simply pack the services required for each area into a minimumnumber of ensembles We call such an assignment, which supplies each area exactly withthe requested services, a strict ensemble assignment It is easy to see that this solution re-quires five frequency blocks in order to avoid interference and, in fact, one can show that

in this example each strict ensemble assignment needs at least five frequencies

It is possible to do better than this, but only if we allow a certain degree of ply,” as shown in the second solution Here, we packed the entire service collection intothree different ensembles, which can then be transmitted using three frequency blocks Itcan be shown that this solution is indeed optimal Thus we see that in DAB networks it ispossible to save frequencies through oversupply

It is not difficult to see that the ensemble planning problem is NP-complete; in fact it tains both the graph coloring and the bin packing problem as special cases (For the graph

con-coloring problem, take S = V, R v= {v} ᭙ v  V and ␮⬅ 1 = M; for the bin packing lem, let G be a one-vertex graph.) So we know that Problem 1 is not only NP-complete,

prob-but also difficult to approximate and, hence, we will be interested in heuristic solutions.How can we approach this problem? We have already mentioned that the problem re-duces to an ordinary graph coloring problem once we have obtained a suitable ensembleassignment Thus a straightforward approach is to solve the problem in two optimizationstages:

12.3 BASIC SOLUTION TECHNIQUES 271

Figure 12.2 Two ensemble/block assignments

앫 Find an admissible ensemble assignment B.

앫 Color G Busing some heuristic graph coloring procedure

For the second stage, a plethora of different graph coloring algorithms is already able In fact, it turns out that fairly simple “sequential” coloring methods like Brélaz’DSATUR [2] or Matula/Beck’s smallest-last algorithm [16] usually perform very well onthe kind of “geometric” graphs arising as models of broadcast networks So we can con-centrate on the first step, which is a kind of simultaneous bin packing problem The diffi-cult part here is to devise a packing strategy that reduces the global number of requiredcolors instead of merely optimizing the packings for individual areas Of course, no singlestrategy will work with all area graphs equally well, so let us take a look at the two ex-treme cases:

avail-앫 Independent (edgeless) graphs In this case the problem decomposes into |V | pendent bin packing problems, one for each vertex of G, and we have that ␹M R (G) = max{p M (R v) : v  V}.

inde-앫 Complete graphs In this case, all distinct ensembles will have to be assigned ent frequency blocks, hence ␹M R (G) = p M (R V)

differ-(Here and in the following, by R Wwe denote the set of all requested services in a set of

areas W
V, i.e., R W= w W R w Thus, in particular, R V is the set of all requested vices.)

ser-This suggests two different types of ensemble packing heuristics that we would pect to work reasonably well with sparse and dense graphs, respectively For sparsegraphs, we will pack the individual requirement sets independently of each other Fordense graphs, we will pack the entire collection of requested services, and then assign toeach vertex those ensembles that are needed to satisfy the supply requirements at thatvertex For an arbitrary graph, we might try both approaches and see which one worksbest

ex-Since the bin packing problem is NP-complete, we need some heuristic for solving it

A simple method, which works reasonably well, is the so-called first-fit (FF) algorithm,which considers the items (a.k.a services) to be packed in some order, and puts each iteminto the first “bin” (a.k.a ensemble) into which it fits It is well-known that if the items areordered by decreasing sizes, this method never performs worse than 11/9 times the opti-mum (asymptotically) (We refer the reader to [7] for the details of this algorithm and itsanalysis.)

In the following, we use FF(R, M) to denote the set of ensembles returned by the fit algorithm when applied to a set of required services R
S and the ensemble size M.

first-We can employ the first-fit algorithm for packing ensembles using the two strategiessketched out above, as follows:

Simultaneous First-Fit (SFF) Algorithm:

B = FF(R , M) ᭙ v  V (12.5)

Global First-Fit (GFF) Algorithm:

B v = {B  FF(R V , M) : B  R v ⫽ 0/} ᭙ v  V (12.6)Incidentally, the two algorithms also correspond to the two solution approaches taken

in Example 1 The SFF algorithm computes strict solutions (ensemble assignments

exact-ly satisfying the requirements), aiming at “local optimality” by avoiding oversuppexact-ly Incontrast, the GFF algorithm strives for “global optimality,” producing overlap-free solu-tions in which the constructed ensembles are mutually disjoint (i.e., each service will only

be supplied in a single ensemble for all areas where it is requested) As already indicated,

we would expect SFF to work best with sparse, and GFF with dense graphs In fact, theperformance bounds for the first-fit bin packing algorithm directly carry over to SFF andGFF solutions That is, a (properly colored) SFF (resp GFF) assignment on an indepen-dent (resp complete) area graph using a service order by decreasing bandwidths will atmost be about 22% off the optimum (asymptotically)

As pointed out by Schmeisser [22], SFF (when ordering services by decreasing widths) produces ensemble assignments which can be colored using at most (11/9)␹(G)

band-colors (asymptotically), whereas GFF can perform arbitrarily bad even if the area graphsare acyclic (and hence bipartite) However, in experiments with geometrically definedarea graphs [10] we found that GFF usually performs much better than SFF when the areagraphs are dense, or when the services have a high average “circulation” (defined as thepercentage of areas requiring a given service)

When dealing with heuristics for NP-hard minimization problems, one is always

interest-ed in finding a good lower bound on the optimal solution, which allows us to assess thequality of computed solutions For the ensemble planning problem, a useful bound is pro-vided by an appropriate generalization of the clique number, which we discuss in this sec-tion

As in the preceding section, we let R W= w W R w ᭙ W
V Similarly, the set of all sembles in a given area set W
V is denoted B W, i.e., B W= w W B w The quantity we con-sider, which we call the clique packing number, is defined as follows:

en-␲M R (G) = max{p M (R W ) : W clique of G} (12.7)

To see why ␲M R (G) is in fact a lower bound on ␹M R (G), let W
V It is easy to see that,

if B is an admissible ensemble assignment w.r.t R and M, then |B W| ⱖ pM (R W) Now

con-sider the special case that W is a clique of G In this case, the subgraph of G Binduced by B

 (W × B W) always contains a clique of size |B W | (For each B  B W choose some w B  W s.t B  B wB Then {(w B , B) : B  B W} is a clique of the requested size.) Hence

p (R ) ⱕ |B | ⱕ␻(G B) ⱕ␹(G B) (12.8)

12.4 LOWER BOUNDS 273

By taking the maximum over all cliques W of G on the left-hand side of Equation

(12.8), and the minimum over all admissible ensemble assignments B on the right-hand

side, we obtain the desired inequality

␲M R (G) ⱕ␹M R (G) (12.9)The clique packing number is a weighted generalization of the clique number which,instead of merely counting the vertices contained in a clique, weights cliques according tothe packing number of their requirements Although computing the clique packing num-ber is NP-hard, in practice it is much easier to approximate than other lower bounds likethe minimum ␻(G B), for which we would have to consider all admissible ensemble assign-ments Here we can employ a generalized version of the Carraghan/Pardalos algorithm, abranch-and-bound method for computing the clique number of a graph [4], which we havefound to work quite well on not too dense graphs with a few hundred vertices The algo-rithm can also be terminated at any time to give a lower bound on the clique number It

can easily be adapted to any weight function which is “monotonous” in the sense that if W

W⬘, then the weight of W is at most the weight of W⬘.

One complication is that since the bin packing problem is NP-complete, we can onlyapproximate the clique weights That is, we actually compute a lower bound

␸M R (G) = max{ f (R W , M) : W clique of G} (12.10)

on the clique packing number, where f (X, M) is a lower bound on p M (X) that can be

com-puted efficiently A simple method is to employ the “sum bound”

re-numbers, there may be a “dual gap” between ␹M Rand ␲M Rthat can be arbitrarily large

As pointed out in the previous section, the SFF and GFF algorithms represent two basicsolution ideas for the ensemble planning problem (EPP), GFF (in contrast to SFF) takingadvantage of the possibilities to reduce the demand of frequencies needed by oversupply-ing some or all of the vertices v  V of the area graph G = (V, E) with services s  S not necessarily required (s  R ) We will see later that for quite a few problem instances SFF

or GFF (but usually not both) can find optimal or near optimal solutions, yet there aremany problem instances, for which both SFF and GFF yield rather poor results In conse-quence, our goal is to contruct a solution method, the results of which are at least as good

as the better of SFF and GFF in all cases, and which takes into account especially thoseproblem instances for which both SFF and GFF fail to deliver good results In this lattercase, we expect considerable improvements in result quality

We have decided to use tabu search (TS) techniques as the conceptual basis of our lution method due to good experiences made with TS in a wide variety of telecommunica-tion applications and related fields [1, 15, 13], including frequency assignment problems[12] A complete description of the algorithm is beyond the scope of this chapter; we referthe reader to [17] for further details Let us start the development of the solution method

so-by having a brief look at the basic conceptual ideas behind TS, including the slight fications needed for our purposes

modi-Note that in contrast to the previous section, here f denotes the target function (or cost function) of our application, i.e., f ( B) is the number of frequencies needed for the ensem-

ble assignment B

12.5.1 The Basic Framework

TS is a heuristic technique for solving discrete optimization problems and is regarded as amember of the family of so-called metaheuristics [18, 20] It can be understood as a form

of neighborhood search, expanded by a few components, the most crucial of which is theconcept of setting so-called tabus on some transitions, blocking any development in direc-tion of these transitions as long as the tabus set on them are valid The underlying idea is

to avoid a well-known problem arising with classic neighborhood search, which easilygets trapped in a local optimum

As an expanded variation of neighborhood search, the first step in adapting TS to a

en optimization problem is to define a neighborhood suitable for the problem; that is,

giv-en a problem instance characterized by a search space T and a cost function f to be mized, one must define a function N : T씮 2T , where for all t  T the subset N(t)
T is interpreted as the neighborhood of t in T.

opti-With a neighborhood defined, we can now run TS by first choosing a suitable initial

value t0 T and then at each iteration i  ⺞ a permissible value t i  N(t i–1) of greatestpossible gain as the next iteration value, continuing in this fashion until some conditionterminates the process To prevent TS from being trapped in a local optimum, a so-calledtabu list is introduced, consisting of either the most recent transitions or of fundamentaldata specifying the most recent transitions or specifying families of similar transitions foreach of the most recent transitions Any entry in the tabu list and, in consequence, anytransition described by an entry in the tabu list is said to be tabued, meaning that such atransition is not selectable in any TS iteration as long as the tabu set on it has not expired.Expiration is then naturally expressed by removing the according entry from the tabu list.For each entry in the tabu list, the time of expiration is denoted by a value called the tabu

tenure In classic TS framework, the tabu tenure is a constant c ⺞, equally valid for anyentry in the tabu list, expressing in number of iterations the duration of the entry’s tabu va-lidity, and measuring this from the moment (iteration) the entry was added to the tabu list

12.5 A TABU SEARCH METHOD 275

Our adaption of TS to the EPP brought up the necessity for more dynamics in the eration of tabu tenure values, for reasons we will show in detail later Thus, we calculate

gen-the tabu tenure using an adequate periodic function g : ⺞ 씮 ⺞, with g(i), i  ⺞, being the tabu tenure value to be used when TS has reached the i-th iteration.

As a further component of TS, the aspiration criterion is somewhat the dual of the cept of working with tabus, allowing, under certain circumstances, a transition of somekind of global improvement to be made even though it might be prohibited by a tabu Inthe most common interpretation of this idea, the aspiration criterion ignores a tabu on anytransition that yields an improvement over the best solution obtained so far in a TS search.The criteria for TS termination are usually kept simple TS terminates if for a givennumber of iterations there is no further improvement in the cost function This terminationcondition is sufficient in most cases, and we use it as is in the current development state ofour TS

con-12.5.2 Definition of the Neighborhood

As already pointed out in the description of the basic TS framework, we must first give adefinition of the neighborhood That is, for each instance of Problem 1 and each B ⺒,where ⺒ denotes the set of all possible ensemble assignments, we must define a set N(B)

⺒ representing the neighborhood of B in ⺒ To simplify matters, we do not define N(B) directly, but rather introduce a set M( B) of admissible elementary moves m on B, with m(B) denoting the resulting ensemble assignment when applying m to B, and then set N(B)

:= {m( B) | m  M(B)}.

In our approach, an elementary move m  M(B), for an ensemble assignment B  ⺒, is specified as a quadrupel m = ( v, u, z, s), with v  V, u  {0, , |B v |}, z  {0, , |B v| +

1}, u ⫽ z and s  S Furthermore, in the case that u ⫽ 0 we demand that s  B u , where B u

 B vis the ensemble of vertex v with index u (Here we assume for each v ⬘  V that the

ensembles of B v⬘carry indices from 1 to |B v⬘| Then, for i = 1, , | B v⬘|, B i  B v⬘is the semble of v ⬘ with index i.)

en-Under these assumptions, an elementary move m = ( v, u, z, s)  M(B) is interpreted as the transfer of service s, which is an element of the “source ensemble” B u  B v of m, to the

“target ensemble” B z  B v of m The following special cases have to be considered

sepa-rately:

앫 If u = 0 then s is understood to be a nonrequired service not yet an element of any

ensemble of v (not broadcasted), which is to be an element of ensemble B z  B v

af-ter the move m has been made.

앫 If z = 0 then s is a service of the ensemble B u  B v, and is not to be an element ofany ensemble of v (not broadcasted) after the move m has been made.

앫 If z = |B v | + 1 then the move m opens a new ensemble B znot yet existing for vertex v,

which consists exactly of element s after the move m has been made

Remark: Given an elementary move m = ( v, u, z, s)  M(B) with u = 0 or z = 0, for sons of convenience, we introduce a virtual ensemble B of v consisting of all the services

rea-not broadcasted on v Thus, if u = 0, we remove the service s from the virtual source

en-semble On the other hand, if z = 0, we add the service s to the virtual target enen-semble

12.5.3 The Problem of Move Evaluation

Now, having defined the neighborhood, we could just use the standard design of the tabuconcept and the aspiration criterion, and should have a working TS algorithm Unfortu-nately, it turns out that the details of a successful TS implementation for the EPP are muchmore complicated The following example illustrates the difficulties we face

Let B ⺒ be an ensemble assignment Consider the case that for each pair of vertices

v1, 2 V with v1v2 E and for each pair of ensembles B1 B v1 , B2 B v2, the followingtwo properties are satisfied:

1 |B1| > 2, |B2| > 2 (This ensures that no single elementary move may cause B1or B2

to become the empty set.)

2 B1⫽ B2, and at least two elementary moves are needed to make B1and B2equal

We then obviously have f ( B) = f (m(B)) for all m  M(B) In effect, if B occurs as an mediate result of a TS search, the cost function f gives no hint how TS should proceed, for

inter-all possible transitions would appear equinter-ally valuable

We have found that the situation described above, even though it appears to be a ratherextreme case, actually arises very frequently Experiments have shown that given a prob-lem instance of EPP, for many ensemble assignments B  ⺒ we have f (B) ⱕ f (m(B)) for all elementary moves m  M(B) and thus no achievable improvement in the cost function.

Furthermore, for many ensemble assignments B  ⺒ we even have f (B) = f (m(B)) for most elementary moves m  M(B) and thus no change in the cost function at all.

This problem-specific phenomenon makes the cost function as is absolutely unsuitablefor any kind of guidance of TS A way to overcome this dilemma is to develop an alterna-

tive system for evaluating the elementary moves m  M(B) Such a system will rely on

characteristic features of the problem at hand, but only those aspects will be consideredthat appear to be relevant for guiding TS into promising directions

One of these characteristic features of EPP is that for any v, w  V, v ⫽ w, and any B

 B v , C  B w , not only may we use the same frequency f ( v, B) = f (w, C) in the case that

vw  E (which, in an analogous form, applies to virtually all kinds of frequency

assign-ment problems), but also in the case that vw  E but B = C Let us call this latter case the

SFN property (SFN = single frequency network) Now, knowing from GFF, which takesexhaustive advantage of the SFN property by oversupplying vertices v  V with services s

 S not explicitly required, that use of this property may lead to significant reduction in

frequency demand, this gives us an idea how to proceed

12.5.4 Heuristic Move Evaluation

Before going into the details, let us give an overview of where we are heading We will troduce an equivalence relation “⬅” and an order relation “䉰” for the elementary moves

in-12.5 A TABU SEARCH METHOD 277