The Skiving Stock Problem and its Application to Resource...

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Technische Universitat DresdenHerausgeber: Der Rektor

The Skiving Stock Problem and its Application toResource Allocation

John Martinovic Eduard JorswieckGuntram Scheithauer

MATH-NM-01-2016

June 2016

Contents

1 Introduction 1

2 Preliminaries on the SSP 3

3 The Spectrum Allocation Problem – Introduction and a First Approach 4

4 An Improved Approach 6

5 A generalized Arcflow Model 8

6 An Assignment Model 10

7 Simulation Results 11

8 Application-oriented Extensions 13

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

8.2 A Solution Strategy based on Connected Patterns . . . . . . . . . . . . . . 14

8.3 The role of interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8.4 Secondary Users with Heteregeneous Bandwidth Requirements . . . . . . . 19

9 Conclusions and Further Research 20

The Skiving Stock Problem and its Application toResource Allocation

John Martinovic Eduard Jorswieck Guntram Scheithauer

Technische Universitat Dresden

August 18, 2016

Abstract

We consider the one-dimensional skiving stock problem (SSP) which is strongly re-lated to the dual bin-packing problem in literature. In the classical formulation,different (small) item lengths l1, . . . , lm and corresponding availabilities b1, . . . , bmare given. We aim at maximizing the number of objects with a certain minimumlength L that can be constructed by connecting the items on hand. Such computa-tions are of high interest in many real world applications, e.g. in industrial recyclingprocesses, wireless communications and politico-economic challenges. After a shortintroduction to the SSP, one of these applications, the spectrum allocation in cogni-tive radio networks, is considered in more detail. As a main contribution, we presentthree modeling formulations for this problem that can cope with a particular addi-tional constraint arising by hardware limitations, and provide numerical simulationsfor one of them. In a final step, we discuss possible extensions of the consideredoptimization problem and show how they can be solved efficiently.

Key words: cutting and packing, skiving stock problem, integer programming, spectrumaggregation, cognitive radio

1 Introduction

We consider the one-dimensional skiving stock problem (SSP) which is strongly related tothe dual bin-packing problem (DBBP): find the maximum number of items with minimumlength L that can be constructed by connecting a given supply of m ∈ N smaller itemlengths l1, . . . , lm with availabilities b1, . . . , bm. Such objectives are of high interest in manyreal world applications, e.g. in industrial production ([Zak03]) or politico-economic topics([AJKL84, LLM95]). Furthermore, also neighboring tasks, such as dual vector packingproblems ([CFGR91]) or the maximum cardinality bin-packing problem ([PD06]), are of-ten associated or even identified with the DBBP. These formulations are of practical use aswell since they are applied in multiprocessor scheduling problems ([ARGA04]) or surgicalcase plannings ([VPS+13]).

1

2 The Skiving Stock Problem and its Application to Resource Allocation

For bi = 1 (i ∈ I := {1, . . . ,m}), the considered optimization problem was firstly men-tioned in [AJKL84]. Based on practical preliminary thoughts ([JRZ97]), a generalizationfor larger availabilities bi (i ∈ I) has been considered in [Zak03], also motivating theterm skiving stock problem. Besides the standard model presented in [Zak03], position-and order-indexed formulations have recently been developed and proved beneficial, see[MS16].

In general, (discrete) optimization problems possess a large area of applications in wirelesscommunications, see [IMF13] or [IFM+15] for some current fields of research.1 One of themis given by spectrum aggregation based resource allocation in cognitive radio networks, see[TZFS13] for a good overview. Due to the significantly increased importance of wirelessconnectivity in the last years, the natural radio spectrum has become a very important andscarce resource. Normally, it is regulated by governmental entities and fixed parts of it areassigned to licensed holders for a long time, see [Hay13]. However, for large parts of the(licensed) spectrum, the utilization is, in most cases, very low ([LBCU+09, LLLL14]) lead-ing to many wasted vacant frequency intervals ([Fed02]). Cognitive Radio (CR), proposedby [Mit00], exploits these spectrum opportunities by adaptive resource allocation based onthe cognition cycle: sensing, deciding, acting. Typically, the spectrum holes are too smallto meet the bandwidth requirements by secondary users. In this regard, Software DefinedRadio (SDR) provides a flexible and programmable transceiver structure that allows tocombine vacant intervals in order to obtain sufficiently large channels ([LVM14]). Thisspectrum aggregation is usually restricted by hardware limitations, such as filter technolo-gies, and the capability of controlling interference. In particular, due to limitations of theRF front ends, only spectrum holes that are – in a certain extent – close enough can beaggregated. This additional constraint separates the considered problem from the ordinarySSP and represents a new modeling challenge for the field of cutting and packing. Notethat such computations might also be relevant when saving data on hard drive disks orwhen managing inventory in storehouses.

Throughout this paper, we assume without loss of generality that all input-data of the SSPare positive integers. Additionally, we expect the item lengths to be smaller than L sinceeach item i ∈ I with li ≥ L already represents a finished product and does not have to beconsidered within the optimization. Finally, we assume that two items can be connecteddirectly, i.e., without any overlap or an intermediate composite.2

The next section deals with a general introduction to the SSP. Afterwards, we add a spe-cific constraint in order to obtain a simplified version of the spectrum aggregation problem.Therefore, we present some approaches on how this closeness condition can be modeled inthe existing frameworks. In a final step, we summarize the main ideas of this paper, pro-vide some simulation results and discuss possible extensions of the considered optimizationproblem. Thereby, we mainly focus on further constraints that have to be added to ourconsiderations in order to achieve fully application-oriented descriptions of the spectrumallocation problem.

1In particular, discrete problems occur naturally at the transmitter for bit-loading ([Cam98]), and atthe receiver for the decoding of multi-stream multi-antenna QPSK signals ([NKDL11]).

2Otherwise, we can shorten or extend the item lengths and L to obtain an equivalent SSP where thisthird assumption is fulfilled.

J. Martinovic, E. Jorswieck, G. Scheithauer. August 18, 2016 3

2 Preliminaries on the SSP

In the following, we refer to a particular SSP with given input-data as an instance E =(m, l, L, b). Here l = (l1, . . . , lm)> ∈ Zm+ and b = (b1, . . . , bm)> ∈ Zm+ contain the m ∈ Ndifferent item lengths and their availabilities, respectively. Remember that L ∈ N denotesthe minimum length that shall be obtained by connecting the given items.

Definition 1. Any feasible arrangement of items leading to length of at least L is called(packing) pattern of E.

We always represent a pattern by a vector a = (a1, . . . , am)> ∈ Zm+ , where ai ∈ Z+ denotesthe number of items of type i ∈ I being contained in the considered pattern. Note thatthis representation only provides information about the total number, but not the specificorder of the corresponding items. For a given instance E the set of all patterns is definedby P (E) :=

{a ∈ Zm+

∣∣ l>a ≥ L}

. Due to |P (E)| = ∞, we only focus on minimal patternsof E.

Definition 2. A pattern a ∈ P (E) is called minimal if a ≤ a =⇒ a = a holds for alla ∈ P ?(E).

Let P ?(E) denote the set of minimal patterns and let J? := J?(E) be a corresponding indexset. Moreover, let xj ∈ Z+ be the number how often pattern aj = (a1j, . . . , amj)

> ∈ P ?(E)is used in the optimization. Then, we obtain a finite model, i.e., the

Standard Model of the SSP

z =∑j∈J?

xj → max

s.t.∑j∈J?

aijxj ≤ bi, i ∈ I, (1)

xj ∈ Z+, j ∈ J?. (2)

The objective function maximizes the total number of connected final products, whereasconstraints (1) guarantee the item supply limitations. In most cases, it is not reasonableor even not possible to have all patterns available prior to the optimization process due tothe huge cardinality of P ?(E). Hence, common solvers, like CPLEX, usually cannot beapplied to tackle this problem. However, at least the LP-relaxation of the standard modelcan be solved efficiently by column generation. In order to solve the ILP, branch-and-pricetechniques can be applied as in the context of one-dimensional cutting, see [BS06] forinstance. Note that, in this case, the computational behavior strongly depends onthe choice of an appropriate branching rule and reasonably tight bounds. Fortunately,practical experience and computational simulations, cf. [Zak03, MS16], have shown thatthere is only a small gap ∆(E) := z?c (E) − z?(E) for any instance E, where z?(E) andz?c (E) denote the optimal values of the integer and the relaxed problem, respectively. Moreprecisely, it is conjectured, see [Zak03], that the inequality ∆(E) < 2 is always satisfied.Hence, a common (approximate) solution approach is also given by the application ofappropriate heuristics to the continuous solution. A further way to tackle the integerproblem consists in considering exact modeling approaches, see [MS16]. In that paper,also many simulation results for instances based on practical meaningful sizes are provided.


3 The Spectrum Allocation Problem – Introduction

and a First Approach

In the next sections, we aim at investigating a simplified version of the spectrum allocationproblem. Therefore, we present three models that can be considered as generalizations ofthe SSP. The spectrum allocation problem is described as follows: consider a frequencyband where some portions are already covered by licensed users, see Fig. 1 for an example.

µ1 ν1 µ2 ν2 µ3 ν3f [Hz]

l1 l2 l3

used spectrum vacant spectrum

Figure 1: A possible frequency band with spectrum holes of bandwidth l1 = ν1 − µ1,l2 = ν2 − µ2 and l3 = ν3 − µ3.

The remaining parts, i.e., the spectrum holes, shall be used by secondary users (SUs)each of them having a required bandwidth of R ∈ N (typically in kHz or MHz). Since,usually, the spectrum holes are too small to meet these requirements, the SUs can accessand aggregate some of them in order to obtain a sufficiently large transmission channel.Thereby we assume:

(A1): Each spectrum hole can only belong to one aggregation, even if it is not usedentirely.

This assumption may not be justified in all practical applications. But, if we restrict ourinvestigations to those cases where, predominantly, small spectrum holes are available andinterference is hard to cope with, placing more than one SU into a spectrum hole wouldcause huge transmission losses due to interference. Possible extensions are discussed inSection 8.

Without further conditions, the above stated problem could be interpreted one-to-one asan ordinary SSP and off-the-shelf models would directly be applicable. However, due tohardware limitations such as filter technologies in the RF chains, we may further assume:

(A2): Only spectrum holes that are in a certain neighborhood to each other, specified bya maximal aggregation range (MAR)3 δ ∈ N, can be aggregated.

In that case, our previous models cannot be applied to tackle this problem due to thefollowing reasons:

i) Due to (A2) intervals of the same width now have to be considered as different whichleads to

∑mi=1 bi (instead of m) item types.

ii) The problem now depends on the specific positionings of the items. Hence, instead ofonly considering the length li, each item type has to be described by two parameters(initial and terminating frequency).

3More precisely: A set of spectrum holes can be aggregated if and only if the difference between theterminating frequency of the last hole and the initial frequency of the first hole is at most δ.


Hence, the previous definition of an instance has to be rephrased in the following way:

Definition 3. A tuple E = (n, µ, ν, R, δ) with

• n ∈ N spectrum holes,

• µ, ν ∈ Zn+ representing the initial and terminating frequencies,

• R ∈ N denoting the SU’s required bandwidth, and

• δ ∈ N representing the MAR

is called generalized instance if and only if the conditions R > max{νi − µi | 1 ≤ i ≤ n}and µ1 < ν1 < . . . < µn < νn are satisfied.

Note that the condition R > max{νi − µi | 1 ≤ i ≤ n} may not be present in all practicalapplications, but it corresponds to the consideration of rather small spectrum holes thatcan only be accessed by one SU each.

Let I ′ := {1, . . . , n}. In order to obtain a workable model based on the above mentionedstandard model we have to find a description for the feasible patterns of a generalizedinstance that could be used within the column generation procedure. Therefore, it isdesirable that only a small number of additional (at best linear) inequalities have to beconsidered.

Definition 4. A vector a ∈ Bn is a generalized pattern if :

i) The aggregated spectrum holes satisfy the SU’s required bandwidth, i.e.,

(ν − µ)> a ≥ R. (3)

ii) The aggregated spectrum holes satisfy ν(a)− µ(a) ≤ δ with

ν(a) := max {ν ∈ Z+ | ∃k ∈ I ′ : ν = νk, ak = 1} ,µ(a) := min {µ ∈ Z+ | ∃k ∈ I ′ : µ = µk, ak = 1} .

Obviously, the second constraint depends on two further optimization problems and is,therefore, rather inappropriate. A first approach to rephrase this property is given bydemanding

aiaj (νi − µj) ≤ δ (4)

for all i, j ∈ I ′ with i > j. Unfortunately, this approach contains two major drawbacks:

i) We need(n2

)additional constraints, while there is one inequality among them imply-

ing all the other ones.

ii) These constraints are non-linear and, therefore, inappropriate for an efficient columngeneration.


At least the second disadvantage can be tackled by introducing additional binary variablesand writing (4) as a system of linear inequalities. To this end, we define αij ∈ B by meansof αij := ai · aj for all i, j ∈ I ′ with i > j. Then, we obtain:

Lemma 1. The nonlinearity in (4) can be replaced by the system

αij ≤ ai, αij ≤ aj, ai + aj ≤ αij + 1 (5)

for each i, j ∈ I ′ with i > j.

Proof. Assume that ai · aj = 0 for some i, j ∈ I ′ with i > j. Then at least one factor, sayai, has to equal zero. Then the first inequality of (5) implies αij = 0, i.e., αij = ai · aj. Onthe contrary, if we assume ai · aj = 1, then we have ai = aj = 1. In this case, the thirdinequality of (5) implies 2 ≤ αij + 1, i.e., αij = 1 = ai · aj. For the sake of completeness,note that, in both cases, the other inequalities of (5) are also satisfied.

In other words, this means:

Lemma 2. A vector a ∈ Bn is a generalized pattern if and only if (3) holds and for alli, j ∈ I ′ with i > j there exists αij ∈ B satisfying (5) and αij (νi − µj) ≤ δ.

In this formulation, a generalized pattern is described by 4 ·(n2

)+ 1 linear inequalities

and(n2

)additional binary variables which represents quite large additional computational

expenses.

4 An Improved Approach

Without loss of generality, we assume the last two spectrum holes to be δ-close (i.e.,νn − µn−1 ≤ δ). Otherwise, the last spectrum hole does not contribute to any generalizedpattern and could be discarded. Due to an analogous argument, the first two spectrumholes can be assumed to be δ-close. Additionally, we claim νn−µ1 > δ in order to separatethe problem from an ordinary SSP. For each i ∈ I ′ we define the index sets

I(i) := {k ∈ {i, . . . , n} | νk − µi ≤ δ} ,I(i) := {k ∈ {i, . . . , n} | νk − µi > δ} .

Observe that the set I(i) (for some i ∈ I ′) contains the indices of all spectrum holes thatcan be aggregated if the i-th spectrum hole is chosen to be the starting interval of such anaggregation. Hence, a generalized pattern a ∈ Bn has to satisfy

ai = 1 =⇒∑k∈I(i)

ak = 0. (6)

for all i ∈ I ′. Since all values of this implication are binary we obtain the following result.

Lemma 3. Implication (6) is equivalent to the linear inequality∣∣I(i)∣∣ · ai +

∑k∈I(i)

ak ≤∣∣I(i)

∣∣ . (7)


Proof. Assuming ai = 1, (7) can be rephrased to∑

k∈I(i) ak ≤ 0 which implies∑

k∈I(i) ak =0 due to the non-negativity of each summand.

Note that for I(i) = ∅ both conditions, (6) and (7), are always satisfied and do not needto be added to the definition of the generalized patterns. Due to

I(i) = ∅ =⇒ I(k) = ∅

for all k ≥ i, it suffices to demand (7) for i ∈ T := {1, . . . , τ − 1} where

τ := inf{i ∈ I ′

∣∣ I(i) = ∅}

holds. Note that, due to I(n) = ∅ the infimum is well-defined and can be replaced by aminimum. Additionally, the particular value of τ can be slightly restricted by means ofthe following observations:

i) Due to νn − µ1 > δ we can conclude τ ≥ 2.

ii) Due to I(n− 1) = {n− 1, n} we obtain τ ≤ n− 1.

Therefore, i ∈ T particularly implies∣∣I(i)

∣∣ ≥ 1 such that (7) can be written as

ai +1∣∣I(i)∣∣ ∑k∈I(i)

ak ≤ 1. (8)

Hence, we obtain the characterization:

Lemma 4. A vector a ∈ Bn is a generalized pattern if and only if (3) holds and inequality(8) is satisfied for each i ∈ T .

This approach contains two main advantages compared to the previous one:

i) We do not need further auxiliary binary variables.

ii) We only have |T | ≤ n − 2, i.e., O(n), additional linear constraints (compared toO(n2) many in Section 3).

These significant savings are accompanied by only minor additional expenses appearingdue to the computation of the index sets I(i). Hence, this description could be used in aformulation based on the standard model.

Generalized Standard Model

z =∑j∈Jg

xj → max

s.t.∑j∈Jg

aijxj ≤ 1, i ∈ I, (9)

xj ∈ B, j ∈ Jg. (10)


There, Jg := Jg(E) represents an index set of the set Pg := Pg(E) of all generalizedpatterns. Due to the additional δ-condition, we already have |Pg| <∞ and do not have toconsider minimal patterns to obtain a finite model.

Note that also this model cannot be tackled directly by means of standard ILP solvers sinceit is not recommendable or even not possible to compute all generalized patterns prior tothe optimization. Nevertheless, at least the

LP-Relaxation of the Generalized Standard Model

z =∑j∈Jg

xj → max

s.t.∑j∈Jg

aijxj ≤ 1, i ∈ I, (11)

xj ∈ [0, 1], j ∈ Jg. (12)

can be solved efficiently by means of column generation. Observe that, in this case, theLP bound may be of worse quality as noticed in the standard SSP scenario. Theoretically,the LP bound of this generalized SSP can be arbitrarily large.

Theorem 5. Let z and zc be the optimal objective values of the generalized standard modeland its continuous relaxation, respectively. Then, the generalized gap ∆g(E) := zc − z isunbounded.

Proof. For the sake of contradiction, let us assume that there exists a constant C ∈ N with

sup {∆g(E) | |E is a generalized instance } ≤ C.

Let E = (m, l, L, b) be an arbitrary non-IRDP-instance of the ordinary skiving stock prob-lem. Then, this instance can be reformulated as an instance E of the generalized skivingstock problem by choosing δ = νn−µ1 such that all packing patterns of E remain general-ized patterns of E and vice versa. If we now start at the frequency f = νn+δ and copy thespectrum holes of the instance E, we obtain a generalized instance E ′ (with n′ = 2n, δ′ = δand R′ = R) with ∆g(E

′) = 2 ·∆g(E). By means of induction we can obtain generalizedgaps larger than C which proves the assertion by contradiction.

Note that this result is, in some sense, only of theoretical value since spectrum allocationproblems normally consider a given range of the frequency band. This means that, inpractical problems, there is not enough space to apply the above mentioned construction,and, hence, generalized gaps may probably not become arbitrarily large.

5 A generalized Arcflow Model

The previous investigations did not lead to models that can directly be tackled by ILPsolvers due to the unavailability of all generalized patterns. In order to obtain a modelwhere all necessary information are given prior to the optimization, we present a modelbased on the general idea of an arcflow formulation, see [MS16]. Thereby, we also have to


(1,0,0) (2,1,0)

(1,1,4)

(2,2,2)

(1,2,4)

(1,2,6)

(2,3,5)

(1,3,7)

(2,3,2)

(2,4,6)

Figure 2: Arcflow graph G1 for R = 5 and I(1) = {1, 2, 3} with ν1 − µ1 = 4, ν2 − µ2 = 2and ν3− µ3 = 3. In blue: the next arcflow graph G2 with I(2) = {2, 3, 4} and ν4− µ4 = 4.

pay attention to the new δ-condition. As an initial idea, we can formulate directed graphsGk = (V k, Ek) for all k ∈ T ′ with

T ′ :={j ∈ I ′

∣∣∣∑i∈I(j) νi − µi ≥ R},

and demand a certain compatibility among them, see Fig. 2 for two directed graphs ofan instance with small input-data. The restriction to T ′ is reasonable since we only needto model a subgraph with j ∈ I ′ as a starting interval if and only if the total bandwidthappearing in this subgraph is large enough to obtain R. For fixed k ∈ T ′ we defineV kk−1 = {0}, V k

k := {νk − µk} and

V ki :=

{y = y′ + j(νi − µi)

∣∣ y′ ∈ V ki−1, j ∈ B, y ≥ Ck

i

},

with Cki := max

{0, R−

∑σ∈I(k),σ>i(νσ − µσ)

}, for i ∈ I(k) \ {k}. Note that the condition

y ≥ Cki ensures that there are enough spectrum holes to be possibly added afterwards in

order to obtain a feasible generalized pattern. In particular, the elements of V kρ(k) thereby

satisfy y ≥ R, where ρ(k) denotes the maximum element of I(k).

Then the vertex set of graph Gk is given by the disjoint set union

V k :=⊔

i∈I(k)∪{k−1}

V ki :=

⋃i∈I(k)∪{k−1}

{(k, i, y)

∣∣ y ∈ V ki

}.

Similarly, the auxiliary arc sets are given by

Eki :=

{(y′, y) ∈ V k

i−1 × V ki

∣∣ y − y′ ∈ {0, νi − µi}, y′ < R}

for i ∈ I(k). Observe that for a given node y′ ∈ V ki−1 with y′ < R the only ways to continue

are either choosing the next spectrum hole (i.e., y = y′+ νi−µi) or not (i.e., y = y′). Thusthe arc set Ek of Gk results to

Ek :=⊔i∈I(k)

Eki :=

⋃i∈I(k)

{(k, i, y′, y)

∣∣ (y′, y) ∈ Eki

}.


Note that the condition y′ < R is important to reduce the number of vertices belongingto non-minimal generalized patterns. Let E :=

⋃k∈T ′ E

k denote the set of all arcs. Then,for each arc (k, i, p, q) ∈ E we define a decision variable xkipq ∈ B stating whether theconsidered arc (p, q) ∈ Ek

i is used or not. Let I?(k) := I(k) \ ρ(k) for k ∈ T ′. In order toease the notation, we introduce the sets

A+k,i(v) :=

{u ∈ V k

i−1∣∣ (u, v) ∈ Ek

i

}for k ∈ T ′, i ∈ I(k) and v ∈ V k

i ,

A−k,i(v) :={w ∈ V k

i+1

∣∣ (v, w) ∈ Eki+1

}for k ∈ T ′, i ∈ I?(k) and v ∈ V k

i , and

Lki :={v ∈ V k

i

∣∣ v ≥ R}

for k ∈ T ′ and i ∈ I(k). Note that for some fixed v ∈ V ki , the sets A+

k,i(v) and A−k,i(v)

describe the incoming and outgoing arcs, respectively. The sets Lki collect all vertices thatcorrespond to the total bandwidth of a generalized pattern. Altogether, we obtain the

Generalized Arcflow Model

zaf =∑k∈T ′

xk,k,0,νk−µk → max

s.t.∑k≤i

∑(p,q)∈Ek

i

q − pνi − µi

· xkipq ≤ 1, i ∈ I ′, (13)

∑p∈A+

k,i(q)

xkipq =∑

r∈A−k,i(q)

xk,i+1,q,r,

k ∈ T ′,i ∈ I?(k),q ∈ V k

i \ Lki ,(14)

xkipq ∈ B, (k, i, p, q) ∈ E. (15)

Constraints (14) can be interpreted as flow-conservation constraints in the graph Gk

(k ∈ T ′). In particular, this implies that if and only if a variable xk,k,0,νk−µk is equalto one (for some k ∈ T ′), the index k represents the initial spectrum hole of a generalizedpattern. Hence, the objective function maximizes the total number of generalized patterns.Constraints (13) can be seen as a compatibility condition stating that each spectrum holecan be used at most once in all graphs. This model possesses O(n+ n2R) constraints andO(n2R2) variables and, therefore, is of pseudo-polynomial complexity. In contrast to theapproach from the previous section, this model directly can be solved by means of ILPsolvers.

6 An Assignment Model

Although the arcflow model consists of a pseudo-polynomial number of variables and con-straints, it is very complex and rather hard to cope with from the modeling point of view.


Instead of focusing on the specific positioning of a spectrum hole within a subgraph Gj, aformulation based on assignment variables

yjk =

{1, if the k-th interval is used in subgraph Gj,

0, otherwise.

for j ∈ T ′ and k ∈ I(j) can reduce the complexity significantly. This idea leads to the

Assignment Model for Spectrum Allocation

za =∑j∈T ′

yjj → max

s.t.∑

j∈T ′: k∈I(j)

yjk ≤ 1, k ∈ I ′, (16)

∑k∈I(j)

yjk · (νk − µk) ≥ R · yjj, j ∈ T ′, (17)

yjk ∈ B,{j ∈ T ′,k ∈ I(j).

(18)

Constraints (16) ensure that each spectrum hole k ∈ I ′ is used in at most one subgraph Gj.Inequalities (17) can be interpreted as follows: the spectrum holes which are active in thej-th subgraph shall sum up to at least the required bandwidth R whenever subgraph Gj

is used at all within the optimization. Note that, in the terminology of the arcflow model,the subgraph Gj is used if and only if its first arc variable xj,j,0,νj−µj is equal to one. In ourassignment model, this corresponds to satisfying the equality yjj = 1. Hence, if we haveyjj = 1 the subgraph Gj has to contain a generalized pattern.

Remark 6. For yjj = 0 (with j ∈ T ′) condition (17) does not forbid the j-th subgraphGj to contain a generalized pattern. In that case the objective value differs from the actualnumber of generalized patterns. Note that this slight inconsistency may not occur in optimalsolutions of the assignment model. However, in order to ensure that also each feasible pointis treated correctly in the objective function, it is possible to add further inequalities suchas ∑

k∈I(j)

yjk · (νk − µk) ≤ 2R · yjj for j ∈ T ′.

This model possesses n+ |T ′| ≤ 2n linear constraints and∑j∈T ′|I(j)| ≤

∑j∈T ′

(n− j + 1) ≤n∑j=1

(n− j + 1) =n

2(n+ 3)

binary variables which represents significant savings compared to the arcflow model. Hence,this model appears to be the most promising regarding numerical computations.

7 Simulation Results

For our numerical simulations we consider the TV band (470MHz − 862MHz) which isknown to be quite static, see [LBCU+09]. Due to the continuous broadcasting of TV sta-tions, spectrum holes lying in this range can be considered as almost time-independent.


The total ratio q ∈ [0, 1] of unoccupied frequencies in this band mainly depends on thespecific geographical location. Measurements have shown that q ≈ 0.2 holds for urbanregions [LBCU+09, Tab. 2], whereas q ≈ 0.8 holds in suburban areas [VFM+09, Tab. 1].Reasonable choices of the further parameters are given by [LLLL14, Tab. 3], hence weconsider the following setting: δ = 40 [MHz], R = 15 [MHz] and n ∈ [10, 70] ∩N.

For each scenario we randomly generated 100 generalized instances and solved the cor-responding assignment formulation by means of the CPLEX-interface (version 12.5.1) ofMATLAB R2013a on an Intel i5-2450M CPU with 2.5 GHz and 6 GB RAM. Thereby, wedid not forbid intervals larger than R, but those spectrum holes were assigned to an SUprior to the optimization (i.e., not all n intervals had to be considered in the model). Inthe following tables the averages of the computation time t (in sec.) and the numbers ofvariables nvar and constraints ncon are listed. Besides this numerical behavior it is alsoimportant to focus on the ratio u = (za ·R)/(q · 492) ∈ [0, 1] of the total vacant bandwidththat is used by SUs. Obviously, this is a good measure to evaluate the practical potentialsof spectrum aggregation based spectrum allocation.

Table 1: Simulation results for an urban arean = 10 n = 20 n = 30

t 0.0117 0.0155 0.0203u 0.4245 0.2640 0.1605nvar 7.66 38.05 62.69ncon 7.20 21.32 24.08

Table 2: Simulation results for a suburban arean = 20 n = 30 n = 50 n = 60 n = 70

t 0.0118 0.0194 0.0822 0.2466 1.7464u 0.5377 0.6803 0.8392 0.9008 0.9349nvar 7.74 32.21 182.48 269.08 367.08ncon 6.85 24.56 97.08 118.18 137.82

Clearly, with growing n, also the numbers nvar and ncon increase which leads to a morecomplex optimization problem and, hence, to longer computation times t. In the urbansetting, the utilization u decreases by adding more spectrum holes. This behavior is quitenatural since, for q = 0.2, the density of the spectrum holes within the given TV band israther small. If there are many spectrum holes each of them is likely to be very small. Dueto the δ-condition, this sparsity leads to many intervals that cannot be aggregated. For asmall number of intervals, such as n = 10, it is more likely that one single spectrum holecan meet the SU’s required bandwidth.

In contrast, the utilization u increases in the suburban scenario (q = 0.8) if n grows. Inthis setting, the spectrum holes are very dense in the TV band. Hence, many of them areδ-close to other intervals, so almost all of them can be used in appropriate aggregations.For n = 20 it is more likely that a single interval is larger than R which excludes theremaining part of this interval from further aggregations. In general, spectrum allocation


problems of practical meaningful sizes can be solved in short time. Moreover, in mostcases, they provide very good results in terms of using the vacant portions of the spectrumefficiently.

8 Application-oriented Extensions

8.1 Introduction

As mentioned earlier in Section 3 the condition that only one SU is allowed per spectrumhole may not be given in each intended application. As a worst-case example, in practice itis also possible that at a certain time and location the total considered part of the frequencyband is completely vacant since it is not used at all by licensed holders. In our currentmodel this would lead to a single (very large) spectrum hole which can only be accessedby one SU although many of them could theoretically be placed therein. Hence, in thissection, we will deal with the question how several SUs per spectrum hole can be copedwith. We will refer to this new question as the generalized spectrum allocation problem(GSAP). Based on the given worst-case example we will have to discard our conditionR > max {νi − µi | i ∈ I ′}.In what follows we assume that a given spectrum hole [µ, ν] can be shared between severalSUs as long as they operate in pairwise distinct subsets of [µ, ν].4 Hence, generalizedpatterns may now start at every point of the set Rs and end at every point of the set Re,where these sets are given by

Rs :=n⋃i=1

([µi, νi) ∩N) and Re :=n⋃i=1

((µi, νi] ∩N) .

Consequently, we need to rephrase the definition of a (generalized) pattern since, in partic-ular, the modeling of the δ-condition has to be modified. In addition, each pattern has tocontain information about the fact, whether a certain spectrum hole is (partly) used and,if so, which particular subset of this interval is occupied.

Definition 5. A triple (a, p, q) ∈ Bn × Zn+ × Zn+ is called (allocation) pattern, if thefollowing conditions hold:

i) µ ≤ p, p+ a ≤ q, and qi ≤ aiνi + (1− ai)µi for all i ∈ I ′,

ii) e>(q − p) ≥ R,

iii) q − p ≤ δ where q := max{qi | qi > pi} and p := min{pi | qi > pi}.

Remark 7. The triple (a, p, q) shall be understood as follows: ai = 1 (i ∈ I ′) if andonly if the i-th spectrum hole is (partly) used in the pattern. Moreover, the interval [pi, qi)describes the specific subset of the i-th interval [µi, νi] that is occupied by the allocationpattern (a, p, q). Due to conditions (i) we obtain [pi, qi) = [µi, µi) = ∅ if and only if ai = 0.

4The role of interference between SUs operating in the same spectrum hole is briefly discussed at theend of this section.


Otherwise (i.e., for ai = 1), these conditions ensure µi ≤ pi < qi ≤ νi.Obviously, the second condition states that the considered pattern satisfies the band-width requirement of a secondary user. Without loss of generality, we can also demande>(q − p) = R since arbitrary subsets of the spectrum holes can be aggregated.The third condition represents the adapted formulation of the δ-closeness. Here, the vari-ables p and q can be interpreted as the starting point and the end point, respectively, of thepattern (a, p, q).Note that an allocation pattern (a, p, q) could also be fully described by its components pand q since we have ai = 1 if and only if qi > pi. However, it is somehow more practicalto directly carry the indicator a ∈ Bn along in the definition (for instance, when trying torephrase condition (iii)).

Observe that, also in this case, the δ-condition depends on two further optimization prob-lems, and is, hence, rather inappropriate for a good description of the pattern set. Unfor-tunately, our idea from Section 4 (of using conditions like (6)) will not work in the newscenario since we are confronted with a very large number of potential starting points p fora pattern (a, p, q). However, at least the possibility of rephrasing (iii) as

aiaj(qj − pi) ≤ δ

for all i, j ∈ I ′ with j ≥ i remains, but is still quite unfavorable due to its nonlinearity.Even the method of Lemma 1 would not lead to a linear description in this case. Hence,we might focus on another approach to solve the given problem.

8.2 A Solution Strategy based on Connected Patterns

As we have seen in the previous paragraph, the formulation of a pattern-based ILP israther difficult due to the fact that the pattern set itself does not possess a practical (atbest linear) description that could efficiently be used in a column generation algorithm.Hence, in this subsection, we would like to focus on some theoretical observations in orderto find a solution strategy that exploits the problem-specific properties. As mentionedbefore, we may reduce our considerations to those patterns that satisfy condition (ii) ofDefinition 5 with equality. Therefore we need the following definitions:

Definition 6. Let (a, p, q) be an allocation pattern. A point σ ∈ Rs is called interruptionpoint of (a, p, q) if

p ≤ σ < q and [σ, σ + 1) ∩n⋃i=1

[pi, qi) = ∅.

An allocation pattern (a, p, q) that does not contain any interruption point is called con-nected.

Example 1. We consider the situation given by Fig. 1. Assuming that the areas paintedred in Fig. 3 represent a pattern (a, p, q) we can notice that this pattern has the interruptionpoint σ = µ2 in the upper picture. In the lower picture, an example for a connected patterncan be seen. Note that this pattern was constructed from the first one by left-shifting someparts of this pattern such that all interruption points disappear.


µ1 ν1 µ2 ν2 µ3 ν3f [Hz]

µ1 ν1 µ2 ν2 µ3 ν3f [Hz]

Figure 3: An example for an interruption point (upper picture) and a connected pattern(lower picture) in the situation of Fig. 1.

The given example illustrates an important principle of our theoretical study.

Lemma 8. Let (a, p, q) be an allocation pattern. Then there exists a unique connectedpattern (a, p, q) with p = p.

Proof. The pattern (a, p, q) can be easily obtained by left-shifting the necessary parts of(a, p, q) until all interruption points disappear. In particular, this movement does notinfluence the conditions (ii) and (iii) of Definition 5 since, obviously, we have

e>(q − p) = e>(q − p) = R and q − p = q − p ≤ q − p ≤ δ.

Based on this concept we can conclude:

Theorem 9. There exists a solution of the generalized spectrum allocation problem thatonly consists of connected patterns.

Proof. Consider a solution of the generalized spectrum allocation problem that contains atleast one non-connected pattern. Then the whole allocation possesses at least one interrup-tion point. Let (a, p, q) denote the pattern that belongs to the minimal interruption pointσmin. Then all patterns that are completely positioned to the left of σmin are connected.Hence, we may encounter a situation as given in Figure 4.

µi−1 νi−1 µi νi µi+1 νi+1f [Hz]

Figure 4: An exemplary scheme with σmin = νi−1 − 1. The red pattern possesses theminimal interruption point. The blue and green parts may belong to other non-connectedpatterns or can be unoccupied as well.

As it was done in Lemma 8, we replace the pattern (a, p, q) by its unique connected repre-sentation (a, p, q) with p = p. Those parts that did possibly belong to other non-connectedpatterns are placed directly after q of (a, p, q), as depicted in Fig. 5. Note that


µi−1 νi−1 µi νi µi+1 νi+1f [Hz]

Figure 5: Construction of (a, p, q) with p = p. The parts that possibly belong to otherpatterns were placed directly after (a, p, q). After the rearrangement σmin has increased.

• the minimum interruption point σmin increases by this procedure,

• the described rearrangement of the subintervals (see Fig. 5) is always possible since,in total, they need the same bandwidth as before,

• the rearrangement does not influence the δ-condition of other patterns. This is dueto the fact that σmin was the mininum interruption point which means that thosesubsets belonging to other patterns (for instance the green and the blue one in Fig. 5)have to be starting intervals of certain other allocation patterns. Hence, by shiftingthem to the right, it is impossible to violate their δ-condition.

Altogether, the proposed procedure leads to a feasible solution with the same number ofallocation patterns, and increases the value of σmin. Hence, after a finite number of steps,we obtain a solution that only consists of connected patterns.

Consequently, considering connected patterns is sufficient to solve the generalized spectrumallocation problem. As a final step, we need the following observation: let (a, p, q) be aconnected pattern with p /∈ {µ1, . . . , µn}. Obviously, we can shift the whole pattern oneunit to the left (by respecting the given spectrum holes) and obtain a new feasible pattern(a′, p′, q′) with p′ = p− 1, see Fig. 6.

Definition 7. The pattern (a′, p′, q′) is called left-shift of (a, p, q).

µ1 ν1 µ2 ν2 µ3 ν3f [Hz]

µ1 ν1 µ2 ν2 µ3 ν3f [Hz]

Figure 6: The pattern in the lower picture describes a left-shift of the pattern in the upperone.

Note that, after a finite number of steps, we obtain a pattern (a′, p′, q′) with p′ = µi forsome i ∈ I ′. Then, a further left-shift (into the previous spectrum hole) does not have tobe possible. This mainly depends on the specific distribution of the spectrum holes andthe value of δ. Nevertheless, we can state the following:


Theorem 10. There exists a solution of the generalized spectrum allocation problem(containing only connected patterns), where each pattern either starts at some point of{µ1, . . . , µn} or at the endpoint q of the previous pattern.

Proof. Consider a solution of the generalized spectrum allocation problem that violates theassertion. Since all pattern are connected, we can find a first pattern that does neither startat some point of {µ1, . . . , µn} nor at the endpoint of the previous pattern. By means ofthe proposed method, we can left-shift this whole pattern, until it either reaches the initialpoint µi of the current spectrum hole or the endpoint q of the previous pattern. Hence,after a finite number of steps, we obtain a solution that satisfies the requirements.

Hence, a solution of this problem can be obtained by placing connected patterns succes-sively into the given spectrum holes and respecting the δ-closeness. In order to ease thenotation, let U(x, y) for x, y ∈ [µ1, νn + δ] ∩ N with y ≥ x denote the vacant bandwidthbetween x and y, i.e.,

U(x, y) =n∑i=1

|[x, y] ∩ [µi, νi)| ,

where |[a, b)| = max{b − a, 0} represents the length of the interval. Then, the followingalgorithm leads to a solution of the considered allocation problem:

Algorithm 1 Finding a solution of the GSAP

Input: Instance E = (n, µ, ν, R, δ).1: Set k := 1, j := 1, xj := µ1, µn+1 := νn and z := 0.2: while k ≤ n and U(xj, νn) ≥ R do3: if U(xj, xj + δ) ≥ R then4: Compute yj ∈

⋃ni=k(µi, νi] with U(xj, yj) = R.

5: Save (xj, yj) and set j := j + 1, xj := yj−1, z := z + 1.6: else7: Set k := k + 1, xj := µk.8: end if9: end while

Output: optimal value z, saved pairs (xj, yj) for j = 1, . . . , z.

Note that, besides the optimal value, this algorithm does also provide the pairs (xj, yj) =(p, q) for each allocation pattern. Since all these patterns are connected, these pieces ofinformation are sufficient to reconstruct the specific allocation patterns (a, p, q) that havebeen used in the given solution.

8.3 The role of interference

An issue that has not yet been dealt with is the influence of interference when several SUsare allowed to operate in the same spectrum interval.


Remark 11. Note that interference does also occur between secondary users and licensedholders. In practice, this problem can be tackled by leaving some units at the left andthe right margin of each spectrum hole unoccupied. Mathematically, this corresponds tochanging the vectors µ (to µ+ ε · e) and ν (to µ− ε · e) for e = (1, . . . , 1)> and some smallε ∈ N prior to the optimization. Hence, interference between primary and secondary usersis not problematic for any approach of this manuscript.

Our above stated algorithm provides a solution where (possibly) two allocation patterns(i.e., two secondary users) are placed directly next to each other in some given spectrumhole. Due to the fact that a signal in some subset [pi, qi) of [µi, νi] is not ideal in practice,it needs a certain additional range [pi− ε, p) and [qi, qi + ε) (for some small ε ∈ N) outsideof [pi, qi) to decay completely, see Fig. 7.

ε ε ε ε

pji qji = pki qkif

Figure 7: A schematic of interference between the signals of neighboring SUs in the i-thspectrum hole.

Hence, if two allocation patterns are positioned directly one after the other, the signals willinterfere in a small neighborhood of their joint borderline. Nevertheless, also this problemcan be solved by the method described in Remark 11, i.e., by leaving a certain vacantspace (for instance an interval of width ε) between two neighboring allocation pattern, ifand only if they operate in the same spectrum hole. Such unoccupied frequency rangesare called guard bands, see Fig. 8 for a schematic and [BKM14] or [UAK14] for a moredetailed explanation.

guard

pji qji pki qkif

Figure 8: The situation of Fig. 7 with a guard band of width ε.

Fortunately, we only need to change our theoretical considerations slightly by rephrasingTheorem 10 as:


Theorem 12. There exists a solution of the generalized spectrum allocation problem (con-taining only connected patterns) with guard bands, where each pattern either starts at somepoint of {µ1, . . . , µn} or with a distance of ε to the endpoint of the previous pattern.

Consequently, Algorithm 1 can also be applied after some minor modifications:

Algorithm 2 Finding a solution of the GSAP with guard bands

Input: Instance E = (n, µ, ν, R, δ), guard band width ε ∈ N.1: Set k := 1, j := 1, xj := µ1, and z := 0.2: Compute µi, νi for all i ∈ I, and define µn+1 := νn.3: while k ≤ n and U(xj, ν)n ≥ R do4: if U(xj, xj + δ) ≥ R then5: Compute yj ∈

⋃ni=k(µi, νi] with U(xj, yj) = R.

6: Save (xj, yj) and set j := j + 1, z := z + 1.7: if νk − yj−1 ≤ ε then8: Set k := k + 1, xj = µk.9: else

10: Set xj = yj−1 + ε.11: end if12: else13: Set k := k + 1, xj := µk.14: end if15: end whileOutput: optimal value z, saved pairs (xj, yj) for j = 1, . . . , z.

8.4 Secondary Users with Heteregeneous Bandwidth Require-ments

So far we did only consider cases where all SUs possessed the same required bandwidthR. Due to standardization in wireless communications this situation is not completelyimplausible in practice. Nevertheless, this assumption may represent a too strong restric-tion when focusing on an application-oriented solution. Hence, this subsection deals withthe question how the GSAP can be solved, if there are N ∈ N secondary users each ofwhich having its specific bandwidth Ri ∈ N. Without loss of generality we may assumeR1 ≤ R2 . . . ≤ RN . Since the treatment of guard bands was described in the previoussubsection we will discard them in the considerations of this subsection.

In order to solve the generalized spectrum allocation problem for specific bandwidths ofthe SUs, the following result is of importance:

Theorem 13. Let z represent the optimal value of the generalized spectrum allocation prob-lem with specific bandwidths. Then there exists a solution where the bandwidth requirementsR1, . . . , Rz are satisfied.

Proof. For the sake of contradiction, we assume that each optimal allocation is differentfrom satisfying the SUs 1, . . . , z. Without loss of generality, we may choose an optimal


solution that only consists of connected patterns. Then there exists an index γ1 ∈ {1, . . . , z}such that the SU with index γ1 is not contained in the allocation and γ1 is chosen minimalwith respect to this property. Equally, there exists an index γ2 > z such that the SUwith index γ2 is satisfied in the considered allocation. Due to Rγ1 ≤ Rγ2 , each connectedallocation pattern (a, p, q) with e>(q − p) = Rγ2 can be reduced to an unique connectedpattern (a′, p′, q′) with e>(q′ − p′) = Rγ1 and p = p′. Hence, by exchanging these twopatterns in the considered solution we obtain a new solution where at least the requirementsof the SUs 1, . . . , γ1 are satisfied. After a finite number of steps this procedure stops at anoptimal solution that satisfies R1, . . . , Rz which gives the contradiction.

However, note that finding the concrete positionings of the chosen secondary users is muchmore difficult than in the previous cases, since it requires new modeling approaches.

Hence, we have seen how some practical meaningful constraints can be tackled in thespectrum allocation framework. Although it is much harder to find an appropriate mod-eling approach, the problems become somehow more managable, i.e., the problem-specificproperties can easily be exploited to find a possible solution by means of the presentedalgorithms.

9 Conclusions and Further Research

In this paper, we gave a short introduction to the one-dimensional SSP. As a main contri-bution, we dealt with the problem of spectrum allocation which is of practical relevancedue to the scarcity of the natural radio spectrum. We showed how a simple version ofthis allocation problem can be interpreted as a generalization of the SSP, where ordinarypatterns are supposed to satisfy an additional δ-closeness condition. For this challeng-ing constraint, we presented possible characterizations and introduced three correspondingmodels: a pattern-based model which can be tackled by column generation, as well asa generalized arcflow model and an assignment model of pseudo-polynomial complexity.Moreover, we proved the unboundedness of the gap of the given optimization problem, andprovided numerical simulations that underlined the huge practical potentials of spectrumaggregation based spectrum allocation.

Additionally, we added further constraints to our spectrum allocation scenario to obtain amore application-oriented version of this optimization problem, i.e.:

i) Allowing the SUs to share spectrum holes.

ii) Introducing guard bands in order to tackle the problem of interference.

One main challenge for our future work consists in the consideration of heteregenenousbandwidth requirements or other optimization tasks in the field of CR networks, see[TZFS13], such as maximizing the total utilization of the spectrum holes. As long aseach SU possesses the same bandwidth requirement R, this problem is equivalent to themaximization of the total number of satisfied SUs. But, if we focus on specific bandwidths,we obtain a completely new optimization criterion. Furthermore, also the energy efficiencyof the obtained resource allocation could be optimized. This objective is a main research


area of the Collaborative Research Center HAEC5, and therefore, in general, of high rele-vance in our current and future research.

Another practical meaningful extension of the considered problem is given by investigatinga time-dependent scenario. There, spectrum holes may change over time depending onhow many licensed users are present at a certain time in the considered frequency range.Such investigations will likely correspond to the wide area of scheduling problems.

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