Fairness and Sum-Rate Maximization via Joint Subcarrier and

1

Fairness and Sum-Rate Maximization via JointSubcarrier and Power Allocation in Uplink SCMA

TransmissionJoao V.C. Evangelista, IEEE Student Member, Zeeshan Sattar, IEEE Student Member, Georges Kaddoum, IEEE

Member, and Anas Chaaban, IEEE Member

Abstract—In this work, we consider a sparse code multipleaccess uplink system, where J users simultaneously transmitdata over K subcarriers, such that J > K , with a constrainton the power transmitted by each user. To jointly optimize thesubcarrier assignment and the transmitted power per subcarrier,two new iterative algorithms are proposed, the first one aimsto maximize the sum-rate (Max-SR) of the network, whilethe second aims to maximize the fairness (Max-Min). In bothcases, the optimization problem is of the mixed-integer non-linear programming (MINLP) type, with non-convex objectivefunctions, which are generally not tractable. We prove thatboth joint allocation problems are NP-hard. To address theseissues, we employ a variant of the block successive upper-boundminimization (BSUM) [1] framework, obtaining polynomial-timeapproximation algorithms to the original problem. Moreover,we evaluate the algorithms’ robustness against outdated channelstate information (CSI), present an analysis of the convergenceof the algorithms, and a comparison of the sum-rate and Jain’sfairness index of the novel algorithms with three other algorithmsproposed in the literature. The Max-SR algorithm outperformsthe others in the sum-rate sense, while the Max-Min outperformsthem in the fairness sense.

Index Terms—SCMA, 5G, Power Allocation, Multiple Access.

I. INTRODUCTION

THE fifth generation (5G) of wireless networks is ex-pected to deliver better coverage and a higher capacity

to massively connected users. One of the fundamental as-pects to achieve this goal is the design of multiple accesstechniques. Orthogonal multiple access (OMA) techniquesallocate different users into orthogonal network resources, tominimize the interference between users. For instance, time di-vision multiple access (TDMA), code division multiple access(CDMA) and orthogonal frequency division multiple access(OFDMA) assign orthogonal time slots, codes and subcarriersto users, respectively. However, due to the increasing demandfor data communications and the introduction of new data-hungry technologies, such as virtual and augmented reality

c© 2019 IEEE. Personal use of this material is permitted. Permission fromIEEE must be obtained for all other uses, in any current or future media,including reprinting/republishing this material for advertising or promotionalpurposes, creating new collective works, for resale or redistribution to serversor lists, or reuse of any copyrighted component of this work in other works.

J.V.C. Evangelista, G. Kaddoum and Z. Sattar were with the De-partment of Electrical Engineering, cole de Technologie Suprieure, Mon-treal, QC, H3C 1K3 CA, e-mail: {joao-victor.de-carvalho-evangelista.1, zee-shan.sattar.1}@ens.etsmtl.ca and georges.kaddoum@etsmtl.ca

Anas Chaaban is with the School of Engineering, University of BritishColumbia, Kelowna, BC, V1V 1V7 CA, email: anas.chaaban@ubc.ca

(VR/AR) and massively deployed internet of things (IoT)devices, a tenfold increase in traffic is expected by 2020 [2]. Asthe number of orthogonal network resources available is finite,this design paradigm is incompatible with the massive trafficand connectivity requirements of 5G networks. Recently, earlyinformation-theoretic works on multi-user communications[3], [4] have reemerged under the name non-orthogonal mul-tiple access (NOMA) as a potential solution to deal withthis requirement. Although NOMA methods are rooted in theinformation-theoretic literature, the recent interest has beenfocused on communication-theoretic aspects such as develop-ing efficient NOMA coding and modulation schemes, withdesired error-rate performance and multi-user communicationcapabilities. Differently from OMA, in NOMA techniques,multiple users are allocated to the same network resources,permitting the allocation of more users and more efficientuse of the available resources. In NOMA, each receiver mustperform multiuser detection (MUD) to recover the intendedtransmitted signal. NOMA techniques can be classified intotwo different groups, power division NOMA (PD-NOMA),code division NOMA (CD-NOMA). Recently, power domainsparse code multiple access (PSMA) [5] has been proposed asa hybrid of PD-NOMA and CD-NOMA. An extensive perfor-mance comparison of NOMA methods in a single cell systemis found in [6], while the comparison between PD-NOMA andCD-NOMA in heterogeneous network is presented in [5]

In CD-NOMA, the transmitter introduces redundancy tothe transmitted symbol, via code and/or spreading, to enablereceivers to perform MUD and separate signals from differentusers. Furthermore, CD-NOMA has additional advantagesin comparison to PD-NOMA [7], such as the coding gainand the shaping gain (i.e., methods using multidimensionalconstellations) [8]. Motivated by these advantages, this paperis focused on one of the promising CD-NOMA techniques,named sparse code multiple access (SCMA) [8]. In SCMA,sparse multidimensional codebooks are assigned to each user,and each user’s data layer is sparsely spread throughoutthe network resources. In comparison to OFDMA, SCMAallows for more users than subcarriers available to be servedsimultaneously, while reducing the peak average power ratio(PAPR) due to the sparsity of the subcarrier allocation. SCMAwas first proposed in [8], as a multidimensional generalizationof the low density spreading code division multiple access(LDS-CDMA) that yielded better results regarding detectionerror. In [9], a method to design SCMA codebooks based

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on lattice coding was proposed. In [10], a downlink SCMAsystem is considered, and an algorithm for user pairing alongwith rate adjustment and a detection strategy is proposed fora multiuser SCMA scheme. It is shown that this scheme canachieve robustness to mobility and high data rates.

A. Related WorkRegarding resource management and allocation in SCMA

networks, an algorithm to maximize the rate of successfulaccesses on a random access massive machine communi-cations network is suggested in [11]. In [12], a resourceallocation and subcarrier pairing scheme combining OFDMAand SCMA for a dual-hop multiuser relay network is proposed.The problem of assigning SCMA subcarriers to maximize sumrate in uplink transmission is formulated as matching game in[13]. A grant-free contention based uplink SCMA scheme wasproposed in [14]. In [15], the capacity of an SCMA cell witha Gaussian input is derived and a joint subcarrier and powerallocation algorithm is proposed. In [16], three algorithms fordynamic subcarrier allocation are presented and their link-levelperformance is evaluated, one of which takes user fairnessin consideration. However, the system-level capacity of thesealgorithms is not investigated and their fairness is compared interms of the bit error rate (BER) difference between the bestand the worst user. In [17], a low complexity bisection-basedpower allocation algorithm, aiming to maximize the capacityof the SCMA system with a finite alphabet is proposed.A stochastic geometry framework to obtain the system-widearea spectral efficiency of underlaid and overlaid device-to-device (D2D) SCMA networks is developed in [18] and apower allocation strategy to minimize cross-tier interferencein underlaid mode and an optimal subcarrier allocation forthe overlaid mode are presented. In [19], a joint subcarrierand power allocation algorithm to maximize the proportionalfairness utility function of the downlink SCMA system isproposed. The subcarrier and power allocation are split intotwo problems. The power allocation problem is transformed toa convex equivalent and the remove-and-reallocate algorithmis proposed to solve the combinatorial subcarrier problem.A similar technique of convexification and alternating op-timization is employed in [20] to solve a SCMA resourceallocation problem taking into account content caching, energyharvesting and physical layer security.

Despite the extensive body of literature regarding the ana-lytical characterization and resource management in SCMA, afew core issues are yet to be properly addressed. Firstly, thenetwork overloading achieved by non-orthogonal schedulingalso results in an additional source of interference. Hence, it isfundamental to approach the resource allocation problem froma fairness perspective, as algorithms that maximize the sum-rate do so at the expense of users with poor channel condition.In this vein, one of the algorithms proposed in this paperfollows a fairness maximization path. Secondly, as shownin this manuscript, the joint subcarrier and power allocationproblem is NP-hard. The algorithms currently proposed in theliterature propose heuristics to achieve sub-optimal solutions.In face of that, we propose a more systematic approach byrelaxing the problem and following the BSUM framework.

To the best of our knowledge, no previous works haveinvestigated the fairness in joint subcarrier and power allo-cation in uplink SCMA transmission in depth. Furthermore,the algorithms proposed in this paper have stronger optimalityguarantees in comparison with algorithms proposed in previ-ous works.

B. Contributions

In this paper, we formulate two optimization problems forjoint subcarrier and power allocation in SCMA networks,one aiming to maximize the sum-rate and another one formaximizing the fairness and propose two algorithms to solvethem. The first algorithm’s goal is to maximize the sum-rateof the network. While this is an essential criterion in cellularnetworks, fairness between users is equally important. Thus,to include fairness in the optimization, we propose the Max-Min algorithm aimed to maximize the minimum rate amongthe users. The obtained results demonstrate better performancethan the former algorithm in terms of fairness, at the cost ofa lowered sum rate.

Both problems are of the non-convex mixed integer nonlin-ear programming (MINLP) type. We prove that both problemsare NP-hard 1. Then, we propose two algorithms based onthe BSUM framework, proposed in [1]. The proposed algo-rithms maximize a lower bound approximation of the objectivefunctions by updating the optimization variables in blocks. Asshown in [1], if the lower bound approximation satifies someconditions, this approach has guaranteed convergence to a sta-tionary point, assuring a locally optimal solution. Additionally,we compare both algorithms and the ones proposed in [16] inthe sum-rate and the Jain’s fairness index sense. Results showthat the Max-SR algorithm outperforms all other algorithmsregarding sum-rate, while the Max-Min algorithm outperformsall others regarding fairness. Furthermore, we evaluate thefairness and sum-rate performance of the algorithms underoutdated CSI. Finally, we compare the BER performance ofthe two proposed algorithms.

To summarize, the list below presents the main accomplish-ments in this work:• We prove that the joint power and subcarrier allocation

problem is NP-hard.• We propose a Max-SR algorithm which achieves a better

sum-rate in comparison to the ones proposed in [16].• We propose a Max-Min algorithm which achieves better

fairness, in terms of the Jain’s fairness index, in compar-ison to the ones proposed in [16].

• We evaluate the robustness of the algorithms againstoutdated channel state information (CSI)

This paper is organized as follows: Section II contains abrief overview of the SCMA encoder and decoder structure.Also, a description of SCMA signals, and the derivation of itssum-rate is presented. In Section III, the optimization problems

1As shown in [21], [22] for the energy-efficiency and heterogeneous cloudradio access networks in PD-NOMA networks respectively, this problems canbe reformulated as a monotonic optimization problem, and the optimal jointallocation can be found using the polyblock outer-approximation algorithm,albeit, the algorithm complexity grows exponentially with the size of theproblem.

3

are formulated, and an algorithm for sum-rate maximization,and, another for fairness maximization are proposed. Further-more, in Section IV numerical results are shown, and theperformance of the algorithm is evaluated. Also, a numericalanalysis of the convergence is presented. Finally, in Section Vthe conclusions are presented.

C. Notation

Throughout this paper, italic lowercase letters denote realand complex scalar values, and x∗ denotes the complexconjugate of x. Lower case boldface letters denote vectors,while upper case boldface denote matrices. A lowercase letterwith one subscript, xi , represents the i-th element of the vectorx, while both xi, j and [X]i, j are used to denote the element onthe i-th row and j-th column of matrix X. The operators xH

and XH denote the hermitian conjugate of a vector and of amatrix, respectively. The operator det(X) is the determinant ofthe square matrix X and tr(X) is its trace. The operator diag(x)denotes a square matrix with its diagonal components given byx. The operator E(·) denotes the expected value of a randomvariable. The function p(·) represents the probability densityfunction (PDF) of a random variable and x ∼ CN(µ,K), whereK ∈ Rn, denotes that x is a complex Gaussian random vector,with mean µ and covariance matrix K. The sets R, C and B arethe sets of the real, complex and binary numbers, respectively.A calligraphic uppercase letter, such as X, denotes a set and|X| is its cardinality. The function ln(·) denotes the naturallogarithm of its argument, while the function I(·; ·) is themutual information between two random variables.

II. SYSTEM MODEL

A. SCMA Overview

Consider a system consisting of one base station (BS), andlet K be the set of available resources (OFDMA subcarriers,MIMO spatial layers and so on), with |K | = K , and J bethe set of users served by the BS, with |J | = J. Each usertransmits a symbol from a multidimensional constellation withorder M . The SCMA encoder is a mapping f : Blog2(M) → Sj ,with sj = f (bj), where bj ∈ Blog2(M) is a vector of bits takenat the output of a channel encoder, Sj ⊂ CK , |Sj | = Mand sj is a sparse vector with N < K nonzero elements forall j ∈ J . Each user encodes its transmitted signal froma different multidimensional constellation Sj . Therefore, theBS serves up to J =

(KN

)users simultaneously and up to

df =(K−1N−1

)users are allocated on the same resource. The

overloading factor of the cell is given by λ = J/K . Figure 1shows an example of codebooks and a multiplexed codewordfor an SCMA system with K = 4, J = 6, N = 2 and df = 3. Inthis figure, each square represents a subcarrier and the differentcolors represents the codebook of a different user. The textureof the squares is a different symbol from the user’s motherconstellation, while the blank square indicates that no signalis transmitted at the subcarrier by the user. In the second row,we give an example of the resulting received signal, which isa superposition of the transmitted symbol by each user, for thetransmission of an arbitrary pair of bits by each user.

Optimal SCMA decoding is achieved by maximum a pos-teriori (MAP) decoding. However, due to the complexityof MAP decoding, message passing algorithms (MPA) thatachieve near-optimal decoding, such as belief propagation(BP) [23] are employed, resulting in a complexity of O(Md f ).In order to reduce the decoding complexity of SCMA, alter-native receiver architectures have been proposed, such as theSIC-MPA decoder [24], [25] which is a hybrid of the SICand MPA procedure, and the list spherical decoding (LSD)algorithm [26].

The structure of the SCMA code can be neatly conveyedthrough a factor graph representation. Let F ∈ BK×J bethe factor graph matrix, each element fk, j indicates if anyinformation from the user j is transmitted on resource k.Figure 2 illustrates a factor graph with J = 6, K = 4, N = 2and df = 3 corresponding to the codebook shown in Figure1, where the circular vertices represent each user, the squaredvertices denote the resources and the edges between them theallocation of a user to specific resources. The reader may referto [8] for more details on the encoder/decoder structure ofSCMA.

In a SCMA system, the signal received by the BS at theresource k can be written as

yk =∑j∈J

fk, jhk, j sk, j + nk, (1)

where hk, j is the channel coefficient, sk, j is the symboltransmitted from user j on the k-th resource, with averagepower pk, j = E(|sk, j |2), and nk is the k-th component ofn ∼ CN(0, σ2

nI). Here, we assume that hk, j =gk, j√1+rαj

. Without

loss of generality, we assume gk, j is a Rayleigh distributedrandom variable representing the small scale fading of thechannel of user j on subcarrier k, rj is the distance of userj from the BS and α is the path loss exponent. Throughoutthis work, it is assumed that the users send a pilot sequenceperiodically, and, the BS is able to perfectly estimate the CSI.From (1), the received signal vector at the BS is written as

y = Hx + n, (2)

where y ∈ CK is a complex vector, H ∈ CK×KJ is a matrixcomposed of submatrices, such that, H = [H1,H2, · · · ,HJ ],where, Hj = diag([h1, j, h2, j, · · · hK, j]T ) ∀ j ∈ J . The vectorx ∈ CKJ is given by x =

[xT1 xT2 · · · xTJ

]T , where xj =[f1, j s1, j f2, j s2, j · · · fK, j sK, j

]T ∀ j ∈ J .In this paper, we consider a centralized resource allocation

architecture, where K users periodically transmit a pilot signalto the BS. We assume the BS obtains perfect CSI, solves theoptimization problem described in Section III, and tells eachuser which subcarriers, power, and code-rate to use for thenext period of time. We assume the channels are quasi-static,so that users can encode at a fixed rate for a period of time.The process is repeated periodically, where the allocations arechanged, and the BS tells users to change their transmissionaccordingly.

The sum-rate of a SCMA system is defined as the maximummutual information between the received and transmitted sig-

4

Fig. 1. Example of an SCMA uplink system with J = 6, K = 4, N = 2 and d f = 3. The square arrays demonstrate the codebook of each user and eachsquare represent the available resource elements (RE). An empty square indicates that no signal is transmitted in the RE and different filling patterns indicatea different complex value.

Fig. 2. Example factor graph with J = 6, K = 4, N = 2 and d f = 3. Thecircles denote user nodes and the squares denote resource nodes.

nals. Therefore, assuming channel knowledge at the receiverwe have

RsumSCMA = max

p(x)I(x; y|H = H′)

= maxp(x)

h(y|H = H′) − h(n)

(a)≤ ln[(πe)K det(σ2

nIK +H′KxH′H )] − K ln[πeσ2n]

= ln[det

(IK +

1σ2n

H′KxH′H)]. (3)

In (3), the inequality in (a) follows since a Gaussian inputmaximizes the entropy of a random vector, under a covarianceconstraint [27]. In this paper, we are concerned with maxi-mizing this upper bound in the Max-SR algorithm which isreferred henceforth as CSCMA. It is worth noting that for anincrease on df the distribution of y approaches a multivariateGaussian, due to the central limit theorem. Furthermore,Kx ∈∈ CKJ×KJ is the covariance matrix of x and is givenby

Kx =

E(x1xH

1 ) E(x1xH2 ) · · · E(x1xH

J )...

.... . .

...E(xJxH

1 ) E(xJxH2 ) · · · E(xJxH

J )

, (4)

where, each E(xixHj ) is given by

E(xixHj ) =

E

(x1,i x

∗1, j

)E

(x1,i x

∗2, j

)· · · E

(x1,i x

∗K, j

)...

.... . .

...

E(xK,i x

∗1, j

)E

(xK,i x

∗2, j

)· · · E

(xK,i x

∗K, j

).

(5)As concluded in [15], if each nonzero coordinate of s is drawnfrom centrally symmetric constellations, the cross correlation

between the multidimensional symbols from different usersis equal to zero, hence, E(xixH

j ) is equal to a K × K zeromatrix for any i , j. Thus, (4) is diagonal. For a moregeneric derivation, not relying on the central symmetry of theconstellation the reader may refer to [28]–[30].

As Hj is diagonal for all values of j, we have

HKxHH =

J∑j=1

HjE(xjx

Hj

)HH

j

=

J∑j=1

h1, jE(x1, j x∗1, j

)h∗1, j · · · 0

.... . .

...

0 · · ·J∑j=1

hK, jE(xK, j x∗K, j

)h∗K, j

(6)

It is worth noting that such codebook satisfies the designprinciples established in [9]. With that being said, the SCMAsum-rate in (3) can be simplified as shown in (7), at the topof the next page. Furthermore, by assuming a decoding orderstarting from user J to user 1 and using the logarithm identitylogb(a + c) = logb(a) + logb(1 + c

a ), it is possible to obtain theachievable rate of user j on resource k, Ck, j , as shown in (8),at the top of the next page. Therefore, the achievable rate ofeach user, Cj , is given by

Cj =∑k∈K

Ck, j =∑k∈K

ln©­­­­«1 +

|hk, j |2 fk, jpk, j

σ2n +

j−1∑i=1|hk,i |2 fk,ipk,i

ª®®®®¬. (9)

In the next section, we formulate the joint subcarrier and powerallocation problems and present our proposed method to solvethem.

III. JOINT SUBCARRIER AND POWER ALLOCATION

we formulate and propose two joint subcarrier and powerallocation algorithms to solve two optimization problems:maximizing the sum-rate (PMax-SR) and maximizing the min-

5

CSCMA = ln det(IK +

1σ2n

HKxHH

)= ln det

1 +

∑j∈J|h1, j |2E(x2

1, j )

σ2n

· · · 0...

. . ....

0 · · · 1 +

∑j∈J|hK, j |2E(x2

K, j )

σ2n

=

∑k∈K

ln©­­«1 +

∑j∈J|hk, j |2E(x2

k, j)

σ2n

ª®®¬ =∑k∈K

ln©­­«1 +

∑j∈J|hk, j |2 fk, jpk, j

σ2n

ª®®¬ . (7)

ln©­­«1 +

∑j∈J|hk, j |2 fk, jpk, j

σ2n

ª®®¬ = ln©­­«σ2n +

∑j∈J|hk, j |2 fk, jpk, j

σ2n

ª®®¬= ln

(σ2n + |hk,1 |2 fk,1pk,1

σ2n

)+ ln

(σ2n + |hk,1 |2 fk,1pk,1 + |hk,2 |2 fk,2pk,2

σ2n + |hk,1 |2 fk,1pk,1

)+ · · · + ln

©­­­­«σ2n +

J−1∑i=1|hk,i |2 fk,ipk,i + |hk,J |2 fk,J pk,J

σ2n +

J−1∑i=1|hk,i |2 fk,ipk,i

ª®®®®¬=

∑j∈J

ln©­­­­«1 +

|hk, j |2 fk, jpk, j

σ2n +

j−1∑i=1|hk,i |2 fk,ipk,i

ª®®®®¬=

∑j∈J

Ck, j . (8)

imum rate of users (PMax-Min). The former can be formulatedas

PMax-SR :

maxP,F

CSCMA =∑k∈K

ln©­­«1 +

∑j∈J|hk, j |2 fk, jpk, j

σ2n

ª®®¬ (10)

s.t.∑k∈K

fk, j ≤ N ∀ j ∈ J (11)∑j∈J

fk, j ≤ df ∀ k ∈ K (12)∑k∈K

fk, jpk, j ≤ P(j)max ∀ j ∈ J (13)

fk, j ∈ {0, 1} ∀ k ∈ K and ∀ j ∈ J, (14)

where P ∈ RK×J is the matrix of allocated power, (10) isthe sum-rate, and (11) is the constraint on the number ofsubcarriers allocated per user. The constraint on the numberof users per subcarrier is given by (12), while (13) is theconstraint on the maximum transmitting power available peruser. Finally, (14) is a binary constraint on the values of fk, j .Furthermore, the problem PMax-Min is formulated as

PMax-Min :

maxP,F

minj∈J

∑k∈K

ln©­­­­«1 +

|hk, j |2 fk, jpk, j

σ2n +

j−1∑i=1|hk,i |2 fk,ipk,i

ª®®®®¬(15)

s.t. (11), (12), (13), (14),

where (15) is the max-min utility function of the rate peruser. The objective function of this problem is non-concaveand, similarly to PMax-SR, also has integer constraints on F.Consequently, we can prove the following statement

Theorem 1. Both the PMax-SR and the PMax-Min problems areNP-hard.

Proof. See Appendix A. �

In order to solve both these problems, we relax the integerconstraint on matrix F, given in equation (14), to a continuousone. Afterwards, we apply the block successive lower boundmaximization (BSLM), which is the maximization variant ofthe approach proposed in [1], which converges to a localminimum of the relaxed problem [1].

For the sake of clarity, we list the conditions for the con-vergence of the BSLM algorithm. First consider the problembelow

maxx

f (x)s.t. x ∈ X,

where f : Rn → R is a non-concave and possibly non-smooth function. Let X be the cartesian product of m closedconvex sets: X = X1 × · · · × Xm, with Xi ⊆ Rni and∑i

ni = n. Furthermore the optimization variable x ∈ Rn can

be decomposed into m vectors x = (x1, · · · , xm), such thatxi ∈ Xi . At iteration t of the BSLM algorithm, the blocks of

6

optimization variables are updated cyclically, where for eachblock the following problem is solved

maxxi

f (xi, x(t−1))

s.t. xi ∈ Xi,

where i = t mod m, and x(t−1) is the previous value of x.The convergence of the BLSM algorithm is guaranteed if thefollowing conditions hold for f (xi, x(t−1)):

f (xi, x) = f (x), ∀ x ∈ X, ∀ i (16)f (xi, y) ≤ f (y1, · · · , yi−1, xi, yi+1, · · · , ym),

∀ xi ∈ Xi, ∀ y ∈ X, ∀ i (17)∇ f (xi, x) = ∇ f (x) (18)

f (xi, y) is continuous in (xi, y), ∀ i (19)

In the rest of this section, we propose lower bound convexapproximations to the objective functions of both PMax-SRand PMax-Min satisfying conditions (16)-(19), and finalize byproviding a description of the block update algorithm thatconverges to locally optimal solutions.

A. Solving PMax-SR

To solve this problem, we first relax the integer constraintin (14), and add a penalty term to the objective function, suchthat, non-integer solutions to F are penalized. The PMax-SRbecomes

P′Max-SR :

maxP,F

∑k∈K

ln©­­«1 +

∑j∈J|hk, j |2 fk, jpk, j)

σ2n

ª®®¬ + γ (F) (20)

s.t. 0 ≤ fk, j ≤ 1 ∀ k ∈ K and ∀ j ∈ J (21)(11), (12), (13) (22)

where γ (F) = λ ∑k∈K

∑j∈J

(f 2k, j− fk, j

)is the penalty function 2.

Notice that γ (F) < 0 for all non-integer solutions and γ (F) =0 for integer ones. This gives incentive for the algorithm toobtain solutions that minimize γ (F), hence leading to integersolutions of F.

There are two issues that make P′Max-SR a hard problem tosolve:• The presence of multi-linear terms of the form fk, jpk, j in

(20) and (13).• Even if F and P are updated cyclically, the objective

function in (20) is non-concave, due to the addition ofγ (F), which is convex.

Now we present two Lemmas that are instrumental to thealgorithm that finds a locally optimal solution to P′Max-SR inpolynomial time.

2By selecting moderately high values for λ (around 20), integer solutionsare obtained. As a matter of fact, higher values for λ results in fasterconvergence to an integer solution, however, it renders the optimization solveriterations more unstable as it contributes to the ill-conditioning of the problem.On the other hand, smaller λ leads to more conservative updates of F at eachBSLM step, resulting in slower convergence, but better optimizers.

Lemma 1. If F and P are updated cyclically in P′Max-SR,the feasible set of the problem solved in each update stepis convex.

Proof. All constraints in P′Max-SR are linear functions of F andP, with the exception of constraint (13) which involves a multi-linear term. However, if F and P are updated cyclically, onlyone of the matrices is updated while the other is kept constantand the multi-linear terms in (13) become linear in the variablebeing updated. Therefore, the feasible sets are convex. �

Lemma 2. Let the function∑k∈K

ln©­­«1 +

∑j∈J|hk, j |2 fk, jpk, j

σ2n

ª®®¬ +γ (F′) + tr

[∇γ (F′)T (F − F′)

], (23)

where ∇γ (F′) ∈ RK×J is a matrix such that

[∇γ (F′)]k, j =∂γ (F)∂ fk, j

����F=F′

, (24)

be an approximation to (20) in the neighborhood of F′ for fixedP. Notice that (23) is a lower bound concave approximationto (20) satisfying conditions (16)-(19).

Proof. Firstly, notice that γ (F′)+ tr[∇γ (F′)T (F − F′)

]is the

first order linear approximation of γ (F) in the neighborhoodof F′. So (16), (18), and (19) are satisfied. Furthermore, asγ (F) is a convex function, we have

γ(F) ≥ γ (F′) + tr[∇γ (F′)T (F − F′)

].

As the linear approximation is globally less than γ (F), wehave that (23) is a lower bound of (10). Thus, (16) is alsosatisfied. �

With Lemmas 1 and 2 in hand, we can derive the conver-gence of the relaxed problem P′Max-SR to a local optimum, asstated in the Theorem below.

Theorem 2. By updating F and P cyclically with the solutionsto P(F)Max-SR and P(P)Max-SR presented below, we can obtain alocally optimal solution to the relaxed problem P′Max-SR.

P(P)Max-SR : maxP

(10) s.t. (13),

P(F)Max-SR : maxF

(23) s.t. (11), (12), (13), (21),

where F′ = F(t−1), i.e the previous value of F.

Proof. From Lemma 1, the feasible set of both P(F)Max-SR andP(P)Max-SR are convex. Moreover, from Lemma 2, we have that(23) is a concave lower bound approximation to (20) satisfyingthe conditions in (16)-(19). Therefore, from the result shown inTheorem 2 in [1], the solution obtained by iteratively updatingF and P cyclically is a local optimum of P′Max-SR. �

As problems P(F)Max-SR and P(P)Max-SR are concave maximiza-tions over a convex set and are readily solvable. Algorithm1 shows the pseudocode of the Max-SR algorithm, usingP(F)Max-SR and P(P)Max-SR as subroutines.

7

Algorithm 1: Maximization of sum-rateVariable Definition1. F(t) is the subcarrier allocation matrix at the t-th

iteration.2. P(t) is the power allocation matrix at the t-th iteration.Initialization1. Set the initial values for the power allocation matrix P(0)

randomly, within the set defined by constraint (13), and,the subcarrier allocation matrix F(0) within the setdefined by constraints (11), (12) and (21).

2. Set the convergence tolerance for the subcarrierallocation εF and for the power allocation εP .

3. t ← 0Output1. Optimized power allocation P∗.2. Optimized subcarrier allocation F∗.while

F(t) − F(t−1) > εF and P(t) − P(t−1) > εP do

t ← t + 1;F(t) ← arg P(F)Max-SR

(F(t−1),P(t−1)

); (see Theorem 2)

P(t) ← arg P(P)Max-SR

(F(t−1),P(t−1)

); (see Theorem 2)

endP∗ ← P(t);F∗ ← F(t);

B. Solving PMax-Min

Before solving the problem, notice that its objective functionin (15) can be rewritten as

minj∈J

∑k∈K

ln©­­­­«1 +

|hk, j |2 fk, jpk, j

σ2n +

j−1∑i=1|hk,i |2 fk,ipk,i

ª®®®®¬= min

j∈J

[∑k∈K

ln

(σ2n +

j∑i=1|hk,i |2 fk,ipk,i

)−

ln

(σ2n +

j−1∑i=1|hk,i |2 fk,ipk,i

)]. (25)

Both expressions inside the minimum function,

ln(σ2n +

j∑i=1|hk,i |2 fk,ipk,i

)and ln

(σ2n +

j−1∑i=1|hk,i |2 fk,ipk,i

)are concave functions, which implies that their difference isnon-concave. The summation of non-concave functions isalso non-concave, and, by function composition rules [31],the minimum of a non-concave function is non-concave aswell.

Similarly to PMax-SR, the first step in solving PMax-Min isrelaxing the integer constraint on the entries of F and addingthe same penalty function to its objective, leading to problem

P′Max-Min, given below

P′Max-Min :

maxP,F

minj∈J

[∑k∈K

ln

(σ2n +

j∑i=1|hk,i |2 fk,ipk,i

)−

ln

(σ2n +

j−1∑i=1|hk,i |2 fk,ipk,i

)]+ γ (F) (26)

s.t. (11), (12), (13), (21).

After the relaxation, we have the non-concave term from(25) and the convex penalty function in the objective. Thechallenges involved in solving P′Max-Min are

• Just as in P′Max-SR, the presence of multi-linear terms ofthe form fk, jpk, j in (26) and (13).

• Even if F and P are updated cyclically, the objectivefunction in (20) is non-concave.

To address these issues, we follow the procedure used to solveP′Max-SR, with a few extra steps. Firstly, notice that the feasiblesets of P′Max-SR and P′Max-Min are the same, and thereforeLemma 1 also holds. We introduce a concave lower boundto (26) in the Lemma below.

Lemma 3. Let θ j (F) =∑

k∈Kln

(σ2n +

j−1∑i=1|hk,i |2 fk,ipk,i

). The

function

minj∈J

[∑k∈K

ln

(σ2n +

j∑i=1|hk,i |2 fk,ipk,i

)−

θ j (F′) − tr(∇θ j (F′)T (F − F′)

)]+

γ (F′) + tr[∇γ (F′)T (F − F′)

], (27)

where ∇θ j (F′) ∈ RK×J is a matrix such that[∇θ j (F′)

]k,n

=∂θ j (F)∂ fk,n

����F=F′

=

|hk,n |2pk,n

σ2n+

j−1∑i=1|hk, i |2 f ′k, i pk, i

, ∀ n < j

0 , otherwise,

is a lower bound concave approximation of (26) for fixed P inthe neighborhood of F′ which satisfies conditions (16)-(19).

Proof. Firstly, as θ j (F) is a concave function we have that

θ j (F) ≤ θ j (F′) − tr(∇θ j (F′)T (F − F′)

)Therefore, the argument of the minimum function in (27)is less than the argument of the minimum function in (26),which implies that (17) holds. Furthermore, (27) is obtainedby approximating the non concave terms in (26) by their firstorder linear approximation in the neighborhood of F′ and (27)is continuous, and hence (16), (18), and (19) also hold. Finally,

8

(27) consists of the minimum of the summation over a concavefunction plus an affine term. So,∑

k∈Kln

(σ2n +

j∑i=1|hk,i |2 fk,ipk,i

)−

θ j (F′) − tr(∇θ j (F′)T (F − F′)

),

is concave. By composition rules [31], the minimum ofconcave functions is also concave, which proves that (27) isconcave, and coimpletes the proof. �

Lemma 4. Let θ j (P) =∑

k∈Kln

(σ2n +

j−1∑i=1|hk,i |2 fk,ipk,i

). The

function

minj∈J

[∑k∈K

ln

(σ2n +

j∑i=1|hk,i |2 fk,ipk,i

)−

θ j (P′) − tr(∇θ j (P′)T (P − P′)

)]+

γ (P′) + tr[∇γ (P′)T (P − P′)

], (28)

where ∇θ j (P′) ∈ RK×J is a matrix such that[∇θ j (P′)

]k,n

=∂θ j (P)∂pk,n

����P=P′

=

|hk,n |2 fk,n

σ2n+

j−1∑i=1|hk, i |2 f ′k, i pk, i

, ∀ n < j

0 , otherwise,

is a lower bound concave approximation of (26) for fixed F inthe neighborhood of P′ which satisfies conditions (16)-(19).

Proof. See the proof of Lemma 3. �

With the results from Lemmas 1, 3 and 4 we can establishthe convergence to a local optimum of an algorithm to solveP′Max-Min.

Theorem 3. By updating F and P cyclically with the solutionsto P(F)Max-Min and P(P)Max-Min presented below, we can obtain alocally optimal solution to the relaxed problem P′Max-Min.

P(P)Max-Min : maxP

(28) s.t. (13),

where P′ is the value of P after the previous update.

P(F)Max-Min : maxF

(27) s.t. (11), (12), (13), (21),

where F′ is the value of F after the previous update.

Proof. From Lemma 1, the domains of P(F)Max-SR and P(P)Max-SRare convex. Also, from Lemmas 3 and 4, we have that (27)and (28) are concave lower bound approximations to (20)satisfying the conditions in (16)-(19). Therefore, from theresult in [1], the solution obtained by iteratively updating Fand P cyclically is a local optimum of P′Max-SR. �

The complete algorithm to solve the Min-Max problem,PMax-Min, is described in Algorithm 2. The algorithm usesP(F)Max-SR and P(P)Max-SR as subroutines.

Algorithm 2: Fairness MaximizationVariable Definition1. F(t) is the subcarrier allocation matrix at the t-th

iteration.2. P(t) is the power allocation matrix at the t-th iteration.Initialization1. Set the initial values for the power allocation matrix P(0)

randomly, within the set defined by constraint (13), and,the subcarrier allocation matrix F(0) within the setdefined by constraints (11), (12) and (21).

2. Set the convergence tolerance for the subcarrierallocation εF and for the power allocation εP .

3. t ← 0Output1. Optimized power allocation P∗.2. Optimized subcarrier allocation F∗.while

F(t) − F(t−1) > εF and P(t) − P(t−1) > εP do

t ← t + 1;F(t) ← arg P(F)Max-Min

(F(t−1),P(t−1)

); (see Theorem 3)

P(t) ← arg P(P)Max-Min

(F(t−1),P(t−1)

); (see Theorem 3)

endP∗ ← P(t);F∗ ← F(t);

IV. NUMERICAL RESULTS

In this section the performance of the algorithms proposedin Section III is presented. Additionally, we compare our re-sults with the three algorithms proposed in [16]: the fixed userorder (FUO), opportunistic allocation (OA) and proportionalfair (PF) algorithms. In these three algorithms, the users aresorted according to a different criteria, and the resources areallocated sequentially by picking the best available resourcefor the user in the sorted order. The FUO algorithm performsthe allocation in a random order, while the OA algorithm sortsthe users according to their overall channel qualities, prior tothe channel allocation. The PF algorithm takes into accountthe L past channel qualities when sorting the allocation orderin order to improve fairness. In our evaluation we considerL = 10 and α = 0.9.

We consider a scenario where one BS is serving 6 usersover 4 subcarriers, with N = 2 and df = 3, in a circular cellof radius R = 300 m and the users are uniformly distributedinside the cell. It is worth mentioning that an increase in Nwould result in higher diversity, as each user would transmitits signal on more subcarriers. However, as the value of df istied to N (i.e. df =

(K−1N−1

)), df would also increase, resulting

in an exponential increase in the decoding complexity. Weconsider a path loss exponent α = 4. We consider a noisepower density of −174 dBm/Hz and a bandwidth of 180 kHz.Also, we consider a normalized slow fading Rayleigh channel,such that the channel remains constant for the duration of eachtransmitted symbol. Furthermore, we simulate the algorithms’performance for a maximum transmit power per user varyingbetween 3 dBm and 10 dBm. We evaluate the performancesaccording to two metrics: the sum-rate and the Jain’s fairness

9

Fig. 3. Sum-rate comparison for J = 6, K = 4, d f = 3 and N = 2.

index [32]. The former, is a measure of the overall achievablethroughput of the network and the latter is a measure of thefairness of the resource allocation between the users basedon their individual achievable throughputs. Let c ∈ RJ be avector, such that, its i-th coordinate, ci , corresponds to thethroughput of the i-th user. The Jain’s fairness index for agiven rate vector, c = [c1, · · · , rJ ]T , is

J(c) =

(J∑j=1

cj

)2

JJ∑j=1

c2j

. (29)

This index varies from 1J , meaning no fairness, to 1, mean-

ing perfect fairness. Furthermore, we consider a normalizedRayleigh fading channel, and the performances are averagedover several channel realizations.

To implement the proposed algorithms, we used the convexoptimization modeling language CVXPY [33], [34], withthe open-source ECOS solver [35], due to its support ofexponential cones [36]. The comparison of the sum-rate ofthe five algorithms is shown in Figure 3. The sum-rate ofthe Max-SR algorithm outperforms the Max-Min, FUO, OA,and PF algorithms for the whole range of transmitted powerevaluated. The three algorithms proposed in [16] present sim-ilar performances with the proportional fairness being slightlyworse than the other two. Finally, the Max-Min algorithm isgreatly outperformed by the other ones. This result is expectedsince the Max-Min gives up on maximizing the sum-rate infavor of improving the fairness.

Figure 4 shows the Jain’s fairness index achieved by eachalgorithm. The Max-Min algorithm greatly outperforms thealternatives for the whole range of transmitted powers. It isworth noting that the Jain’s fairness index is bottleneckedby the rate of the user with the worst channel. Therefore,an increase in the maximum transmit power results in ahigher throughput for the worst user, consequently, increasing

Fig. 4. Jain’s fairness index comparison for J = 6, K = 4, d f = 3 andN = 2.

the overall fairness. Furthermore, with increasing maximumtransmit power, the fairness of the Max-Min algorithm ap-proaches one. The other algorithms achieve similar fairnessperformance, with the PF algorithm achieving slightly betterfairness than the others.

In order to evaluate the link level performance of the Max-SR and Max-Min allocation, another simulation, evaluatingthe BER of both allocation algorithms, is presented. In thesimulation, each user transmits a multidimensional symbolwith M = 4 using a quadrature amplitude modulation (QAM)mother constellation [8]. We assume that the users’ channelgains are ordered, such that, ‖H1‖F ≤ ‖H2‖F ≤ · · · ≤ ‖HJ ‖F ,i.e., the first user has the worst channel gain, while the last userhas the best one. Figure 5, shows a comparison between theBER of the 6 users using Max-SR and Max-Min allocation.As expected, with Max-SR allocation, the error probabilitiesare small, but there is a large gap between the best and theworst user. On the other hand, Max-Min allocation results inoverall larger error probabilities when compared to the Max-SR, but the performance gap between users with differentchannel quality is smaller.

A. Performance with Outdated CSI

In this section, we investigate the effect on the performanceof the algorithms, under an outdated CSI regime. In the resultsshown so far, we considered that for every new channel real-ization the users would send a pilot sequence to the BS, whowould run the optimization routine and send the allocationsback to the users. This approach requires a large overhead asit requires pilots and allocations to be sent constantly betweenthe users and the BS. Hence, we consider a system where thepilots are sent periodically with period T , and the users reusethe same allocation during the period, as illustrated in Figure6.

In this experiment, we model the temporal relationshipbetween two successive channel realizations as an i.i.d first-

10

Fig. 5. BER comparison for J = 6, K = 4, d f = 3 and N = 2.

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Fig. 6. Resource allocation procedure

order Gauss-Markov process [37], for each k ∈ K and j ∈ J ,given by

h(n+1)k, j

= ρh(n+1)k, j

+ wk, j, (30)

where h(n)k, j∼ CN(0, 1) and w

(t)k, j∼ CN(0, 1 − ρ2) is the

innovation component. Moreover The correlation betweensuccessive fading components is given by

ρ = J0 (2π fmaxTs) , (31)

where fmax is the maximum Doppler frequency, Ts is thetime between channel updates, and J0 is the Bessel function.Figure 7 shows the performance deterioration for T ∈ [1, 50],Pmax = 10 dBm, TS = 0.01 s, and four values of fmax resultingin ρ2 ∈ {0.95, 0.62, 0.22, 0.01}. Tables I and II summarizesthe effects of outdated CSI in the performance of Max-SRand Max-Min algorithms respectively. From the table resultswe conclude that the Max-SR algorithm is more robust tooutdated CSI as for the worst case (T = 50 and ρ2 = 0.01), itstill achieves 85% of the sum-rate with T = 1, while the Max-Min only achieves 75% of the original fairness. Moreover,as in the proposed model the same allocation is reused forsubsequent transmissions, and the less correlated the current

Fig. 7. Performance of the Max-SR (solid line) and Max-Min (dashed line)allocations for T ∈ [1, 50] and different values of fmax.

TABLE IPERFORMANCE DETERIORATION WITH OUTDATED CSI (MAX-SR)

Fairness Sum-Rateρ2 = 0.95 102.38% 98.81%ρ2 = 0.62 96.45% 87.88%ρ2 = 0.22 96.47% 85.85%ρ2 = 0.01 94.11% 85.89%

TABLE IIPERFORMANCE DETERIORATION WITH OUTDATED CSI (MAX-MIN)

Fairness Sum-Rateρ2 = 0.95 83.58% 119.73%ρ2 = 0.62 77.22% 114.51%ρ2 = 0.22 76.66% 112.38%ρ2 = 0.01 74.99% 115.34%

CSI is with the one used to obtain the allocation (i.e. longerperiods between updates), the more it resembles a randomallocation, resulting in a performance decrease. For instance,the Max-Min algorithm maximizes the fairness of the system,hence, its fairness index decreases with longer periods. On theother hand, we observe in Figure 7 that its sum-rate increaseswith longer periods, thus, we conclude that the Max-Minalgorithm increases fairness at the expense of the sum-rate.The same does not happen in the Max-SR algorithm as thefairness remains roughly the same under the effect of outdatedCSI.

B. Convergence Analysis and Algorithm Complexity

In this section, the convergence rate of the proposed al-gorithms and their complexity is investigated. The stopping

11

Fig. 8. Convergence of the PMax-SR, with maximum transmit power of 10dBm, algorithm for three different initial conditions.

Fig. 9. Convergence of the PMax-Min algorithm, with maximum transmitpower of 10 dBm, for three different initial conditions.

criteria for the procedure is based on the difference betweensuccessive updates of the optimization variables F and P.When this difference falls below a threshold, εF and εP ,respectively, the algorithm has converged. In order to illustratethe convergence, Algorithms 1 and 2 are simulated withP(j)max = 10 dBm ∀ j ∈ J , with the same channel gains andusers location, but with three randomly chosen initial condi-tions (F(0),P(0)). Figure 8 shows the convergence of Algorithm1. Each iteration in Figure 8 consists of a full cycle of updates.In Figure 9, the convergence of Algorithm 2 is shown. Thealgorithm converges in five steps or less for all three initialconditions. Furthermore, Algorithm 1 consistently convergesto a higher sum-rate than Algorithm 2, while Algorithm 2converges to a higher fairness index as expected.

TABLE IIICOMPLEXITY OF THE MAX-SR AND MAX-MIN ALGORITHMS

Procedure Number of Steps

Subcarrier Allocation Update O(√

2J + K + KJ log2

(1ε

))Power Allocation Update O

(√J log2

(1ε

))Total Number of Updates O

(1ε

)

1) Algorithm Complexity: Each update of the Max-SR algo-rithm involves solving a convex optimization problem, namelyP(F)Max-SR for the subcarrier allocation update and P(P)Max-SR forthe power allocation update. We consider that both updatesteps are solved using a primal-dual interior point algorithm[38] with a logarithmic barrier function.

Definition 1 (ε-optimal solution). Let

maxx

f (x) s.t. x ∈ X,

where f (x) is a concave function and X is a convex set be anoptimization problem such that x∗ is the unique maximizer tothe problem. The vector x′ ∈ X is an ε-optimal solution to theproblem if

f (x∗) − f (x′) ≤ ε . (32)

As derived in [39], the number of steps required to obtainan ε-optimal solution to a convex optimization problem usingthe interior point algorithm with logarithmic barrier functionis O

(√n log2

(1ε

)), where n is the number of inequality

constraints. The problems solved in the update steps P(F)Max-SRand P(P)Max-SR have 2J + K + K J and J inequality constraints,respectively. Therefore, the complexity of each subcarrierallocation update is O

(√2J + K + K J log2

(1ε

))and the com-

plexity of each power allocation update is O(√

J log2

(1ε

)).

In order to establish the total number of iterations requiredfor the convergence of the algorithm, we use the result fromTheorem 3.1 in [40]. Let

maxx

f (x) s.t. x ∈ X,

where f (x) might be non-concave and non-smooth, be ageneric non-convex optimization problem, and x∗ ∈ X∗, whereX∗ is the set of the problem’s stationary points. Then, theoptimality gap after the t-th cyclic update of the BLSMalgorithm is given by

∆t = f (x∗) − f(x(t)

)≤ c

t, (33)

where c is a constant. Therefore, to obtain an ε-optimal solu-tion, O

(1ε

)update steps are necessary. The same results apply

to the Max-Min algorithm as the same number of constraintsare involved to solve the power allocation and subcarrierallocation update problems. Both algorithms complexity aresummarized on Table III.

12

V. CONCLUSIONS

In this paper, two joint channel and power allocation al-gorithms are proposed: the Max-SR and the Max-Min algo-rithms. The former aims for sum-rate maximization, while thelatter aims for maximizing fairness. The BSUM framework isemployed to obtain algorithms converging to locally optimalpoints of the relaxed problems. We compare the performanceof the Max-SR and the Max-Min algorithms with the onesproposed in [16]. The results show that the Max-SR algorithmhas better performance on the sum-rate sense, while the Max-Min has better performance on the fairness sense. Moreover,a numerical analysis of the convergence of the algorithms ispresented. Finally, we derive the worst-case time complexityof both algorithms. The results show that the Max-SR consis-tently achieve a better sum-rate, while the Max-Min achievesbetter fairness. Furthermore, there is a tradeoff between thefairness and the sum-rate. For future works, we intend toinvestigate new algorithms that can achieve a compromisebetween sum-rate and fairness.

APPENDIX APROOF OF THEOREM 1

To prove that both problems are NP-hard, we show thatthe subcarrier assignment subproblem can be reduced inpolynomial time to the hypergraph assignment problem (HAP)which is shown to be NP-hard in [41].

We start by briefly introducing the HAP. The HAP takes asinput a bipartite graph G = V ∪ U such that V ∩ U = ∅,a set of hyperedges E, and a cost function c : E → R. Inaddition to that, we have that |e ∩V| ≥ 1 and |e ∩U| ≥1 ∀ e ∈ E. A hyperassignment is a set H ⊆ E suchthat every element v ∈ V ∪ U appears exactly once inH . The output of the HAP is an optimal hyperassignmentH ∗ = min{c(H)|H is an hyperassignment of G}.

Now consider the problem PMax-SR in the HAP context, witha fixed power allocation matrix P. Let U = {SC1, · · · , SCK }and V = {U1,1 · · ·U1,N,U2,1, · · ·U2,N, · · ·UJ,N } be the vertexset of available subcarriers and allocated subcarriers, respec-tively. Notice that SCk denotes the k-th available subcarrier,while Uj,i denotes the i-th subcarrier allocated to user j, with1 ≤ i ≤ N . In this context, a hyperassignment determineswhich subcarriers are allocated to each user. For instance, ahyperedge e = {SC1,U1,1,U2,2} indicates that users 1 and 2are allocated to the first subcarrier. As at most N subcarrierscan be allocated to each user, every hyperassignment satisfies(11). Notice that the subcarrier allocation matrix F correspondsto the hyperedge incidence matrix of the hypergraph, forinstance, the hyperedge e = {SC1,U1,1,U2,2} would result inf1 =

[1 1 0 · · · 0

], where f1 is the first row of F. Fur-

thermore, consider the hyperedge set E = {e ∈ 2U∪V | 1 <|e| ≤ df + 1} and let FH ∈ BK×J be the incidence matrix of ahypermatch H ⊆ E. As |e| ≤ df + 1 ∀ e ∈ E constraint (12)is always satisfied. Furthermore, the cost function is given as

c(H) =∑k∈K

ln©­­«1 +

∑j∈J|hk, j |2 fk, jpk, j

σ2n

ª®®¬ .

So, solving PMax-SR with P fixed, is equivalent to solving anHAP, therefore, PMax-SR is NP-hard.

Finally, we also conclude PMax-Min is NP-hard using thesame HAP formulation, but using the cost function

c(H) = minj∈J

∑k∈K

ln©­­­­«1 +

|hk, j |2 fk, jpk, j

σ2n +

j−1∑i=1|hk,i |2 fk,ipk,i

ª®®®®¬.

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