Assignment of Frequencies

The frequency assignment problem (FAP), or channel assignment problem, plays an important role in all wireless networks that use the frequency division multiple access (FDMA) technology, the most prominent example being the second genera-tion of cellular networks based on the GSM technology. Other applicagenera-tions include satellite communication, TV broadcasting, military wireless networks, WLANs, and Orthogonal Frequency Division Multiplexing (OFDM) in future wireless networks.

Due to this wide variety of applications, there does not exist a single frequency assignment problem, but many variations. The problem was first mentioned in the 1960s [75] when individual frequencies in the radio spectrum were licensed (and charged) by the authorities, and operators of the first cellular phone networks could financially benefit from intelligent frequency planning. Later, frequencies were li-censed in blocks and the objective of the operators changed to minimize the

dif-ference between the highest and lowest frequency to be used. In the 1990s most frequencies were licensed and the popularity of mobile communication forced op-erators to increase the number of antennae, significantly causing high interfer-ence levels whenever the frequency planning was not done carefully. Accordingly the objective changed to the minimization of interference or the maximization of interference-free operated antennae.

All these different objectives have been addressed in a vast number of scientific publications. For an overview on the state of the art of frequency assignment we refer to Aardal et al. [3] and the FAP website [44]. In this section, we will restrict ourselves to the basic modeling of the frequency assignment problem and its relation with the vertex coloring problem in undirected graphs.

Frequency Assignment and Vertex Coloring

All variations of frequency assignment have two features in common.

1. A set of wireless transmitters must be assigned frequencies. For each transmitter there is a set of available frequencies given.

2. The frequencies assigned to two transmitters may incur interference of one an-other, resulting in quality loss of the signal. Two conditions must be fulfilled in order to have interference of two signals:

• The two frequencies must be close on the electromagnetic band. Harmonics may also interfere due to the Doppler effect, but the parts of the electromag-netic band that are generally selected prevent this type of interference.

• Connections must be geographically close to each other, so that interfering signals are powerful enough to disturb the quality of a signal.

Usually, data on the level of interference is provided for every quadruple of two transmitters and two frequencies.

This rather general description is complemented by an objective to be optimized.

The problem can be modeled as a graph-theoretical problem by defining an undi-rected interference graph G= (V,E)describing the interference relations. To sim-plify the presentation we assume in this section that no transmitters are colocated, as is usually the case in cellular phone networks (the description can be easily adapted to such situations; see, for example, [41]). The set V now represents all transmitters.

The planner can choose a threshold value representing an acceptable level of inter-ference. An edge{i,j} ∈E exists if and only if there exists a frequency pair(f,g) such that the interference level for the quadruple(i,f,j,g)exceeds the threshold.

In the most simplified case we only consider so-called co-channel interference, i.e., an edge exists if and only if the interference level exceeds the threshold for as-signing the same frequency to both transmitters. We further assume that the set of frequencies is equal for all transmitters and the objective is to minimize the num-ber of frequencies under the condition that all interference levels remain below the threshold. In this case, FAP reduces to the well-known vertex coloring problem. In

the vertex coloring problem, we have to color the vertices of a graph G= (V,E)such that no two adjacent vertices get the same color. The chromatic numberχ(G)is the minimum number of colors needed in a feasible coloring of G. It is not difficult to see that in the above-described case the minimum number of frequencies needed equalsχ^(G).

From an algorithmic point of view, this problem can be either treated as a deci-sion problem or an optimization problem. If the number of frequencies is fixed, say K, the planner has to answer the question: “Does there exist a solution without unac-ceptable interference using at most K frequencies?” which is equivalent to solving the K-COLORABILITYdecision problem. This problem is not onlyNP-hard [66] but also hard to approximate [14]. In the case where the number of frequencies is not yet fixed but must be determined, the optimization version of K-COLORABILITY

has to be solved.

unlimited spectrum

vertex coloring

T -coloring list coloring

list T -coloring

Minimum Span Frequency Assignment

(MS-FAP)

fixed spectrum

k-colorability

min-k-partition max-k-colorable induced subgraph

Minimum Interference Frequency Assignment

(MI-FAP)

Minimum Blocking Frequency Assignment

(MB-FAP)

Fig. 1.13 Classification of vertex coloring and frequency assignment problems

This differentiation between choice of frequencies (unlimited spectrum) and fixed frequencies (predetermined spectrum) also holds for more realistic frequency assignment problems; see Figure 1.13. If we include the sets of available frequen-cies to the vertex coloring problem, a list coloring problem has to be solved, whereas the inclusion of interference levels between nonidentical frequencies results in a T -coloring problem. Both these variants on the vertex -coloring problem have origi-nally been studied in the context of frequency assignment, providing perfect show-cases of the interaction between discrete mathematics and applications.

The combination of T - and list coloring is known as List-T -coloring. If not the number of frequencies but the difference between highest and lowest frequency has to be minimized, the problem is known as the minimum span FAP.

In the case of a fixed spectrum, K-COLORABILITYis generalized in two ways in case the problem is infeasible. If the planner needs to assign a frequency to each transmitter, some unacceptable interference has to be permitted. Otherwise, the planner might consider leaving some transmitters unassigned to avoid unaccept-able levels of interference. If only co-channel interference is considered, the first case is known as the minimum K-partition problem: We associate a value c^co_e ∈Q+

with all edges e∈E. The minimum K-partition problem now consists of a partition of the vertices in K subsets S1, . . . ,SK such that the sum over all subsets of the to-tal weight c^co_e of the edges in G[Si]is minimized. Stated otherwise, all vertices in a subset are assigned the same frequency, and so the interference incurred by this assignment is given by the sum of the weights c^co_{i j} for all i,j∈Si. A solution with total weight 0 exists if and only if the graph is K-colorable. See [42] for a solution approach using semidefinite programming.

The second case is known as k-colorable induced subgraph. In this problem we have to color as many vertices as possible with K colors. A solution where all ver-tices are colored implies that the graph is K-colorable, whereas non-K-colorable graphs have at least one vertex uncolored in any K-colorable induced subgraph so-lution.

The inclusion of potential interference between nonidentical frequencies for transmitter pairs results in the study of the so-called minimum blocking FAP. In-clusion of the same information in the minimum K-partition problem results in the minimum interference FAP.

Frequency Assignment in GSM Networks

We conclude the discussion of FAP with the formulation as integer linear program of the minimum interference problem with co- and adjacent-channel interference, as it occurs frequently in GSM networks.²Usually an operator of a GSM cellular phone network has acquired the right to use a certain spectrum of frequencies[fmin,fmax]in a particular geographical region, e.g., a country. The frequency band is—depending on the technology utilized—partitioned into a set of channels, all with the same bandwidthΔ. The available channels are here denoted by F={1,2, . . . ,N}, where N= (fmax−fmin)/Δ. A transmitter pair is exposed to adjacent-channel interference if the assigned frequencies are consecutive numbers in the spectrum.

In GSM networks, communication between mobile and base station (up-link) as well as between base station and mobile (down-link) must be established. To simplify the management of these networks, two separate, but paired, spectrums of frequencies are licensed, one for up-link and one for down-link. For FAP one can re-strict assigning down-link frequencies to the base stations, as the paired frequencies can then be used for up-link communication.

We associate with the edges of the interference graph G= (V,E) two values c^co_{i j},c^ad_{i j} ∈Q+denoting the level of co- and adjacent-channel interference between

2Some details are ignored; see, for example, [43] for a more detailed discussion

the vertices. For ease of notation we introduce two subsets:

E^co ={i j∈E|c^co₍ i j)>0}, and E^ad ={i j∈E|c^ad₍ i j)>0}.

Given a subset Fi⊆F for all i∈V , the formulation uses binary variables xi f, indi-cating whether frequency f ∈Fiis assigned to vertex i∈V . We further introduce variables z^co_{i j} and z^ad_{i j} to denote violation of the co-channel and adjacent-channel constraints, respectively. Then, the integer linear program reads

min

∑

i j∈E^co

c^co_{i j}z^co_{i j} +

∑

i j∈E^ad

c^ad_{i j}z^ad_{i j} (1.27a)

s.t.

∑

f∈Fi

xi f=1 ∀i∈V (1.27b)

xi f+xj f−z^coi j ≤1 ∀i j∈E^co,f ∈Fi∩Fj (1.27c) xi f+xjg−z^adi j ≤1 ∀i j∈E^ad,f∈Fi,g∈Fj:|f−g|=1

(1.27d) xi f,z^co_{i j},z^ad_{i j} ∈ {0,1} (1.27e) If i j∈E^co and frequency f can be assigned to both vertices, constraint (1.27c) assures that such a mutual assignment incurs a penalty of c^co_{i j}. Similarly, con-straint (1.27d) guarantees that the adjacent-channel interference is comprised of the objective if i j∈E^adand the frequencies f and g differ by 1.

A major drawback of the above formulation is the difficulty to solve these coloring-like integer linear programs to optimality. A wide range of different ap-proaches have therefore been proposed to compute close-to-optimal solutions and/or lower bounds on the minimum total interference; see [3] for a survey. In Chapter 11 a similar frequency assignment model is studied in more detail for the case of wire-less local area networks (WLANs). Due to the limited number of frequencies the drawbacks of the above formulation have less impact in this case.

Figure 1.14 shows an example of a carrier network, where the so-called DSATUR heuristic [24] followed by a very simple local search heuristic (1-opt) is ap-plied. This procedure improved the solution provided by a network operator by 96% [23, 43]. In the plots, transmitters are represented by dots in the Euclidean plane according to their geographical coordinates. Two transmitters are connected by a colored line if the assignment of frequencies results in interference. If the line is drawn in a pale color, then the interference is small. With increasing interference, the color of the line turns black.