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Relay Selection for Multi-Channel CooperativeMulticast: Lexicographic Max-Min Optimization

Yitu Wang, Student Member, Wei Wang, Senior Member, IEEE, Lin Chen, Member, IEEE,Pan Zhou, Member, IEEE, Zhaoyang Zhang, Member, IEEE

Abstract—Cooperative multicast has been demonstrated toachieve significant performance gain over the classic source-destination transmission paradigm by exploiting spatial diversitythrough the participation of multiple relay nodes. As a ma-jor technical challenge, the selection of relays for a multicastsession has significant impact on the multicast performance.The challenge is even more pronounced when the number ofchannels are limited as the relay selection is in this contextcoupled with channel allocation. The goal of this paper is todesign a fair multicast relay selection scheme with limited channelresources. Specifically, we establish an analytical frameworkfor this joint relay selection and channel allocation problemand develop a lexicographic max-min multicast relay selectionscheme. Our design consists of two technical steps. 1) We considerthe maximization of the minimal data rate. By decoupling relayselection and channel allocation, the problem is transformed to amax-min-max problem, which is difficult to solve. To make thisproblem tractable, we reformulate it as a convex optimizationproblem via relaxation and smoothing, and prove the asymptoticequivalence from a geometrical perspective. 2) We propose anadjustment algorithm based on the initial max-min solution, andprove that the proposed scheme achieves lexicographic optimality.Finally, our proposed algorithm is evaluated by simulation toshow its superiority over the conventional schemes.

I. INTRODUCTION

Nowadays emerging wireless multimedia applications suchas mobile TV are more and more exigent on data rateover limited spectrum resources. Multicast is a spectrum-efficient paradigm for one-to-many transmissions over wirelesschannels by enabling service providers to send multimediadata to multiple users simultaneously [2], [3], [4]. To furtherproliferate multimedia applications over wireless networks andenable quality of service (QoS)-aware wireless video stream-ing, Scalable video coding (SVC) stands out with its gracefulrate adaptation capabilities to cope with bandwidth scarcity

Part of this work has been presented at IEEE WCNC 2017, San Francisco,U.S.A. [1].

This work is supported in part by National Natural Science Foundation ofChina (Nos. 61571396, 61725104, 61401169), Zhejiang Provincial NaturalScience Foundation of China (No. LR17F010001), Young Elite ScientistSponsorship Program by CAST (No. 2016QNRC001), Talent Project ofZJAST (No. 2017YCGC011), and Fundamental Research Funds for theCentral Universities (No. 2015XZZX001-03). The corresponding author isWei Wang (Email: [email protected]).

Yitu Wang, Wei Wang and Zhaoyang Zhang are with College of InformationScience and Electronic Engineering, Zhejiang Provincial Key Laboratory ofInformation Processing, Communication and Networking, Zhejiang Univer-sity, Hangzhou 310027, China.

Lin Chen is with Laboratoire de Recherche en Informatique (LRI), Univer-sity of Paris-Sud and CNRS, Orsay 91405, France.

Pan Zhou is with School of Electronic Information and Communications,Huazhong University of Science and Technology, Wuhan 430074, China.

and network variation, and is often employed in multicast so asto improve radio resource utilization and provide differentiatedQoS [5]. In a nutshell, SVC divides a multimedia stream intoone base layer and multiple enhancement layers, where thebase layer provides a minimum quality of the multimediawhile the enhancement layers gradually increase the quality.QoS-aware streaming can be achieved by including differentnumber of the enhancement layers during transmission toconform to network variation, hardware heterogeneity or usersrequirement.

As a recently emerged technique, cooperative multicasttakes the merit of spatial diversity to combat the influence ofpath loss and channel fading in order to further improve themulticast capacity [6], [7], [8], [9]. In the canonic two-hopcooperative multicast system, the source node first transmitsdata to relay nodes, then the users requesting the same data canbe logically grouped as multicast groups and served by desig-nated relay nodes respectively. Although cooperative multicasthas the potential to increase the capacity, an improper relayselection scheme will result in an even lower data rate than thatin the non-relay-assisted transmission paradigm. Therefore,the relay selection scheme should be carefully designed so asto fully exploit the performance gain brought by cooperativerelay.

A. Motivation

Due to its importance, a large body of works have con-sidered the relay selection problem in cooperative multicast.For one channel case, maximal ratio combining (MRC) isusually adopted, which combines the signals received fromdifferent transmitters to increase the signal-to-noise ratio [10].However, the MRC technique is generally not compatible withthe SVC paradigm due to the implementation issue. Differenttransmitters may transmit different layers of the broadcastcontent, which cannot be combined by simply applying MRC.With perfect synchronization and coordination, it is possible toincorporate MRC with SVC, however, imperfect synchroniza-tion results in significant performance loss and the associatedcommunication overhead is fairly large, and thus makes MRCcostly to implement for the scenario with multiple relay nodesand multiple destination nodes1.

To address the above problem, a natural approach is touse multiple orthogonal channels for different transmitters.In this research strand, most existing works assume that the

1In Section IV.E, we discuss incorporating MRC to further improve theperformance.

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number of orthogonal channels are sufficient to avoid co-channel interference among relays [11], [12], [13]. However, inmany practical networks, e.g., IEEE 802.11 [14], the numberof available channels is rather limited, thus rendering thisassumption invalid. Under this context with limited numberof channels, it is clear that a subset of relay nodes needto be deactivated to avoid interference, which brings a newdimension on the problem of relay selection.

B. Main Contribution and Results

Motivated by the above analysis, in this paper we addressthe problem of relay selection in cooperative multicast with alimited number of channels. Our goal is to achieve a lexico-graphic max-min optimal solution. Lexicographic optimizationis a well-recognized fair optimility criterion [15], [16], [17]for multi-objective optimization problems. Theoretically, it isproven that the lexicographically optimal vector is uniquelyoptimal over any given convex and compact set, and such asolution is always Pareto optimal [18].

The main technical challenge in our problem comes fromthe limitation of the channel number, which creates complexinterdependence among relays and thus makes the problem ofrelay selection an involved optimization problem. To addressthis challenge, we decompose the problem and proceed by twosteps. 1) We consider the problem of maximizing the minimaldata rate. By decoupling relay selection and channel allocation,we transform the problem to a max-min-max problem. Tomake the transformed problem tractable, we reformulate it asa convex optimization problem via relaxation and smoothing,and establish the asymptotic equivalence from a geometricalperspective. 2) With the solution in the first step, we furtherdevelop an adjustment procedure, which is proved to producea lexicographically optimal solution.

The rest of the paper is organized as follows. Section IIdiscusses the related works. Section III presents the systemmodel. Section IV proposes the details in designing therelay selection algorithm. The performance of the proposedalgorithm is evaluated by simulation in Section V. Finally, thispaper is concluded in Section VI.

II. RELATED WORKS

This paper develops an analytical framework for the prob-lem of relay selection in cooperative multicast with a limitednumber of channels. In this section, we briefly review theexisting works on the receiver design and the fairness con-siderations.

A. Receiver Design

Multicast relay selection has attracted increasing attentionrecently for its spectral efficient nature. It is more efficientand realistic compared with the conventional model in [11],which restricts each relay node to be assigned to only onedestination node for cooperative multicast. Multicast relayingprovides an extra degree of freedom to achieve higher capacity,while brings additional constraint during the scheme design.

For one channel case, MRC is usually adopted and nodestransmit and receive data at the same channel. Most existing

works on the cooperative multicast networks using MRCin SVC scenario investigate the case that a source nodebroadcasts data to multiple relay nodes, and then the selectedrelay nodes broadcast data to one receiver [19], [20], where thePHY layer behavior is analysed, e.g., the outage probability. In[21], the capacity and SNR performance is analysed in a multi-receiver case, but the source node broadcasts the basic layerand one enhancement layer, then the relay nodes transmit thebasic layer only. There are also a few publications investigatingcooperative multicast networks using MRC without SVC [22],[23]. In this case, the relay nodes transmit the same data tothe destination nodes, where the user diversity is not fullyexploited. The MRC technique is generally not compatiblewith the SVC paradigm because different transmitters maytransmit different number of the enhancement layers of thebroadcast content, which cannot be combined by simplyapplying MRC.

To address the above problem, a natural approach is to usemultiple orthogonal channels for different transmitters. Formulti-channel case, in [12] and [13], a group of intendedusers receive the same data from the source, where theformer assumes that the selected multicast relays transmitin orthogonal channels, and investigates the optimal relayscheduling and power allocation strategies to minimize thetotal power consumption, while the latter proposes a dis-tributed energy efficient multicast relay selection scheme. In[24], the authors propose an optimal multicast relay selectionscheme achieving maximal capacity in deterministic relaynetworks without cross-channel interference. The authors in[25] consider cooperative multicast of videos over wirelessnetworks, where the relays use TDMA to forward packets, andadopt layered video coding to provide different video qualitiesto the users according to their channel conditions, which doesnot provide any discussion on the algorithm optimality. Mostexisting publications assume that enough orthogonal channelsare available for cooperative multicast without interferenceamong relays. In many practical networks, e.g., IEEE 802.11[14], the number of available channels is rather limited, thusrendering this assumption invalid. However, few publicationdiscusses multicast relay selection scheme considering a lim-ited number of channels.

B. Fairness Consideration

One of the mostly used criteria for resource allocation ismaximizing the total throughput of the system known as max-sum criterion. The weak point of this criterion lies in thefact that the users achieve less resources due to their poorlink qualities. To achieve the fairness, more resources shouldbe allocated to the users with the poorest channel condition,which is known as the max-min optimization. The main prob-lem of the max-min optimization is that the optimal resourceallocation is not necessarily Pareto optimal. In other words,starting from the max-min optimal resource allocation, onecan increase the utility of one individual without decreasingthe utilities of the others, which is clearly not a desirableproperty of an efficient resource allocation algorithm. Movingone step ahead, the solution of the lexicographic max-min





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Relay node 1

Relay node 2

Relay node 3

Destination nodes

Source node

Fig. 1. Cooperative Multicast System

optimization is a refinement of the standard max-min concept.This ordering takes both fairness and efficiency into account,which is addressed in few previous works [15], [16], [17] incooperative networks. In [15], the cooperative transmissionin cognitive radio networks is considered to provide reli-able communication for secondary users by lexicographicallyoptimizing the received data rates. In [16], a lexicographicresource allocation scheme is proposed by allowing the relaysto transmit their own data frames while performing cooperativetransmission. In [17], the joint optimization of subcarrier-relay assignment and power allocation is investigated to obtaina lexicographically optimal solution in an OFDMA system,without considering the coupling of channel allocation andrelay selection, such that channel allocation and relay selectioncan be optimized separately, which significantly simplifies theproblem. Different to [17], we limit the number of channels inthe system, and address the coupling of the relay selection andthe channel allocation head-on to derive a lexicographicallyoptimal solution.

III. SYSTEM MODEL

A. Cooperative Multicast Model

Consider a wireless cooperative multicast network, con-sisting of a source node S = {s}, M relay nodesR = {r1, r2, · · · , rM} and N destination nodes D ={d1, d2, · · · , dN}. Relay nodes multicast with SVC to improvethe wireless resource utilization and provide differentiatedQoS according to the weakest channel conditions in theircorresponding multicast group. The time is slotted and theduration of each time slot is assumed to be a unit of timewithout loss of generality. Due to the long distance or theshielding effect caused by some barrier between the sourcenode and the destination nodes, the destination nodes are notwithin the communication range of the source node, so that allsignals received at the destination nodes need to be forwardedby the assisting relay nodes2. A two-stage cooperative relay inmulticast transmission paradigm is adopted in this work, whichis commonly used to improve the performance of the users

2Such a two-hop relaying model is practical and has been widely adoptedto extend the communication coverage, e.g., the Type-II relaying in IEEE and3GPP standards [26].

[22], [23], [27]. It takes two time slots to accomplish the coop-erative multicast 3. In the first time slot, as the blue links shownin Fig. 1, the source node s broadcasts the data to the relaynodes according to the weakest source-relay channel conditionγ. To focus on the relay selection problem, γ is assumed to belarge enough to support the transmission of at least the basiclayer [28]. In the second time slot, the relay nodes multicastthe received data to the destination nodes simultaneously,where the multicast data rates are determined according to theweakest channel conditions in their corresponding multicastgroups, as the yellow links in Fig. 1. As in [8], we assumethat K orthogonal channels are available in the network (e.g.,using OFDMA), denoted as C = {c1, c2, · · · , cK}, which areflat and remain constant over time slot [29], [30].

Let G = (V,E) denote the conflict graph of the relay nodes,where each relay node ri ∈ R is one of the vertices in Vof the conflict graph, and (ri, rj) ∈ E implies that ri andrj cannot transmit on the same channel simultaneously sincetheir transmissions interfere with each other. If any destinationnode can receive the signal from both the relay nodes ri andrj , then (ri, rj) ∈ E. Taking Fig. 1 as an example, we obtainthat the relay nodes r1 and r2 conflict with each other, r2 andr3 conflict while r1 and r3 do not conflict.

Both the source node and the relay nodes transmit the signalwith a unit power. For the destination node dj , its receivedsignal from the relay node ri can be written as

yij =√D−αij hijx+ nij , (1)

where hij is the fading coefficient between destination nodedj and relay node ri, α is the path loss exponent dependingon the propagation environment, Dij is the distance betweendestination node dj and relay node ri, and nij is additive whiteGaussian noise with variance N0. Therefore, when the relaynode ri transmits, the recieved signal-to-noise ratio of dj is

γij =|hij |2D−αij

N0. (2)

When γij is greater than a certain threshold, the signal can bereceived and decoded successfully. To this end, the enhance-ment layers are selected according to the channel capacity, i.e.,the total size of the basic layer and the selected enhancementlayers should not be larger than the channel capacity, whilethe number of the enhancement layers should be chosen aslarge as possible to provide the best possible QoS.

Decode-and-forward transmission mode is adopted in thecooperative multicast systems to be compatible with SVC. Therelay node ri decodes the signal received from the source nodes and then transmits the decoded data to the destination nodedj . The capacity from source node s to destination node djwith the assistance of relay node ri is

Cij =W

2min{log2(1 + γ), log2(1 + γij)}. (3)

When the signals from multiple relay nodes reach a desti-nation node, the selection combiner is adopted because these

3By embracing multicast, more relay nodes are able to receive the content,and thus more destination nodes are able to receive the content from a nearbyrelay node, such that better QoS can be potentially achieved.





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signals are transmitted over different channels for avoiding theinterference. The destination node receives the signal fromonly one relay node which has the largest number of theenhancement layers so as to achieve the best QoS. Following[31], it is assumed that the channel state information betweenthe destination node and the relay nodes are known by this des-tination node from the report of the relay nodes. Specifically,relay nodes report a received-signal-strength information viacontrol messages, and the destination nodes derive the channelstate information according to the messages before choosinga relay node.

B. Problem Formulation

Denote the relay selection scheme as µ : D → R, whereµ(dj) = ri indicates the relay node ri is selected to help thetransmission to dj via cooperative multicast. Note that onerelay node can help the transmission of multiple destinationnodes by multicast, i.e., it is possible that µ(di) = µ(dj) fori 6= j, which is different from the models in [8], [11], where arelay node can be assigned to assist only one destination node.The channel allocation matrix of the relay nodes is denotedas τ = {τik}M×K , where τik = 1 represents that relay noderi is activated using channel ck.

Considering the multicast nature of wireless communicationsystems, a relay node ri multicasts data to the destinations inDi = {dj |µ(dj) = ri,∀dj ∈ D} with a maximal rate of

Ri =∑ck∈C

τik mindj∈Di

{Cij}, (4)

such that all the destination nodes in Di can successfullydecode the data.

To provide a unique fair solution that outperforms allpossible classic max-min solutions [18], our goal is to designa lexicographic max-min relay selection scheme, in which thelexicographically optimal rate vector is no lexicographicallyless than that of any other scheme. We define the lexicographicoptimality formally as follows:

Definition 1 (Lexicographic Optimality). Let R =(ν1, ν2, ..., νN ) be an achievable rate vector which is sortedin non-descending order, where νi represents the i-th smallestdata rate. Two vectors R and R′ have the following relation-ships:

• If νi = ν′i, ∀i = 1, 2, · · · , N , then R is lexicographicallyequal to R′.

• If there exist a prefix (ν1, ν2, ...νi) of R and a prefix(ν′1, ν

′2, ...ν

′i) of R′ such that νi > ν′i, and νj = ν′j for

1 ≤ j ≤ i − 1, then R is lexicographically greater thanR′.

A rate vector R is lexicographically optimal if it is nolexicographically less than all other feasible rate vectors.

Remark 1. The idea for finding the lexicographic max-min so-lution is to sequentially identify all the max-min solutions andto sort the vectors in weakly decreasing order to identify thelexicographically optimal one. Specifically, lexicographic max-min optimization takes into account maximizing the second

smallest item, maximizing the third smallest one and furtherto be hierarchically maximized.

The received data rates for destination nodes are lexi-cographically optimized by determining which relay nodesshould be activated and which destination nodes that theserelay nodes forward data to. Based on the definition oflexicographic optimality in Definition 1, we can formulate thelexicographic max-min problem as follows.

lex maxτ ,µ

R

s.t.∑ck∈C

τik ≤ 1

τik + τlk ≤ 1, ∀(ri, rl) ∈ Eτik ∈ {0, 1},

(5)

where lex max indicates the operation of lexicographic max-imization. The first constraint indicates that a relay node canuse at most one channel which depends on the fact that usuallyonly one radio interface is deployed in a device. The secondconstraint represents that if two relay nodes ri and rl interferewith each other, they must engage different channels to avoidcross-channel interference. The optimization problem in Eq.(5) lexicographically maximizes the rate vector by determiningthe relay selection scheme µ and channel allocation schemeτ .

IV. ALGORITHM DESIGN

In this section, we propose an analytical framework forlexicographic max-min multicast relay selection scheme forcooperative multicast with a limited number of channels.As we discussed before, the major technical challenge isinduced by the complicated coupling between relay selectionand channel allocation. It is difficult to decouple them inthe lexicographic optimization problem. To overcome thischallenge, instead of solving the lexicographic optimizationdirectly, we first consider the optimization problem to maxi-mize the minimal data rate, where it is possible to decouplerelay selection and channel allocation. Technically, we addressthe problem in two steps:• Consider only the minimal data rate and solve the max-

min problem to obtain an initial relay selection solution.• Optimize the rates of other nodes by further adjusting the

relay selection to achieve lexicographic optimality.

A. Decoupling in Max-Min SubproblemTo design the lexicographic max-min scheme, we consider

only the minimal data rate first. Maximizing the minimal datarate among the destination nodes is equivalent to maximizingthe minimal multicast rate among the relay nodes accordingto Eq. (4), we can formulate the max-min subproblem fromEq. (5) as follows:

maxτ ,µ

mini

Ri

s.t.∑ck∈C

τik ≤ 1

τik + τlk ≤ 1,∀(ri, rl) ∈ Eτik ∈ {0, 1}.

(6)





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This max-min problem involves both the relay selectionscheme µ and the channel allocation scheme τ , which arecoupled with each other due to the limited number of channels.To decouple these two aspects, we exploit the property of themax-min problem and suggest a capacity-based relay selectionscheme which is independent to channel allocation in thefollowing lemma.

Lemma 1 (Capacity-Based Relay Selection). The max-minsolution of the cooperative multicast system is achievable onlyby adjusting channel allocation if each destination node djjoins the multicast group of the relay ri which has the largestchannel capacity Cij , i.e.,

maxτ ,µ

mini

Ri = maxτ

mini

Ri(µ̂), (7)

whereµ̂(dj) = arg max

ri∈R{Cij}. (8)

Proof: The weakest link in a cooperative multicast systemis the link of the destination node dk satisfying

dk = arg mindj∈D{maxri∈R{Cij}}. (9)

To maximize the minimal data rate of the system, we canmaximize the received data rate for the destination node dkinstead. Under the relay selection scheme µ̂ in Lemma 1, eventhough not all destination nodes receive a data rate as much asthe channel link capacity according to Eq. (4), the destinationnode k indeed receives data at a rate of maxri∈R{Cij}, whichis maximized among all possible choices.

It is noted that such a scheme µ̂ in Lemma 1 does notachieve lexicographic optimality, since the data rates of otherdestination nodes are not taken into consideration. We willlater propose a relay selection scheme µ∗ that achieves lexi-cographic optimality based on µ̂.

To decouple relay selection and channel allocation in Eq.(6), we adopt the relay selection scheme µ̂ in Lemma 1 andtransform the max-min problem in Eq. (6) to a max-min-maxproblem which is a channel allocation problem only.

maxτ

Φ(τ ) = minj

maxi

∑ck∈C

τikCij

s.t.∑ck∈C

τik ≤ 1

τik + τlk ≤ 1,∀(ri, rl) ∈ Eτik ∈ {0, 1}.

(10)

B. Tractable Reformulation for Channel Allocation

Due to the non-smooth structure of max-min-max function,the max-min-max problem in Eq. (10) is very difficult tosolve both in theoretical analysis and in numerical calculation[32]. To solve the max-min-max problem directly by numericalcalculation, the result comes close to the exhausting method,which definitely faces the curse of dimensionality, i.e., thecomplexity exponentially grows with the size of the problem,and is not applicable to a real system [33]. To solve the prob-lem, we transform the max-min-max problem into a convex

one by relaxation and smoothing techniques, and further provethat the relaxation and smoothing are tight.

The max-min-max problem is a combinational optimizationproblem, which is not differentiable due to the 0-1 integralconstraint in Eq. (10), so that traditional optimization tech-niques cannot approach the problem. To obtain the optimalsolution, we relax the 0-1 integral constraint in Eq. (10) to abox constraint as

τik ∈ [0, 1], (11)

so that the objective in Eq. (10) is a continuous function of τ .With the above relaxation, the max-min-max problem is

continuous but still non-differentiable, we further adopt asmoothing technique to approximate the original max-min-max optimization problem, and the transformed approximationproblem is differentiable about τ .

Adopting the smoothing technique in [34], the objectiveΦ(τ ) of the max-min-max problem in Eq. (10) can be ap-proximated by

Φε(τ ) =1

εln

(M∑i=1

1∑Nj=1 e

ε∑Kk=1 τikCij

)+

lnM

ε, (12)

where ε is the approximation parameter. For given ε, theobjective function in Eq. (12) can be safely transformed into

ln

(M∑i=1

1∑Nj=1 e

ε∑Kk=1 τikCij

). (13)

Consider the exponential of the objective function in Eq. (13),the optimality is preserved according to the monotone propertyof the exponential function [36]. The optimization problem inEq. (10) can be transformed to

minτ

M∑i=1

1∑Nj=1 e

ε∑Kk=1 τikCij

s.t.K∑k=1

τik ≤ 1

τik + τjk ≤ 1,∀(ri, rj) ∈ E0 ≤ τik ≤ 1.

(14)

We prove the convexity of the problem in Eq. (14) in thefollowing Theorem

Theorem 1 (Convexity of Problem (14)). The optimizationproblem in Eq. (14) is convex.

Proof: We analyze the Hessian of the objective functionin Eq. (14), which is proved to be positive semi-definite andhence it is convex. Since all the constraints in Eq. (14) arelinear, we prove that the problem in Eq. (14) is a convexoptimization problem. More details of the proof are providedin Appendix A.

C. Geometrical Analysis

To obtain some critical insights of the above convex prob-lem, we consider the relay selection optimization problemin Eq. (14) as a geometrical problem which investigates the





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relationship of positions of a line and multiple points in two-dimensional space. This method can decrease the computa-tional complexity significantly. More importantly, the relax-ation is proved to be tight by adopting geometrical analysis.

We first consider the optimality condition of the problem inEq. (14). According to Lagrangian duality [36], the problemcan be transformed to

minτ

P =M∑i=1

1∑Nj=1 e

ε∑Kk=1 τikCij

+M∑i=1

λi

(K∑k=1

τik − 1

)

+K∑k=1

M∑i=1

∑j,(ri,rj)∈E

βijk(τik + τjk − 1)

s.t. λi

(K∑k=1

τik − 1

)= 0

βijk(τik + τjk − 1) = 0,(15)

where λi is the Lagrangian multiplier for the first constraint inEq. (14) and βijk is the Lagrangian multiplier for the secondconstraint in Eq. (14).

The derivative of P with respect to τ is

∂P

∂τin= − 1∑N

j=1 eε∑Kk=1

τikCij

N∑j=1

εCij + λi +∑

j,(ri,rj)∈E

βijn.

(16)

Based on different cases of optimal τ ∗, the optimalityconditions are provided as

∂P

∂τin|τin=1≤ 0 −→ τ∗in = 1

∂P

∂τin|τin=τ∗

in= 0 −→ 0 < τ∗in < 1

∂P

∂τin|τin=0≥ 0 −→ τ∗in = 0.

(17)

A relay node ri is activated to transmit using the channel cnif τin > 0. When 0 < τ∗in < 1, the optimal solution is achievedif ∂P

∂τin|τin=τ∗

in= 0. When τ∗in = 1, because the Lagrangian

multiplier λ is configurable for each relay node, we can let∂P∂τin

= 0 by adjusting λ. Therefore, the optimal solution of arelay node transmitting with a channel is ∂P

∂τin|τin=τ∗

in= 0 for

τin > 0. Therefore, it is necessary to analyze the relationshipbetween Eq. (16) and zero. We rewrite Eq. (16) by multiplying∑Nj=1 e

ε∑Kk=1 τikCij as

−N∑j=1

εCij +

λi +∑

j,(ri,rj)∈E

βijn

N∑j=1

eε∑Kk=1 τikCij .

(18)Since

∑Nj=1 e

ε∑Kk=1 τikCij > 0, the multiplication does not

affect the relationship between Eq. (16) and zero.To analyze Eq. (18), we rewrite the optimality condition

for the problem in Eq. (18) as an expression of a line in two-dimensional space [38],

yin = Aixin, (19)

wherexin = λi +

∑j,(ri,rj)∈E

βijn

Ai =N∑j=1

eε∑Kk=1 τikCij

yin =N∑j=1

εCij .

From a geometrical perspective, each relay ri using thechannel cn has a corresponding point Sin = (xin, yin) in thetwo dimensional space. Define Si as the set of all points forri, i.e., Si = {Sin,∀n}. For given λ, β and the relay node ri,the coordinates xin and yin are determinate. The problem is todetermine Ai, which is the slope of the line Yi = AiXi of therelay node ri. In such a case, the problem is transformed to ageometrical problem on the position relationship of the pointsSin ∈ Si and the line Yi = AiXi. The geometrical problemis to find a line Yi = AiXi by adjusting τ to let some of thepoints Sin ∈ Si on the line and all the other points under theline.

If only one relay node ri transmits using the channel cn,i.e., the line Yi = AiXi goes through Sin = (xin, yin) andall the other points Sik = (xik, yik), k 6= n are under the line,the optimal conditions are

yin = Aixin (20)

yik < Aixik,∀k 6= n. (21)

In this way, we obtain the optimal conditions for thereformulated convex optimization problem in Eq. (14). Besidesproviding the optimal conditions, the geometrical analysisalso gives important insights on the relationship between thereformulated convex optimization problem and the originalmax-min-max problem in Eq. (10). From the geometricalperspective, we prove that the relaxation and smoothing inthe tractable reformulation step are asymptotically tight in thefollowing Theorem

Theorem 2 (Asymptotic Equivalence). With sufficiently largesmoothing parameter ε, the solution τ ∗approx of the convexoptimization problem in Eq. (14) is asymptotically optimalfor the original max-min-max problem in Eq. (10), where theapproximation error satisfies

Φ(τ ∗)− Φ(τ ∗approx) = o

(1

ε

), (22)

where τ ∗ represents the optimal solution solving the originalmax-min-max problem in Eq. (10).

Proof: Since each relay node can use at most one channelto transmit, and the line of Eq. (20) in a two-dimensional spacecan be found with probability 0 to go through 3 points, weprove that the relaxation of τ in Eq. (11) is tight throughreasoning by contradiction. Then, the approximation error isanalyzed to prove the asymptotically equivalent property. Moredetails of the proof are provided in Appendix B.





7

Relay i is activated

x

y

Base point

Feasible

Region

knyy ikin ,,

Relay j is deactivated

x

y

Base point

Feasible Region

1ix 2ix

knyy jkjn ,,

1jx 2jx

Fig. 2. Relationship between lines and points

D. Proposed Algorithm

Solving the geometrical problem in Eq. (20) is equivalentto finding the slope of a line Yi = AiXi for each relay noderi, such that some of the points Sin = (xin, yin) on the lineand all the other points Sik = (xik, yik), k 6= n under the line,because the choice of τ only poses influence on the slope ofeach line. Accordingly, the following rule can be determined

τik = 1,∃Ai =yikxik

τik = 0,∀Ai >yikxik

,(23)

where k = arg minj xij . To determine the slope of the lineYi = AiXi, we only need to compare Ai with the slope of aline, containing the leftmost point (xik, yik) and the base point(0, 0), for each relay node ri. The feasible region of the slopeAi is determined according to (20) by varying τik from 0 to 1.As the example in Fig. 2, there exists a slope Ai that the linepasses through the left-most point (xi1, yi1). If so, relay noderi is activated to transmit with channel 1. While there doesnot exist a slope Aj that the line passes through the left-mostpoint (xj1, yj1). As a result, relay node rj is deactivated.

Towards the lexicographic optimality, we propose a relayselection scheme µ∗ based on the initial relay selection schemeµ̂ in Lemma 1 and the channel allocation τ . Even thoughµ̂ achieves a max-min data rate of the system, it is notlexicographically optimal. A destination node may actuallyreceives a higher data rate from another relay node thanthat from the relay node which holds the largest channelcapacity between them, because the received data rate is themulticast rate which depends on the channel capacities of allthe destination nodes in the multicast group.

We propose a lexicographically optimal relay selectionscheme µ∗ to improve the performance. We first divide thedestination nodes into two categories, including the node withthe weakest channel quality in each multicast group and otherdestination nodes. Denote the set of the weakest nodes in allmulticast groups as J , i.e.,

J = {d(ri)|d(ri) = arg mindj∈Di

{Cij},∀ri ∈ R}, (24)

where Di = {dj |µ̂(dj) = ri,∀dj ∈ D} and d(ri) is thedestination node which has the weakest channel quality in themulticast group of the relay node ri.

For the destination node dj ∈ J ,

µ∗(dj) = µ̂(dj) = arg maxri∈R{Cij}. (25)

For the destination node dj ∈ D/J ,

µ∗(dj) = arg maxri∈R{min{Cij , Ri}}, (26)

where Ri is the multicast rate of the relay node ri with theinitial relay selection scheme µ̂ and the channel allocation τobtained in the first step.

Remark 2 (Interpretation of the Relay Selection Scheme).The destination nodes in J have the worst channel conditionsof their multicast group, and hence limit the multicast rate ofeach relay node. Their relay selection schemes in µ∗ are thesame as those in µ̂. Each of the other destination nodes selectsthe relay which provides the maximal possible data rate forthis destination node.

Theorem 3 (Lexicographic Optimality). The proposed relayselection scheme µ∗ is lexicographically optimal.

Proof: Reasoning by contradiction, suppose there existsa relay selection scheme µ′ that is lexicographically greaterthan µ∗. Despite the cases whether the weakest links in eachmulticast group are changed or not, we can prove that µ∗ is notlexicographically less than µ′. Therefore, the proposed relayselection scheme µ∗ achieves the lexicographic optimality.More details of the proof are provided in Appendix C.

Algorithm 1 Lexicographic Max-Min Relay Selection1: loop2: Each destination node selects a relay node according to

Eq. (8).3: Adjust λ and β by augmented Lagrange method.4: for each user i and channel k do5: if ∃Ai = yik/xik, where k = arg minn xin, and

τlk = 0,∀(ri, rl) ∈ E. then6: Set τik = 1.7: end if8: end for9: end loop until the approximation gap is within a given

threshold by updating the approximation parameter ε.10: Each destination node selects a relay node according to

Eq. (25) and Eq. (26).11: The multicast rate of each relay node is determined

according to Eq. (4).

We provide the pseudocodes of the proposed algorithm inAlgorithm 1, which is launched at the beginning of each timeslot. In the pseudo-codes, Line 2 presents the max-min relayselection scheme, Lines 3-9 determine the channel allocation,Line 10 deploys the lexicographically optimal relay selectionscheme, and Line 11 determines the scheduled transmissionrate for each activated relay nodes. Any strictly increasingupdating rule of ε can be used [34].

Remark 3 (Computational Complexity). The complexity ismainly brought by channel allocation whose complexity is





8

O(K2M). Therefore, the complexity of the proposed schemeis O(cK2M), where c represents the iteration rounds.

Remark 4 (Compatibility with Dynamic Scenarios). In dy-namic scenarios, the communication overhead could be largeif we execute the proposed algorithm at the beginning ofeach time slot. However, if the resource allocation is updatedless frequently, the performance will slightly deviate from theoptimality. Therefore, there is an inherent trade-off betweenthe communication overhead and the performance.

In slow fading channel, the channel quality can be consid-ered to be unchanged during a whole signal block, which iscommonly considered in cooperative networks [35]. In such acase, we execute the proposed algorithm for lexicographicallyoptimal resource allocation at the beginning of each signalblock instead of each time slot, and the performance will notbe degraded.

Moreover, in dynamic scenarios, we may not able to es-timate the channel quality accurately due to the delay ofestimation process. To address this issue, we budget someallowance to enlarge the average decoding success rate. Theless number of the enhancement layers is transmitted, thelarger average decoding success rate we have.

E. Discussions on Further Improving the Performance

In this section, we discuss two possible approaches tofurther improve the performance, i.e., optimizing the timeintervals of the two cooperative transmission stages and in-corporating MRC by allocating channels to the deactivatedrelay nodes.

1). Optimizing the time intervals of the two stages: Inthis paper, we adopt a two-stage cooperative relay in multicasttransmission paradigm, where each time slot for multicastservices T is divided into two time intervals T1 and T2. Todetermine the lexicographically optimal rate vector, we needto find R for all possible T1 and T2 satisfying T1 + T2 = T .We demonstrate that with the proposed algorithm, we caneasily find the optimal T1 and T2. For given T1 and T2,we can calculate the associated lexicographically optimal ratevector using the proposed algorithm. Then adopting exhaustivesearch, we can find the optimal T1 and T2 as well as theassociated lexicographically optimal rate vector.

2). Incorporating MRC: Consider an example that we haveallocated the channel ck to the relay node ri and its neighborrelay node rj is deactivated due to the limitation of the numberof channels. Observing that the neighboring relay nodes riand rj share a part of the destination nodes in their multicastgroups, who are able to receive the signal from both the relaynodes, it is possible for those destination nodes to receive alarger number of enhancement layers by incorporating MRC.To this end, we allocate the channel ck to the relay node rjwho transmits the same number of the enhancement layers asthe relay node ri, where the exact number of the enhancementlayers can be determined through further coordination betweenthe two relay nodes. In such a case, it is equivalent to mergingthe two relay nodes ri, rj into a new mega-node r(i,j), where∀rl ∈ R, any (rl, ri) ∈ E or (rl, rj) ∈ E, it is satisfied

that (rl, r(i,j)) ∈ E. By doing so, we form a new conflictgraph, where the proposed algorithm can be applied to obtainthe lexicographically optimal resource allocation. However,the problem of finding the optimal MRC combination is NP-hard, because the MRC weights vary according to differentcombinations [40]. It is an interesting problem to derive alow-complexity algorithm which we are willing to investigatein our future work.

V. SIMULATION

In this section, we evaluate the performance of the pro-posed algorithm through simulation. In the simulation, thesource node is located at the center, the destination nodesare randomly distributed in a circular area with a radius of500 meters and the relay nodes are randomly distributed ina circular area with a radius of 100 meters. Following thesimulation parameter settings in [11], we set the bandwidth as22 MHz for all channels. The transmission power is the samefor each node, i.e., Ps = Pri = 1 Watt for the source node sand the relay nodes ri ∈ R. For the transmission model, weassume that the path loss exponent α = 4 and the abient noiseis 10−10.

For performance comparison with the proposed lexico-graphic max-min scheme µ∗, we adopt three baseline schemesas follows:• Random: The relay nodes are activated randomly and

the destination nodes choose the relay node with the bestchannel quality.

• Throughput-based scheme [41]: The channels are allo-cated to the multicast relay nodes to maximize total datarate.

• Max-Min: The max-min scheme µ̂ proposed in Lemma1.

For performance comparison, the deviation from the lexi-cographic optimality is defined as N − i, where i is givensuch that the prefix (ν∗1 , ν

∗2 , ...ν

∗i ) of the lexicographically

optimal rate vector R∗ and the prefix (ν′1, ν′2, ...ν

′i) of R′

under a baseline scheme satisfy that ν∗i > ν′i, and ν∗j = ν′j for1 ≤ j ≤ i− 1.

We first analyze the deviation from the lexicographic op-timality of the four schemes, and then further discuss thedata rates achieved by all destination nodes. We evaluatethe performance with different number of available channels,different number of the relay nodes and different number of thedestination nodes. For each case, 10 instances are generatedfor obtaining the average performance.

Fig. 3(a) demonstrates the deviation from the lexicographicoptimality with different numbers of available channels, where10 relay nodes and 30 destination nodes are deployed. Eventhough the max-min scheme achieves the same minimalcapacity as the proposed scheme, the deviation from thelexicographic optimality is quite large, which verifies thatthe solution of the lexicographic max-min optimization is arefinement of the standard max-min concept. The performanceof the proposed scheme is better than those of the other twobaselines. The throughput-based scheme deviates from thelexicographic optimality the most, because it sacrifices theperformance of the worst user.





9

TABLE ISORTED RATE VECTOR R

Proposed 12.7496 12.7496 12.7496 14.7067 14.7067 15.8894 15.8894 15.8894 17.2415 17.2415Random 12.2972 12.2972 12.2972 14.2155 14.2155 14.2155 15.0306 15.0306 18.2415 18.2415EPSA 11.5488 11.5488 11.5488 13.4623 18.0838 18.0838 18.0838 20.2415 20.2415 20.2415

Max-Min 12.7496 12.7496 12.7496 12.7496 14.7067 14.7067 14.7067 15.8894 15.8894 17.2415

2 4 6 8 1 00

3

6

9

1 2

1 5

1 8

2 1

2 4

2 7T h r o u g h p u t - b a s e d s c h e m e

M a x - M i n

R a n d o m

P r o p o s e d

Devia

tion f

rom th

e Lex

icogra

phic o

ptima

lity

N u m b e r o f C h a n n e l s

(a) Varying the number of channels

6 8 1 0 1 2 1 40

5

1 0

1 5

2 0

2 5

P r o p o s e d

M a x - M i n

T h r o u g h p u t - b a s e d s c h e m e

R a n d o m

Devia

tion f

rom th

e Lex

icogra

phic o

ptima

lity

N u m b e r o f R e l a y N o d e s

(b) Varying the number of relay nodes

4 0 6 0 8 0 1 0 0 1 2 0 1 4 00

2 0

4 0

6 0

8 0

1 0 0

1 2 0

1 4 0

P r o p o s e d

R a n d o m

T h r o u g h p u t - b a s e d s c h e m e

M a x - M i n

Devia

tion f

rom th

e Lex

icogra

phic o

ptima

lity

N u m b e r o f D e s t i n a t i o n N o d e s

(c) Varying the number of destination nodes

Fig. 3. The deviation from the lexicographic optimality

Fig. 3(b) provides the deviation from the lexicographicoptimality with different numbers of relay nodes when thereare 5 channels and 30 destination nodes. The proposed schemeoutperforms the three baseline schemes. The number of therelay nodes has small influence to the deviation from thelexicographic optimality, because the number of channels issmall and thus limits the choice of relay selection.

Fig. 3(c) provides the deviation from the lexicographicoptimality with different numbers of destination nodes whenthere are 10 relay nodes and 5 channels. As the number ofthe destination nodes is large, the performance gain of theproposed algorithm is significant.

Besides the deviation from the lexicographic optimality, wefurther analyze the data rates of all destination nodes for thelexicographic optimality. Table I provides the rate vector R forall four schemes when there are 10 destination nodes and 10relay nodes with 3 available channels. It can be found that therate vector of the proposed scheme is lexicographically greaterthan those of three baseline schemes. It is worth mentioningthat the proposed scheme lexicographically dominates themax-min one, even though the minimal capacities of the twoschemes are the same. As a result, we can safely draw theconclusion that the proposed scheme achieves superior max-min fairness than the three baseline schemes and providesrelatively homogeneous service quality to all the destinationnodes in the cooperative multicast system.

VI. CONCLUSION

In this paper, we construct an analytical framework forlexicographic max-min multicast relay selection for coopera-tive multicast with a limited number of channels. Specifically,we design the algorithm in two steps. 1) We consider themaximization of the minimal data rate. By decoupling relayselection and channel allocation, the problem is transformed

to a max-min-max problem, which is difficult to solve. Tomake this problem tractable, we reformulate it via relaxationand smoothing, and prove the asymptotic equivalence from ageometrical perspective. 2) We propose an adjustment algo-rithm based on the initial max-min solution, and prove thatthe proposed scheme achieves lexicographic optimality. Thesimulation results show that the proposed scheme achieves thedata rate lexicographically greater than those of the conven-tional schemes.

APPENDIX APROOF OF THEOREM 1

Denote the objective function in Eq. (14) as f(τ ). To ana-lyze the convexity of f(τ ), we take the first-order derivativeof f(τ ) with respect to τln as

∂f(τ )

∂τln= − 1∑N

j=1 eε∑Kk=1 τlkClj

N∑j=1

εClj < 0, (27)

which is a strictly decreasing function of τ .Then we consider the second-order derivative of f(τ )

∂2f(τ )

∂2τln=

( ∑Nj=1 εClj∑N

j=1 eε(∑Kk=1 τlkClj)

)2

> 0

∂2f(τ )

∂τln∂τmp=

∂2f(τ )

∂τln∂τmn= 0

∂2f(τ )

∂τln∂τlp=

( ∑Nj=1 εClj∑N

j=1 eε(∑Kk=1 τlkClj)

)2

=∂2f(τ )

∂2τln.

(28)

We derive the Hessian matrix of f(τ ) as





10

A1 0 0 00 A2 0 0...

.... . .

...0 0 0 AM

, (29)

where

Al =

al . . . al...

. . ....

al . . . al

, (30)

and al =(

1∑Nj=1 e

ε(∑Kk=1

τlkClj)

)2. Note that the Hessian of

f(τ ) is a symmetric matrix. Each sub-block Al is also asymmetric matrix. It can be derived according to [37] that

E = (N, 0, · · · , 0), (31)

where E is the eigenvector of the matrix in Eq. (30).Therefore, the eigenvalue of the Hessian of f(τ ) is non-

negative. According to [36], [37], the Hessian is positive semi-definite, and f(τ ) is convex with respect to τ . Since all theconstraints in Eq. (14) are linear, we prove that the problemin Eq. (14) is a convex optimization problem.

APPENDIX BPROOF OF THEOREM 2

For the optimality, each relay node can use at most onechannel to transmit, which makes the relaxation tight.

Reasoning by contradiction, suppose there exists more thanone channels used by a relay node. Assume there are twochannels cn and cm used by the relay node ri, i.e., τin > 0 andτim > 0. Base on Eq. (11), we have 1 ≥ τin ≥ 0, 1 ≥ τim ≥ 0.For the optimality, the following equations need to be satisfiedaccording to the optimality conditions,

Aixin − yin = 0

Aixim − yim = 0.(32)

Thus, it follows from Eq. (32) thatxinyin

= Ai =ximyim

. (33)

According to Eq. (20),

yin = yim. (34)

Thus, it comes from Eq. (33) that

xin = xim. (35)

Observing the intercept of the line in Eq. (20) is zero, i.e., theline must go through the base point. To allocate the channelsto the relay nodes efficiently, we assume that a relay nodealways wants to use a channel with smaller index as in [39],such that the lagrange multipliers are adjusted differently fordifferent channels. Channels cn and cm are different, and aline in a two-dimensional space can be found with probability0 to go through all these 3 points, which is a contradiction.Therefore, the relaxation of τ in Eq. (11) is tight.

According to [34], the approximation error satisfies

2 lnK

ε≤ Φε(τ )− Φ(τ ) ≤ ln(MK)

ε. (36)

With sufficiently large smoothing parameter ε, the value ofthe approximating function converges to a stationary pointwith the approximation error Φε(τ ) − Φ(τ ) = o

(1/ε), such

that the smoothing in Eq. (12) is tight. Then, we analyse theapproximation error. According to Eq. (36), we have

2 lnK

ε≤ Φε(τ

∗approx)− Φ(τ ∗approx) ≤ ln(MK)

ε, (37)

2 lnK

ε≤ Φε(τ

∗)− Φ(τ ∗) ≤ ln(MK)

ε. (38)

Let Eq. (37) minus Eq. (38),

lnK − lnM

ε≤ Φ(τ ∗)− Φ(τ ∗approx)

+ Φε(τ∗approx)− Φε(τ

∗) ≤ lnM − lnK

ε.

(39)

Observing that Φε(τ∗approx) − Φε(τ

∗) ≥ 0 and Φ(τ ∗) −Φ(τ ∗approx) ≥ 0, we have

Φ(τ ∗)− Φ(τ ∗approx) ≤ lnM − lnK

ε= o(1

ε

). (40)

Since the relaxation of τ (t) in Eq. (11) and the smoothingin Eq. (12) are tight, the optimization problem in Eq. (12)and the one in Eq. (10) are asymptotically equivalent with theapproximation error o

(1/ε).

APPENDIX CPROOF OF THEOREM 3

Reasoning by contradiction, suppose there exists a relayselection scheme µ′ that is lexicographically greater than theoptimal µ∗ at the position k, i.e., ν′k > ν∗k and ν′i = ν∗i ,∀i < k.

To prove this theorem, we respectively consider the caseswhether the weakest links in each multicast group are the sameunder µ′ and µ∗, i.e., µ′(dj) = µ∗(dj),∀dj ∈ J .

1) At least one of the weakest links in each multicastgroup is not the same. First, let us consider the case whereone of the weakest links is not the same under µ′ and µ∗.Suppose that one destination node dj ∈ J belongs to themulticast group of rl under µ∗, and belongs to the multicastgroup of rn under µ′. The received data rates a∗j = R∗l underµ∗ and a′j = R′n under µ′ should satisfy a∗j > a′j due to theoptimality of µ∗.

Since the weakest link is removed from the multicast groupof rl under µ∗, the multicast rate of rl under µ′ is not lessthan that under µ∗, i.e., R′l ≥ R∗l . As for any destination nodedq ∈ D, the received data rates under µ′ and µ∗ have thefollowing relationship:

a′q ≥ a∗q , if R∗l ≤ a∗q < R′l,

a′q ≤ a∗q , if a∗q = R∗n,

a′q = a∗q , if a∗q < R∗n, or R∗n < a∗q < R∗l , or a∗q ≥ R′l.

(41)

For the first equation, the destination nodes in the multicastgroup of rl under µ∗ (i.e., a∗q = R∗l ) receive higher data ratesunder µ′ than those under µ∗, and the destination nodes whosedata rates satisfy R∗l < a∗q < R′l under µ∗ may reselect rl astheir relay node under µ′ for achieving higher data rates. Inthis case, their position numbers in the rate vector sorted in





11

non-descending order under µ′ are larger than those of thedestination nodes in the multicast group of rl under µ∗. Forthe second equation, the destination nodes in the multicastgroup of rn under µ∗ (i.e., a∗q = R∗n) receive lower data ratesunder µ′ than those under µ∗ due to the reduction of the datarate of the multicast group of rn. Note that some destinationnodes in the multicast group of rn under µ∗ may reselect theirrelay nodes under µ′, but their data rates under µ′ are stilllower than R∗n due to the optimality of µ∗. In this case, theirposition numbers in the rate vector sorted in non-descendingorder under µ′ are smaller than those of the destination nodesin the multicast group of rl under µ∗. Apart from the abovetwo cases, the received data rates of the destination nodesunder µ′ and µ∗ are the same, which is described in the thirdequation.

According to the definition of lexicographic optimality inDefinition 1, we focus on the first position (i.e., the smallestposition number) that results in different data rates of therate vectors sorted in non-descending order under µ′ and µ∗,which is caused by the difference between µ′(dj) and µ∗(dj).Next, we discuss the difference between µ′ and µ∗ from theperspective of the position numbers of the destination nodesin the multicast group of rn = µ′(dj) under µ′ and µ∗ in therate vectors.

a). The position numbers of the destination nodes in themulticast group of rn under µ′ are the same as those underµ∗, which is achieved if the multicast rate of rn under µ′ andµ∗ are the same despite of dj , or the reduction of the multicastrate of rn under µ′ is small enough. To explicitly describe suchcircumstances, we consider two sub-cases as follows:a-1). The multicast rate of rn under µ′ is the same as that underµ∗. Denote the largest position number of the destination nodesin the multicast group of rn under µ∗ as p, i.e., ν∗p ≤ ν∗p+1.Here, the largest position number of the destination nodes inthe multicast group of rn under µ′ becomes p+ 1 due to thejoining of dj , i.e., ν′p = ν′p+1 ≤ ν′p+2. Since the multicast rateof rn under µ′ is the same as that under µ∗, we have ν∗p = ν′p.Since µ′ is assumed to be lexicographically greater than µ∗ atthe position k, and the first position that results in different datarates under µ′ and µ∗ is p+ 1, we have k = p+ 1. Therefore,we have ν′k = ν′p+1 = ν′p = ν∗p ≤ ν∗p+1 = ν∗k , hence µ∗ islexicographically greater than or equal to µ′, which contradictsthe premise that µ′ is lexicographically greater than µ∗.a-2). The multicast rate of rn under µ′ is smaller thanthat under µ∗. Denote the smallest position number of thedestination nodes in the multicast group of rn under µ∗ asg, i.e., ν∗g ≥ ν∗g−1. Here, the multicast rate of rn underµ′ is smaller than that under µ∗, i.e., ν′g < ν∗g . Since theposition numbers of the destination nodes in the multicastgroup of rn under µ′ are the same as those under µ∗, wehave ν∗g−1 ≤ a′j = ν′g < ν∗g . Since the first position thatresults in different data rates under µ′ and µ∗ is k = g,and it is satisfied that ν′k = ν′g < ν∗g = ν∗k , hence µ∗

is lexicographically greater than µ′, which contradicts thepremise that µ′ is lexicographically greater than µ∗.

b). The position numbers of the destination nodes in themulticast group of rn under µ′ are smaller than those underµ∗, hence it is satisfied that a′j < ν∗g−1 ≤ ν∗g . According to

(42), for all destination nodes dq whose data rate under µ∗

satisfies a∗q < R∗n = ν∗g , their data rates are the same underµ′ and µ∗, i.e., a′q = a∗q . Since the rate vector under µ′ issorted in non-decreasing order, the smallest position numberof the destination nodes in the multicast group of rn underµ′ is determined as h that satisfies ν′h > ν∗h−1 and ν′h ≤ ν∗h.In this context, the first position that results in different datarates under µ′ and µ∗ is k = h. Therefore, we have ν′k =ν′h ≤ ν∗h = ν∗k , hence µ∗ is lexicographically greater thanµ′, which contradicts the premise that µ′ is lexicographicallygreater than µ∗.

Then, let us consider the case where multiple weakest linksare not the same under µ′ and µ∗. Denote the destination nodesin J under µ∗ that belong to different multicast groups underµ′ and µ∗ as W , and denote their selected relay nodes underµ′ as Q and that under µ∗ as T . Due to the optimality of µ∗,it is satisfied that a′j < a∗j ,∀dj ∈ W . Since the weakest linksare removed from the multicast group of ri ∈ T under µ∗,their multicast rates under µ′ in not less than those under µ∗,i.e., R′i > R∗i ,∀ri ∈ T . As for any destination node dq ∈ D,the received data rate under µ′ and µ∗ have the followingrelationship:

a′q ≥ a∗q , if minri∈T{R∗i } ≤ a∗q < max

ri∈T{R′i},

a′q ≤ a∗q , if a∗q = R∗i ,∃ri ∈ Q,a′q = a∗q , if a

∗q < min

ri∈Q{R∗i } or min

ri∈Q{R∗i } < a∗q < min

ri∈T{R∗i }

&a∗q 6= R∗i ,∀ri ∈ Q, or a∗q ≥ maxri∈T{R′i}.

According to the definition of lexicographic optimality inDefinition 1, we focus on the first position that results indifferent data rates of the rate vectors sorted in non-descendingorder under µ′ and µ∗, which are caused by the differencebetween µ′(dj) and µ∗(dj), where dj = arg mindi∈W{a′i}.Here, we only have to consider the influence brought by dj ,while the influence brought by the rest of the destination nodesin W will not affect the result of the analysis between µ′ andµ∗ in view of lexicographic optimality, because any destinationnode dz ∈ W/{dj} has a higher data rate under µ′ thanthat of dj under µ′, i.e., a′z ≥ a′j , and thus their positionnumbers under µ′ will be larger than that of dj under µ′,hence they do not account for the first position that results inthe different data rates of the rate vectors under µ′ and µ∗.In this context, the case where multiple weakest links are notthe same is reduced to the case where one weakest link isnot the same. Following the same analysis when discussingthe difference between µ′ and µ∗ from the perspective of theposition numbers of the destination nodes in the multicastgroup of µ′(dj) under µ′ and µ∗ in the rate vectors, it isproved that under all circumstances, we have ν∗k ≥ ν′k, andthus µ∗ is lexicographically greater than µ′, which contradictsthe premise that µ′ is lexicographically greater than µ∗.

2) The weakest links in each multicast group are thesame under µ′ and µ∗. If ∀dj ∈ J it is satisfied thatµ′(dj) = µ∗(dj), then we have ∀dj ∈ D that a′j ≤ a∗j . Becausethe proposed relay selection scheme µ∗ selects the relay nodewhich provides the maximal possible data rate for the rest of





12

the destination nodes. Thus, when µ′(dj) = µ∗(dj),∀dj ∈ J ,it is not possible that ∃dj ∈ D/J such that a′j > a∗j .

Therefore, there does not exist a relay selection scheme µ′

that is lexicographically greater than µ∗, which means that theproposed relay selection scheme µ∗ achieves the lexicographicoptimality.

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Yitu Wang (S’16) received the B.S. degree fromZhejiang University, Hangzhou, China, in 2013.From August to November 2014, he was a visitingstudent with the University of Paris-Sud, Orsay,France. He is currently a Ph.D candidate at ZhejiangUniversity, Hangzhou, China. His research interestsmainly focus on stochastic optimization for cross-layer resource allocation in wireless networks andcache-enabled networks.

Wei Wang (S’08-M’10-SM’15) received the B.S.and Ph.D. degrees from the Beijing University ofPosts and Telecommunications, China, in 2004 and2009, respectively. From 2007 to 2008, he was aVisiting Student with the University of Michigan,Ann Arbor, USA. From 2013 to 2015, he was aHong Kong Scholar with the Hong Kong Universityof Science and Technology, Hong Kong. He iscurrently an Associate Professor with the Collegeof Information Science and Electronic Engineering,Zhejiang University, China. His research interests

mainly focus on stochastic optimization for cross-layer resource allocationin wireless networks, and caching and computing in wireless networks. Heis the Editor of the book entitled Cognitive Radio Systems, and serves as anEditor of the IEEE Access, Transactions on Emerging TelecommunicationsTechnologies, and KSII Transactions on Internet and Information Systems.

Lin Chen (S’07-M’10) received his B.E. degree inRadio Engineering from Southeast University, Chinain 2002 and the Engineer Diploma from TelecomParisTech, Paris in 2005. He also holds a M.S.degree of Networking from the University of Paris6. He currently works as associate professor in thedepartment of computer science of the University ofParis-Sud. He serves as Chair of IEEE Special Inter-est Group on Green and Sustainable Networking andComputing with Cognition and Cooperation, IEEETechnical Committee on Green Communications and

Computing. His main research interests include modeling and control forwireless networks, distributed algorithm design and game theory.

Pan Zhou (S’07-M’14) is currently an associateprofessor with School of Electronic Information andCommunications, Wuhan, P.R. China. He receivedhis Ph.D. in the School of Electrical and ComputerEngineering at the Georgia Institute of Technology(Georgia Tech) in 2011, Atlanta, USA. He receivedhis B.S. degree in the Advanced Class of HUST,and a M.S. degree in the Department of Electronicsand Information Engineering from HUST, Wuhan,China, in 2006 and 2008, respectively. He heldhonorary degree in his bachelor and merit research

award of HUST in his master study. He was a senior technical memberat Oracle Inc, America during 2011 to 2013, and worked on Hadoop anddistributed storage system for big data analytics at Oralce could Platform.His current research interest includes: big data analytics and machine learning,security and privacy, and information networks.

Zhaoyang Zhang (M’10) received the Ph.D. degreein communication and information systems fromZhejiang University, Hangzhou, China, in 1998.He is currently a Full Professor with the Collegeof Information Science and Electronic Engineering,Zhejiang University. He has coauthored more than150 refereed international journal and conferencepapers, as well as two books in his areas of interest.His research interests are mainly focused on infor-mation theory and coding theory, signal processingtechniques, and their applications in wireless com-

munications and networking.Dr. Zhang coreceived three conference Best Paper Awards/Best Student

Paper Award. He is currently serving as an Editor for the IEEE TRANSAC-TIONS ON COMMUNICATIONS, IET Communications, and several otherinternational journals. He has served as a Technical Program CommitteeCochair or a Symposium Cochair for many international conferences, such asthe 2013 International Conference on Wireless Communications and SignalProcessing and the 2014 IEEE Global Communications Conference WirelessCommunications Symposium.



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