18
Coding-Aware Proportional-Fair Scheduling in OFDMA Relay Networks Bin Tang, Student Member, IEEE, Baoliu Ye, Member, IEEE, Sanglu Lu, Member, IEEE, and Song Guo, Senior Member, IEEE Abstract—In recent years, OFDMA relay networks have become a key component in the 4G standards (e.g., IEEE 802.16j, 3GPP LTE- Advanced) for broadband wireless access. When numerous bidirectional flows pass through the relay stations in an OFDMA relay network that supports various interactive applications, plenty of network coding opportunities arise and can be leveraged to enhance the throughput. In this paper, we study the proportional-fair scheduling problem in the presence of network coding in OFDMA relay networks. Considering the tradeoff between performance and overhead, we propose two models, global approach (GA) and local approach (LA), under which the corresponding problems are shown both NP-hard. For the GA model, we show that it cannot be approximated within some constant factor. Hence, we propose a heuristic algorithm with low time complexity. For the LA model, we propose a theoretical polynomial time approximation scheme (PTAS), and also present a practical greedy algorithm with approximation factor of 1 2 . Simulation results show that our algorithms can achieve significant throughput improvement over a state-of-the-art noncoding scheme. Index Terms—Network coding, OFDMA relay networks, proportional-fair scheduling, approximation algorithm Ç 1 INTRODUCTION O RTHOGONAL Frequency Division Multiplexing (OFDM) is a digital modulation scheme that can combat multipath fading/interference robustly, achieve high spec- tral efficiency, and be easily implemented using the Fast Fourier Transform (FFT) Algorithm 1. In the last decade, as a multiuser version of OFDM, Orthogonal Frequency Division Multiple Access (OFDMA) has become a promis- ing technology for supporting broadband wireless access and been adopted by fourth generation standards including IEEE 802.16e and 3GPP Long Term Evolution (LTE). Generally, a typical OFDMA-based cellular network has a high bandwidth and is expected to support various bandwidth-intensive applications; however, it usually has a limited communication range, and often suffers from coverage holes. A popular cost-effective approach for extending the range and filling up the holes is to add relay stations (RSs) between the base station (BS) and mobile stations (MSs, or users), thereby making the network multihop in essence. Recently, the two-hop OFDMA relay networks (see Fig. 1a, e.g.) have become a dominant component in some emerging fourth generation standards such as IEEE 802.16j [2] and 3GPP LTE-Advanced [3]. In two-hop OFDMA relay networks, the prescribed frequency band is divided into multiple narrow subchan- nels, and time is partitioned into frames of multiple time slots. The scheduling, carried out by the BS, is to allocate subchannels across the two hops over time slots on a frame basis, such that all MSs can be served in an efficient and fair manner. The two-hop nature, on one hand, makes the scheduling problem difficult to deal with [4]; on the other hand, it also brings opportunities to employ network coding, which has been proved an effective approach to improve the network throughput [5], [6]. As various interactive applications (e.g., online gaming, peer-to-peer streaming, and so on) are supported by OFDMA relay networks, numerous bidirectional flows involved in these applications pass through RSs, resulting in a plenty of network coding opportunities that can be leveraged to enhance the throughput. Such benefit of network coding is illustrated in Figs. 1b and 1c via a simple packet exchange scenario. However, most existing solutions just simply schedule downlink and uplink traffic separately, thereby missing such coding opportunities. While network coding could potentially improve the network throughput, it would hurt the throughput if blindly used. Taking Fig. 1c as an example, we consider the coded data to be broadcasted to both the BS and MS over a given subchannel. To ensure that both the BS and MS can receive the data successfully, the achieved broadcast rate is bounded by the lower one of rates over links (RS, BS) and (RS, MS), leading to a diminished throughput if one of the link rates is very low. Therefore, the scheduling decision should be made under a well-devised network coding mechanism. In this paper, we address the scheduling problem in the presence of network coding in two-hop OFDMA relay networks. We adopt the widely used proportional-fair scheduling policy [7], which provides a good balance between network throughput and system fairness. Con- sidering the tradeoff between performance and overheads, we consider two models for supporting the coding-aware scheduling decision making, which have been accepted by IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 9, SEPTEMBER 2013 1727 . B. Tang, B. Ye, and S. Lu are with the National Key Laboratory for Novel Software Technology, Nanjing University, Xianlin Campus, 163 Xianlin Avenue, Qixia District, Nanjing 210046, China. E-mail: [email protected], {yebl, sanglu}@nju.edu.cn. . S. Guo is with Performance Evaluation Lab, School of Computer Science and Engineering, The University of Aizu, Tsuruga, Ikki-machi, Aizu- Wakamastu City, Fukushima 965-8580, Japan. E-mail: [email protected]. Manuscript received 25 Oct. 2011; revised 8 June 2012; accepted 2 Sept. 2012; published online 11 Sept. 2012. Recommended for acceptance by M. Guo. For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number TPDS-2011-10-0789. Digital Object Identifier no. 10.1109/TPDS.2012.269. 1045-9219/13/$31.00 ß 2013 IEEE Published by the IEEE Computer Society

IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

  • Upload
    others

  • View
    2

  • Download
    0

Embed Size (px)

Citation preview

Page 1: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

Coding-Aware Proportional-Fair Schedulingin OFDMA Relay Networks

Bin Tang, Student Member, IEEE, Baoliu Ye, Member, IEEE,

Sanglu Lu, Member, IEEE, and Song Guo, Senior Member, IEEE

Abstract—In recent years, OFDMA relay networks have become a key component in the 4G standards (e.g., IEEE 802.16j, 3GPP LTE-

Advanced) for broadband wireless access. When numerous bidirectional flows pass through the relay stations in an OFDMA relay

network that supports various interactive applications, plenty of network coding opportunities arise and can be leveraged to enhance the

throughput. In this paper, we study the proportional-fair scheduling problem in the presence of network coding in OFDMA relay networks.

Considering the tradeoff between performance and overhead, we propose two models, global approach (GA) and local approach (LA),

under which the corresponding problems are shown both NP-hard. For the GA model, we show that it cannot be approximated within

some constant factor. Hence, we propose a heuristic algorithm with low time complexity. For the LA model, we propose a theoretical

polynomial time approximation scheme (PTAS), and also present a practical greedy algorithm with approximation factor of 12 . Simulation

results show that our algorithms can achieve significant throughput improvement over a state-of-the-art noncoding scheme.

Index Terms—Network coding, OFDMA relay networks, proportional-fair scheduling, approximation algorithm

Ç

1 INTRODUCTION

ORTHOGONAL Frequency Division Multiplexing (OFDM)is a digital modulation scheme that can combat

multipath fading/interference robustly, achieve high spec-tral efficiency, and be easily implemented using the FastFourier Transform (FFT) Algorithm 1. In the last decade, asa multiuser version of OFDM, Orthogonal FrequencyDivision Multiple Access (OFDMA) has become a promis-ing technology for supporting broadband wireless accessand been adopted by fourth generation standards includingIEEE 802.16e and 3GPP Long Term Evolution (LTE).

Generally, a typical OFDMA-based cellular network hasa high bandwidth and is expected to support variousbandwidth-intensive applications; however, it usuallyhas a limited communication range, and often suffers fromcoverage holes. A popular cost-effective approach forextending the range and filling up the holes is to add relaystations (RSs) between the base station (BS) and mobilestations (MSs, or users), thereby making the networkmultihop in essence. Recently, the two-hop OFDMA relaynetworks (see Fig. 1a, e.g.) have become a dominantcomponent in some emerging fourth generation standardssuch as IEEE 802.16j [2] and 3GPP LTE-Advanced [3].

In two-hop OFDMA relay networks, the prescribedfrequency band is divided into multiple narrow subchan-

nels, and time is partitioned into frames of multipletime slots. The scheduling, carried out by the BS, is toallocate subchannels across the two hops over time slots ona frame basis, such that all MSs can be served in an efficientand fair manner. The two-hop nature, on one hand, makesthe scheduling problem difficult to deal with [4]; on theother hand, it also brings opportunities to employ networkcoding, which has been proved an effective approach toimprove the network throughput [5], [6]. As variousinteractive applications (e.g., online gaming, peer-to-peerstreaming, and so on) are supported by OFDMA relaynetworks, numerous bidirectional flows involved in theseapplications pass through RSs, resulting in a plenty ofnetwork coding opportunities that can be leveraged toenhance the throughput. Such benefit of network coding isillustrated in Figs. 1b and 1c via a simple packet exchangescenario. However, most existing solutions just simplyschedule downlink and uplink traffic separately, therebymissing such coding opportunities.

While network coding could potentially improve thenetwork throughput, it would hurt the throughput ifblindly used. Taking Fig. 1c as an example, we considerthe coded data to be broadcasted to both the BS and MSover a given subchannel. To ensure that both the BS and MScan receive the data successfully, the achieved broadcastrate is bounded by the lower one of rates over links (RS, BS)and (RS, MS), leading to a diminished throughput if one ofthe link rates is very low. Therefore, the schedulingdecision should be made under a well-devised networkcoding mechanism.

In this paper, we address the scheduling problem in thepresence of network coding in two-hop OFDMA relaynetworks. We adopt the widely used proportional-fairscheduling policy [7], which provides a good balancebetween network throughput and system fairness. Con-sidering the tradeoff between performance and overheads,we consider two models for supporting the coding-awarescheduling decision making, which have been accepted by

IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 9, SEPTEMBER 2013 1727

. B. Tang, B. Ye, and S. Lu are with the National Key Laboratory for NovelSoftware Technology, Nanjing University, Xianlin Campus, 163 XianlinAvenue, Qixia District, Nanjing 210046, China.E-mail: [email protected], {yebl, sanglu}@nju.edu.cn.

. S. Guo is with Performance Evaluation Lab, School of Computer Scienceand Engineering, The University of Aizu, Tsuruga, Ikki-machi, Aizu-Wakamastu City, Fukushima 965-8580, Japan. E-mail: [email protected].

Manuscript received 25 Oct. 2011; revised 8 June 2012; accepted 2 Sept. 2012;published online 11 Sept. 2012.Recommended for acceptance by M. Guo.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TPDS-2011-10-0789.Digital Object Identifier no. 10.1109/TPDS.2012.269.

1045-9219/13/$31.00 � 2013 IEEE Published by the IEEE Computer Society

Page 2: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

the community. One is called the global approach (GA),which has been comprehensively studied in coding-obliv-ious scheduling problems, for example, in [4], [7]. Ourpaper extends this model for the coding-aware schedulingin OFDMA relay networks. The other is named the localapproach (LA), which is essentially the classic round-robinscheduling scheme and has been also widely adopted, forexample, in [8]. We study it with the target of proportionalfairness in this paper. Both models investigate the benefitsof network coding and frequency selectivity. In particular,the GA model achieves a higher performance by exploitingthe multiuser diversity gain, but incurs a significantoverhead due to the collection of link rates over the wholenetwork; in contrast, the LA model processes the schedul-ing in a simplified and local fashion, and thus introduces amuch lower overhead with the cost of sacrificing themultiuser diversity. For both the proportional-fair schedul-ing problems under each model, we establish their hardnessand propose efficient polynomial time algorithms.

Our main contributions are summarized as follows:

1. Under the GA model, we prove that the coding-aware scheduling problem is NP-hard and nopolynomial time approximation scheme (PTAS)1

exits. Then, we propose a heuristic algorithm withlow time complexity.

2. Under the LA model, we show its NP-hardness andpresent a theoretical PTAS. We also propose apractical greedy algorithm with an approximationfactor of 1

2 .3. Our algorithms are evaluated to highlight the benefit

of network coding via simulations. Simulation resultsshow that our algorithms are close to optimal and canachieve about 10-30 percent throughput improve-ment over a state-of-the-art noncoding scheme.

The remainder of the paper is organized as follows:Section 2 discusses about some related work, and Section 3describes the system model. Coding-aware schedulingproblems under the GA model and the LA model areinvestigated in Sections 4 and 5, respectively. Performanceevaluation of our algorithms is presented in Section 6.Finally, Section 7 concludes.

2 RELATED WORK

In the context of OFDMA-based single hop networks, thescheduling problem has been extensively studied, forexample, in [9], [10], [11], [12], [13]. However, theseapproaches cannot be easily applied to OFDMA relaynetworks, where RSs are introduced to improve networkcoverage and capacity [14], [15], making the networkessentially multihop. Therefore, great efforts have beenmade in designing efficient scheduling algorithms forOFDMA relay networks. In [4], the authors propose severalscheduling algorithms to exploit both multiuser diversity,frequency selectivity, as well as spatial reuse in OFDMA-based two-hop relay networks. They also present algo-rithms for low overhead scheduling in [16]. In [17], theauthors study the downlink scheduling problem to exploitthe multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis on IEEE802.16j-based networks. In [18], the authors investigate theQoS-aware scheduling problem for multihop relay net-works. As these efforts focus on the downlink schedulingand consider the uplink scheduling as a symmetric case,none of them exploit the benefit of network coding whenbidirectional traffic is taken into account. In contrast, wepropose network coding-aware scheduling algorithmsbased on a framework in [4] that we extend to accom-modate network coding into OFDMA relay networks. Itssignificant benefit is demonstrated by the supreme perfor-mance of our algorithms over the state-of-the-art coding-oblivious scheduling algorithm DIV1 [4].

Network coding, first proposed by Ahlswede et al. [5],has become a promising approach to improve the perfor-

1728 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 9, SEPTEMBER 2013

1. For a maximization problem, a PTAS is an algorithm which takes aparameter � > 0 as an input, and produces a solution that is within a factor1� � of being optimal in polynomial time.

Fig. 1. Illustration of network model and the benefit of network coding. (a) A two-hop network model. (b) Without network coding, the packet

exchange requires four transmissions. (c) With network coding, RS combines the two received packets by XOR (exclusive or) operation, and

broadcasts the coded packet. Both BS and MS can recover each other’s packet by XORing again with their own packet. This process only takes

three transmissions. The saved transmission can be used for new data to achieve an increased throughput.

Page 3: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

mance of wireless networks. In [6], the authors propose apractical XOR-based network coding scheme named COPEto improve the throughput of multiple unicast flows in802.11-based wireless mesh networks. Compared to theextensive research on coding algorithms (e.g., [19]), coding-aware routing (e.g., [20], [21]), and so on, in 802.11-basedwireless networks, only a few show the potential benefits ofnetwork coding in OFDMA based networks. In [23], theauthors study the network coding-aware scheduling pro-blem while making use of the multiuser diversity andfrequency selectivity in OFDMA-based single-hop cellularnetworks, where coding is performed by BS when messagesare exchanged between MSs. Obviously, this work cannotbe extended to two-hop networks, where coding is used forreducing traffic at RSs. The authors in [22] and [25] showthat network coding can achieve a significant throughputgain when the cooperative diversity is considered as well inrelay-assisted OFDMA networks. The authors in [24] studythe joint routing and resource allocation problem in thepresence of network coding with the objective to maximizethe scaling factor for some given traffic pattern. In [26], theauthors also propose a network coding-based opportunisticscheduling scheme for OFDMA relay networks, but theyallow that data can be stored at RSs over time, making itinconsistent with the simplicity assumption of RSs. Allthese proposals in [25], [24], [26] focus on maximizing theaggregate throughput over all MSs without taking fairnessinto account. Besides, they assume that each schedulingunit can be continuously partitioned with no constraintsimposed by the frame structure in OFDMA relay networks.In contrast, we adopt the proportional fairness as theobjective metric for the frame-based scheduling problemsuch that MSs can be served in an efficient and fair manner.

3 SYSTEM DESCRIPTION

3.1 Network Model

We consider a two-hop OFDMA relay network as shown inFig. 1a, in which the network consists of one BS, a set of RSsR (jRj ¼ R), and a set of MSs M (jMj ¼M). Each MSconnects to the BS directly or via an RS according to somepredetermined routing scheme, such as using the transmis-sion time based metric [27]. The prescribed frequency bandis divided into a set of multiple orthogonal subchannelsC ðjCj ¼ CÞ that are allowed to be used by all stations.

We consider a synchronized, time-slotted, frame-basedOFDMA relay network, where each station has only onetransceiver and, hence, cannot transmit and receive con-currently. As shown in Fig. 2, each frame consists of T timeslots and C subchannels. The scheduling, usually per-formed by the BS at the beginning of each frame, is toallocate the two dimensional resources to users in anefficient and fair manner. To leverage the benefit of networkcoding, the frame is partitioned into three sequentialsubframes named downlink subframe, uplink subframe, andrelay subframe. As the names suggest, the downlinksubframe is used by the BS to transmit data to RSs, theuplink subframe is used by MSs to upload data to RSs, andthe relay subframe is used by RSs to forward the receiveddata to respective MSs and BS with or without networkcoding. Different from the scheme proposed in [25] thatpartitions the whole frame into equal-sized subframes, ourpartition allows these subframes with different sizes. The

optimal partition can be found based on the proposedalgorithms by exhaustive search, with an increased timecomplexity by OðT 2Þ times. The XOR-based network codingoperations are performed as illustrated in Fig. 1c, in whicheach coded packet is attached with the required informationto guarantee the decoding at its intended receivers.

Similar to [12], [4] for ease of presentation, we use the

following equivalent technical treatments:

. We consider each MS connects to the BS via an RS.For those MSs with direct connections with the BS,we can separate their traffic scheduling from othersby assigning different frames to them.

. We consider only one time slot for each subframe. Thesubchannels of the same subframe in other time slotscan be seen as additional subchannels available tothe considered time slot. In such a way, all sub-channels for downlink, uplink, and relay subframesare denoted as Ci (jCij ¼ Ci) with i ¼ 1; 2 and 3,respectively.

Finally, to indicate the subchannel allocation, we intro-duce binary variables Iiðm; cÞ, which is equal to one ifsubchannel c 2 Ci is allocated to MS m, and equal to zerootherwise, in the downlink subframe (i ¼ 1) or uplinksubframe (i ¼ 2). If Iiðm; cÞ ¼ 1, the corresponding link rateis denoted as riðm; cÞ. Similarly, I3ðm; c; kÞ represents aworking mode k, under which subchannel c 2 C3 isallocated to forward downlink traffic (k ¼ 1), to forwarduplink traffic (k ¼ 2), or to broadcast traffic with coded data(k ¼ 3) for MS m with the corresponding rate r3ðm; c; kÞ. Tomake sure both BS and MS m can receive the broadcast datasuccessfully, the broadcast rate is constrained by the lowerachievable rate for unicasts, i.e.,

r3ðm; c; 3Þ ¼ minfr3ðm; c; 1Þ; r3ðm; c; 2Þg: ð1Þ

Note that all rates mentioned above are given by the system

configuration and known beforehand.

3.2 Proportional-Fair Scheduling

Generally, the BS makes the scheduling decision accordingto some optimization objective, and then disseminates theresult in the preamble of each frame. To maintain a goodbalance between network throughput and system fairness,we adopt the proportional-fair scheduling, which is apopular scheduling policy, and has been widely used inOFDMA systems [28], [29], [17]. Under this policy, theoptimization objective in a long run is to maximize

TANG ET AL.: CODING-AWARE PROPORTIONAL-FAIR SCHEDULING IN OFDMA RELAY NETWORKS 1729

Fig. 2. Illustration of frame structure in a two-hop OFDMA relay network.

In this example, both the downlink and uplink subframes have four time

slots, while the relay subframe has six time slots.

Page 4: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

Xm2MðlogDTm þ logUTmÞ; ð2Þ

where DTm and UTm are the long-term downlink and

uplink throughput of MS m, respectively. As proved in [7],

such optimization can be achieved by maximizing the

following objective function in each frame2:

Xm2M

dmDmþ umUm

� �; ð3Þ

where dm (um) is the total downlink (uplink) data amount

in bits for MS m to be allocated in the current frame, and

Dm (Um) denotes the average downlink (uplink) rate for

MS m till the previous frame.In the following, we propose two approaches to

maximize objective (3) in terms of a tradeoff between

performance and overheads:

. GA: It aims at fully exploiting multiuser diversityand frequency selectivity, as well as network codinggain, by acquiring all link rate information over thewhole network. It works as follows: First, all datarates in bits/slot captured through channel stateinformation (CSI) at all links for each subchannel arereported to BS. Then, the BS runs some algorithm tomake the scheduling decision. Finally, the BS broad-casts the scheduling decision during the transmis-sion of the preamble. Such model can been seen as adirect extension of the model for coding-obliviousscheduling [4], [17].

. LA: Inspired by the simplest round-robin scheduling[8], this approach introduces a restriction that eachframe can only be used to serve one MS. With thisrestriction, the scheduling can be processed in asimplified and local fashion as described below. Atthe beginning, the link rates are fed back to theircorresponding RSs. Each RS then calculates themaximum attainable value vm of dm

Dmþ um

Umfor each

associated MS m, under the assumption that thewhole frame is used to serve m. Subsequently, thelargest vm from its MSs is reported to the BS. Finally,the BS picks the MS m with the largest vm, andbroadcasts the decision that m will be served duringthe next whole frame. Compared to the round-robinscheduling, LA exploits the benefit of networkcoding targeting proportional fairness while main-tains almost the same simplicity.

Comparing the above two approaches, we have thefollowing observations. First, all CSI information should befurther forwarded to BS in the GA model, resulting in adoubled CSI overhead, which is the dominant componentof the whole feedback overhead, of the LA model.Furthermore, LA only introduces a constant decisiondissemination overhead, which is significantly less thanthat of GA. On the other hand, the multiuser diversity gainis not exploited in LA, and thus it would lead to someperformance degradation.

4 CODING-AWARE SCHEDULING UNDER GA MODEL

In this section, we study the Coding-Aware proportional-fair scheduling problem under the GA model, denoted asCA-GA. We first formulate CA-GA as an integer linearprogramming, and then establish its hardness. A low-complexity heuristic algorithm is presented at the end ofthis section.

4.1 CA-GA Formulation

Based on the predefined indicator variables defined earlier,we can formulate the CA-GA as an integer linear program-ming problem as follows:

CA-GA: maxXm2M

dmDmþ umUm

� �; ð4Þ

s.t.

dm �Xc2C1

r1ðm; cÞI1ðm; cÞ; 8m 2 M ð5Þ

dm �Xc2C3

Xk2f1;3g

r3ðm; c; kÞI3ðm; c; kÞ; 8m 2 M ð6Þ

um �Xc2C2

r2ðm; cÞI2ðm; cÞ; 8m 2 M ð7Þ

um �Xc2C3

Xk2f2;3g

r3ðm; c; kÞI3ðm; c; kÞ; 8m 2 M ð8Þ

Xm2M

Iiðm; cÞ � 1; 8i 2 f1; 2g; c 2 Ci ð9Þ

Xm2M

X3

k¼1

I3ðm; c; kÞ � 1; 8c 2 C3 ð10Þ

Iiðm; cÞ 2 f0; 1g; 8m 2 M; i 2 f1; 2g; c 2 Ci ð11Þ

I3ðm; c; kÞ 2 f0; 1g; 8m 2M; c 2 C3; k 2 f1; 2; 3g: ð12Þ

For the subchannel allocation of each frame, (5), (6), and(7), (8) characterize the achievable downlink and uplink dataamounts for each MS m, respectively. The standard networkconfiguration [4], [17], where each RS has no per-user bufferdue to its simplicity, requires the flow conservation to bestrictly guaranteed over the whole frame duration. Constraint(9) states that each subchannel in the downlink/uplinksubframes can be allocated to only one MS. Similarly,constraint (10) represents that each subchannel in the relaysubframe can be assigned to only one MS in a specific workingmode. Finally, (11) and (12) are indicator constraints.

4.2 Hardness Analysis

In this section, we analyze the hardness of the CA-GA

problem. The theoretical results are given in the following

theorem.

Theorem 1. The CA-GA problem is NP-hard and no PTAS

exists, i.e., for some positive constant � > 0, it does not admit

any ð1� �Þ-approximation algorithm unless P ¼ NP .

1730 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 9, SEPTEMBER 2013

2. When the data flow have different priorities, we can modify thisobjective function by associating each addition term with a multiplier whichindicates the corresponding flow priority. In this case, our results still holdby integrating these priority multipliers with the denominators.

Page 5: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

Proof. When there is only one MS in the network, the CA-GA

problem degenerates into the CA-LA problem which is

NP-hard as shown in the proof of Theorem 3 in Section 5.2.

Therefore, the CA-GA problem is also NP-hard.To prove the stronger result that CA-GA does not

admit any PTAS, we make an approximation factorpreserving reduction [30] from a scheduling problemunder queuing model (SUQ) [12] to CA-GA, because noPTAS exists for SUQ [12]. We first give the definition ofSUQ for the sake of completeness.

SUQ: Consider a one-hop OFDMA-based cellularnetwork where the BS transmits data to M MSs throughC subchannels directly. Each MS m has a finite backlogQm at the BS, i.e., the data transmitted to m cannotexceed Qm. Then, SUQ is to find a feasible subchannelassignment such that

Pm Qmminf

Pc rðm; cÞIðm; cÞ; Qmg

is maximized, where rðm; cÞ denotes the rate of sub-channel c for MS m and Iðm; cÞ indicates whether c isassigned to m.

Given any instance of SUQ as shown above, we willconstruct an instance of CA-GA with downlink trafficonly, such that they have the same optimal solution.Consider an OFDMA relay network consisting of M MSsand the corresponding associated M RSs, in which Ccommon subchannels and M private subchannels areavailable. A common subchannel can be used by BS totransmit downlink data to any RS, while a privatesubchannel is dedicated to a pair of associated RS andMS. More specifically, for each subchannel c and MS m,we set r1ðm; cÞ ¼ rðm; cÞ if c is a common subchannel,and set r3ðm; c; 1Þ ¼ Qm if c is a private subchannel. Allother rates are set to zero. Finally, we set Dm ¼ 1=Qm andUm ¼ 1 for each MS m.

Under this construction, a frame is partitioned intodownlink subframe and relay subframe, each having asingle time slot, and no benefit of network coding can beobtained. Note that during the relay subframe, only theassociated private subchannel is available for each MSm, which forms a nominal backlog size of Qm forassigning subchannels in the downlink subframe due tothe flow conservation constraints in (5) and (6). It is astraightforward exercise to show that both instanceshave the same optimal solution. Besides, any feasibleassignment of subchannels in the downlink subframefor the CA-GA problem provides a feasible assignmentfor the corresponding instance of SUQ with equalobjective value. In summary, such reduction preservesthe approximation factor. tu

4.3 Heuristic Algorithm

Although the GA model provides a general approach tofully exploit multiuser diversity and frequency selectivity,Theorem 1 reveals that no efficient algorithm for the CA-GAproblem exists in a technical sense. In other words, it ishardly to find exact or even approximate solutions for CA-GA instances. In the following, we propose a practical, lowtime complexity heuristic algorithm Emulated MaxWeightalgorithm (EMW), which can be seen as an emulation of thewell-known MaxWeight algorithm [31], [32], [33].

Note that the data amount dm (um) is decided by a jointsubchannel assignment for the downlink/uplink and relay

sub-frames. To make CA-GA easy to deal with, thisheuristic algorithm decouples the correlation in jointoptimization by allocating the subchannels in the relaysubframe first. Let wrcðm; kÞ denote the weight of assigningrelay subchannel c 2 C3 to MS m in mode k. It is defined as

wrcðm; kÞ ¼

r3ðm;c;1ÞDm

k ¼ 1;r3ðm;c;2Þ

Umk ¼ 2;

r3ðm;c;3ÞDm

þ r3ðm;c;3ÞUm

k ¼ 3:

8>><>>: ð13Þ

Then, each subchannel c 2 C3 is assigned to MS m in modek, where ðm; kÞ ¼ arg maxm;kw

rcðm; kÞ, such that the utility of

assignment could be maximized. Once the assignment of allsubchannels in the relay subframe is determined, accordingto the flow conservation constraints, the upper bounds ofdata amounts dm and um for each MS m will be also known,denoted as Qd

m and Qum, respectively. In the subsequent

subchannel allocation of downlink/uplink subframes, weapply a greedy technique that has been used in [12] as well.The main idea is to assign subchannels in a sequentialmanner. Let Qd

m be the residual data amount for downlinktraffic of MS m. Then, considering the flow conservation, wedefine the weight wdi ðmÞ for assigning downlink subchannelc to MS m as

wdcðmÞ ¼min

�Qdm; r1ðm; cÞ

�Dm

: ð14Þ

Our greedy algorithm will assign subchannel c to MS m,where m ¼ arg maxmw

di ðmÞ. After that, the residual data

volume will be updated as

Qdm max

�0; Qd

m � r1ðm; cÞ�; ð15Þ

and

Qdm Qd

m; 8m 6¼ m: ð16Þ

The assignment of subchannels in the uplink subframe isconducted in a similar way. The formal description of thewhole algorithm is given in Algorithm 1. The followingresult is rather straightforward from Algorithm 1.

Proposition 2. The computational complexity of AlgorithmEMW is OðMCÞ.

Algorithm 1. Algorithm EMW for CA-GA

1: for c 2 C3 do

2: ðm; kÞ ¼ arg maxm;kwrcðm; kÞ.

3: I3ðm; c; kÞ 1.4: I3ðm; c; kÞ 0, 8m 6¼ m, or k 6¼ k.

5: end for

6: for m 2 M do

7: Qdm

Pc2C3

Pk2f1;3g r3ðm; c; kÞI3ðm; c; kÞ.

8: Qum

Pc2C3

Pk2f2;3g r3ðm; c; kÞI3ðm; c; kÞ.

9: end for

10: for c ¼ 1 to C1 do

11: m arg maxmminfQd

m;r1ðm;cÞgDm

.12: I1ðm; cÞ 1; I1ðm; cÞ 0, 8m 6¼ m.

13: Qdm maxf0; Qd

m � r1ðm; cÞg.14: end for

15: for c ¼ 1 to C2 do

16: m arg maxmminfQu

m;r2ðm;cÞgUm

.

TANG ET AL.: CODING-AWARE PROPORTIONAL-FAIR SCHEDULING IN OFDMA RELAY NETWORKS 1731

Page 6: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

17: I2ðm; cÞ 1; I2ðm; cÞ 0, 8m 6¼ m.18: Qu

m maxf0; Qum � r2ðm; cÞg.

19: end for

5 CODING-AWARE SCHEDULING UNDER LA MODEL

In this section, we consider the Coding-Aware schedulingproblem under the LA model (CA-LA). We first formulate itas an integer linear programming and show it NP-hard.Then, we propose a practical 1

2 -approximation algorithm.

5.1 CA-LA Formulation

Under the LA model, the optimizations are conducted foreach MS m. For this reason, we omit dimension-m to allsymbols in this section without incurring any confusion.The corresponding problem formulation can, thus, berewritten as follows:

CA-LA: maxd

Dþ u

U; ð17Þ

s.t.

d �Xc2C1

r1ðcÞ ¼4 Qd ð18Þ

d �Xc2C3

Xk2f1;3g

r3ðc; kÞI3ðc; kÞ ð19Þ

u �Xc2C2

r2ðcÞ ¼4 Qu ð20Þ

u �Xc2C3

Xk2f2;3g

r3ðc; kÞI3ðc; kÞ ð21Þ

X3

k¼1

I3ðc; kÞ � 1; 8c 2 C3 ð22Þ

I3ðc; kÞ 2 f0; 1g; 8c 2 C3; k 2 f1; 2; 3g: ð23Þ

5.2 Hardness Analysis

While this problem seems to be simplified under the LAmodel, the following analysis reveals that CA-LA is stillNP-hard.

Theorem 3. The CA-LA problem is NP-hard.

Proof. The proof follows by reducing the partition problem,which is a well-known NP-hard problem, to an instanceof CA-LA. In the partition problem, we are given a finiteset A and a size sðaÞ 2 Zþ for each a 2 A. For any subsetY � A, we use sðYÞ to denote

Pa2Y sðaÞ. The partition

problem is to decide whether there exists a subsetA0 � A, such that sðA0Þ ¼ sðA � A0Þ.

An instance of CA-LA with three time slots is, thus,constructed as follows: For each element a 2 A, theinstance includes a subchannel ca such that r1ðcaÞ ¼3sðaÞ; r2ðcaÞ ¼ sðaÞ; r3ðca; 1Þ ¼ 4sðaÞ, and r3ðca; 2Þ ¼ 2sðaÞ.The last two settings lead to r3ðca; 3Þ ¼ minfr3ðca; 1Þ;r3ðca; 2Þg ¼ 2sðaÞ. Finally, we set D ¼ 1 and U ¼ 1

2 . Then,the objective is to maximize obj ¼ dþ 2u.

We list all possible scheduling strategies as follows:

. Strategy 1: The whole frame is used for downlinktraffic. It is a straightforward exercise to showthe allocation scheme that the first two slots areused by link (BS, RS) and the last one by link (RS,MS) will maximize the objective to be obj ¼dþ 2u ¼ 4sðAÞ, where d ¼ minf6sðAÞ; 4sðAÞg ¼4sðAÞ and u ¼ 0.

. Strategy 2: The whole frame is used for uplinktraffic. Similarly, we have the maximum obj ¼ d þ2u ¼ 0þ 4sðAÞ ¼ 4sðAÞ.

. Strategy 3: The frame is partitioned into threesubframes, with one time slot for each. Due to theflow conservation, we have d �

Pa2A r1ðcaÞ ¼

3sðAÞ and u �P

a2A r2ðcaÞ ¼ sðAÞ. Therefore, weobtain obj � 5sðAÞ.

Denote OPT as the optimal value of the instance of CA-LA. The above analysis shows OPT � 5sðAÞ. In thefollowing, we prove that A has a feasible partition if andonly if OPT ¼ 5sðAÞ.

If there exists a subset A0 � A such that sðA0Þ ¼sðA � A0Þ ¼ 1

2 sðAÞ, then we adopt the third strategy. Inthe last time slot, subchannels corresponding to elementsin A0 are used for broadcasting with network coding, andother subchannels are assigned only for downlink traffic.Then, we have

d ¼ minXa2A

r1ðcaÞ;X

a2A�A0r3ðca; 1Þ þ

Xa2A0

r3ðca; 3Þ( )

¼ minf3sðAÞ; 4sðA �A0Þ þ 2sðA0Þg¼ minf3sðAÞ; 3sðAÞg ¼ 3sðAÞ;

and

u ¼ minXa2A

r2ðcaÞ;Xa2A0

r3ðca; 3Þ( )

¼ minfsðAÞ; 2sðA0Þg

¼ minfsðAÞ; sðAÞg ¼ sðAÞ:

This leads to obj ¼ 5sðAÞ � OPT . Combining thisresult OPT � 5sðAÞ obtained earlier, we concludeOPT ¼ 5sðAÞ.

Conversely, if OPT ¼ 5sðAÞ, then the third strategymust be used, and furthermore d ¼ 3sðAÞ and u ¼ sðAÞ.Let Bk (k ¼ 1; 2; 3) be the disjoint sets of elementscorresponding to subchannels working on mode k in thelast time slot (i.e., the relay subframe). According to theflow conservation, we can conclude 4sðB1Þ þ 2sðB3Þ �d ¼ 3sðAÞ and 2sðB2Þ þ 2sðB3Þ � u ¼ sðAÞ. CombiningsðB1Þ þ sðB2Þ þ sðB3Þ ¼ sðAÞ and sð�Þ � 0, we can derivesðB1Þ ¼ sðB3Þ ¼ 1

2 sðAÞ and sðB2Þ ¼ 0. In other words, B1

(or B3) forms a feasible partition to set A. tu

Further investigation on the hardness of the CA-LAproblem leads to an important discovery that a theoreticalPTAS exists for the problem as stated in Theorem 4. Thisresult is different from the previous section and implies CA-LA is easier than CA-GA in some technical sense.

Theorem 4. The CA-LA problem admits some PTAS.

1732 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 9, SEPTEMBER 2013

Page 7: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

Proof. The proof follows by constructing a PTAS for CA-LA. The details are given in the supplemental material,which can be found on the Computer Society DigitalLibrary at http://doi.ieeecomputersociety.org/10.1109/TPDS.2012.269. tu

5.3 Half-Approximation Greedy Algorithm (HAG)

Although the PTAS can achieve an almost optimalperformance in polynomial time, it is still too complex tobe implemented in practical systems. Alternatively, wepropose a practical greedy algorithm HAG with guaranteedperformance.

In the greedy algorithm HAG, the subchannels in the relaysubframe, numbered as 1; 2; . . . ; C3, are assigned in sequence.LetQd

c andQuc be the residual data amount for downlink and

uplink traffic, respectively, before subchannel c is allocated.Initially, we have Qd

1 ¼ Qd and Qu1 ¼ Qu. The subchannel

allocation is conducted based on a weight wcðkÞ that isdefined for subchannel c working on mode k as follows:

wcðkÞ ¼

minfQdc ; r3ðc; 1ÞgD

k ¼ 1;

minfQuc ; r3ðc; 2ÞgU

k ¼ 2;

minfQdc ; r3ðc; 3ÞgD

þminfQuc ; r3ðc; 3ÞgU

k ¼ 3:

8>>>>><>>>>>:

ð24Þ

The working mode k is chosen for each subchannel c suchthat k ¼ arg maxkwiðkÞ. After c is assigned, the residual dataamounts become

Qdcþ1 ¼

maxf0; Qdc � r3ðc; 1Þg k ¼ 1;

Qdc k ¼ 2;

maxf0; Qdc � r3ðc; 3Þg k ¼ 3;

8<: ð25Þ

and

Qucþ1 ¼

Quc k ¼ 1;

maxf0; Quc � r3ðc; 2Þg k ¼ 2;

maxf0; Quc � r3ðc; 3Þg k ¼ 3:

8<: ð26Þ

The formal description of HAG is given in Algorithm 2.It is apparent that the running time of Algorithm HAG islinear on the number of subchannels.

Proposition 5. The time complexity of Algorithm HAG is OðCÞ.

Algorithm 2. Algorithm HAG for CA-LA

1: Initialization:

2: Qd1

Pc2C1

r1ðcÞ.3: Qu

1 P

c2C2r2ðcÞ.

4: Sequential Assignment:

5: for c ¼ 1 to C3 do

6: k arg maxkwcðkÞ.7: I3ðc; kÞ 1; I3ðc; kÞ 0; 8k 6¼ k.

8: Update Qdc and Qu

c according to Eqs. (25) and (26),

respectively.

9: end for

Finally, we show the theoretical performance of theproposed algorithm HAG in terms of approximation ratio.The basic idea is to show that HAG can be viewed as a special

case of a greedy algorithm for maximizing a nondecreasing

submodular function over a partition matroid. For the sake

of completeness, we first introduce the definitions of

partition matroid and nondecreasing submodular functions:

. A matroid is an ordered pair ðS; IÞ, where S is a finitenonempty set and I is a set of subsets of S, satisfying

- ; 2 I ,- if B 2 I and A � B, then A 2 I , and- if A;B 2 I , and jAj < jBj, then there is some

element x 2 B �A such that A [ fxg 2 I .. A matroid ðS; IÞ is said to be a partition matroid, if

there exists some partition of S into components�1;�2; . . . , such that A 2 I if and only if jA \ �kj � 1,for all k.

. A function fð�Þ on sets in I is said to be submodularand nondecreasing if it satisfies

- fð;Þ ¼ 0 and- For all a 2 S,A;B 2 I , if A [ fag 2 I and B � A,

then fðAÞ � fðA [ fagÞ and fðA [ fagÞ �fðAÞ � fðB [ fagÞ � fðBÞ:

For the problem of maximizing a nondecreasing submod-

ular function over a partition matroid, the greedy algorithm

proposed in [34] works as follows: Set A is initialized as

empty at the beginning. In each subsequent iterative step,

an element a from component �k is picked up such that

fðA [ fagÞ � fðAÞ is maximized and then A is updated as

A [ fag. It has been shown that this greedy algorithm

achieves an approximation guarantee of 12 [34].

The formal result as well as its proof are given below.

Theorem 6. Algorithm HAG achieves an approximation factor of12 for the CA-LA problem.

Proof. We construct a partition matroid for a given CA-LA

problem. We define S ¼ fðc; kÞ j 1 � c � C3; 1 � k � 3g,and I to be a set of subsets of S as follows: for each

A � S;A 2 I if and only if A satisfies that for any

ðc; kÞ 2 A, ðc; k0Þ 62 A for any k0 6¼ k. Furthermore, S is

partitioned into components �i ¼ fði; kÞ j 1 � k � 3g,i ¼ 1; 2; . . . ; C3. It is a straightforward exercise to show

that ðS; IÞ is a partition matroid. We then define the

function fð�Þ on sets in I as

fðAÞ ¼min

�Qd;

Pk2f1;3g;ðc;kÞ2A r3ðc; kÞ

�D

þmin

�Qu;

Pk2f2;3g;ðc;kÞ2A r3ðc; kÞ

�U

:

It can be verified directly that maximizing this function

corresponds to our scheduling objective. The remaining

work is to show that this function is submodular and

nondecreasing.According to the definition of function fð�Þ, it is

straightforward to see that fð;Þ ¼ 0. Besides, for alla 2 S, A;B 2 I , if A [ fag 2 I and B � A, then fðAÞ �fðA [ fagÞ holds. To see

fðA [ fagÞ � fðAÞ � fðB [ fagÞ � fðBÞ; ð27Þ

TANG ET AL.: CODING-AWARE PROPORTIONAL-FAIR SCHEDULING IN OFDMA RELAY NETWORKS 1733

Page 8: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

we assume, without loss of generality, a ¼ ðca; kaÞ 62 A.We will prove (27) in the case of ka ¼ 1 in the following.For cases ka ¼ 2 and ka ¼ 3, the proof is similar.

According to the definition of fð�Þ and the fact: for anyx; y; z � 0,

minfx; yþ zg �minfx; yg ¼ maxf0;minfx� y; zgg;

we have

fðA [ fagÞ � fðAÞ

¼ min

(Qd;

Xk2f1;3g;ðc;kÞ2A[fag

r3ðc; kÞ)

�min

(Qd;

Xk2f1;3g;ðc;kÞ2A

r3ðc; kÞ)

¼ min

(Qd;

Xk2f1;3g;ðc;kÞ2A

r3ðc; kÞ þ r3ðca; 1Þ)

�min

(Qd;

Xk2f1;3g;ðc;kÞ2A

r3ðc; kÞ)

¼ max

(0;min

(Qd �

Xk2f1;3g;ðc;kÞ2A

r3ðc; kÞ; r3ðca; 1Þ))

:

ð28Þ

Similarly, we can derive

fðB [ fagÞ � fðBÞ

¼ max

(0;min

(Qd �

Xk2f1;3g;ðc;kÞ2B

r3ðc; kÞ; r3ðca; 1Þ))

:

ð29Þ

Since B � A, we haveXk2f1;3g;ðc;kÞ2B

r3ðc; kÞ �X

k2f1;3g;ðc;kÞ2Ar3ðc; kÞ: ð30Þ

By combining (28), (29), and (30), we can see that (27) holds.Finally, recalling that HAG goes through each sub-

channel and assigns a mode to it such that the incrementof the objective is maximized, HAG is exactly the greedyalgorithm for maximizing a nondecreasing submodularfunction over the matroid ðS; IÞ that we just construct.This conclusion completes the proof. tu

We end this section by showing that the approximationfactor 1

2 of HAG is tight.

Theorem 7. For any constant 0 < � < 1, there exists an instanceof CA-LA on which HAG achieves at most a 1=ð2� �Þ fractionof the optimal value.

Proof. We prove it by constructing a tight example. In theexample, only two subchannels are available. Therates for subchannel 1 are r3ð1; 1Þ ¼ 1 and r3ð1; 2Þ ¼ �=2.The rates for subchannel 2 are r3ð2; 1Þ ¼ 1� �=2 andr3ð2; 2Þ ¼ 0. Thus, r3ð1; 3Þ ¼ �=2 and r3ð2; 3Þ ¼ 0. Finally,we set Qd ¼ Qu ¼ 1, D ¼ 1, and U ¼ �

2ð1��Þ .Using exhaustive search, we can obtain the following

optimal value by setting subchannel 1 to work in mode 3and subchannel 2 in mode 1:

minfQd; r3ð1; 3Þ þ r3ð2; 1ÞgD

þminfQu; r3ð1; 3ÞgU

¼ 2� �:

On the other hand, since r3ð1;1ÞD > r3ð1;3Þ

D þ r3ð1;3ÞU , HAG

makes subchannel 1 work in mode 1, and subchannel 2becomes useless. Therefore, it achieves a value ofr3ð1;1ÞD ¼ 1, resulting in an approximation factor of 1

2�� ,which approaches arbitrarily close to 1

2 . tu

6 PERFORMANCE EVALUATION

In this section, we evaluate our proposed coding-awarescheduling algorithms through extensive simulations. Ourgoal is two-fold. One is to demonstrate the potential benefit ofnetwork coding; the other is to show the efficiency of ouralgorithms.

6.1 Methodology

To demonstrate that network coding is indeed helpful inOFDMA relay networks, we would like to compare ourproposed algorithms EMW and HAG against some optimalcoding-oblivious scheduling scheme. However, the coding-oblivious scheduling problem is NP-hard [4], implying thatthere is no efficient algorithm to find its optimal solutionunless P ¼ NP. To cope with this, we compare ouralgorithms in our simulations with 1) DIV1 [4], which is astate-of-the-art noncoding scheduling algorithm and hasbeen shown to be close to the optimum via simulations, and2) OPT-NO-NC, which represents the optimal fractionalsolution of the CA-GA formulation by letting allI3ðm; c; 3Þ ¼ 0 and relaxing all integer constraints. Theresulting LP can be solved in a timely manner by GNULinear Programming Kit (GLPK) [38]. Note that OPT-NO-NC provides a natural upper bound on the performance ofany coding-oblivious scheduling problem, which impliesthat if our algorithms have better performance than OPT-NO-NC, then network coding must be beneficial.

To show the efficiency of our algorithms, we quantifytheir performance gap to the optimal solution of the coding-aware scheduling problem. Similarly, we use its optimalfractional solution instead, denoted by OPT-NC, which isobtained by solving the relaxed CA-GA using GLPK.Clearly, OPT-NC is a natural upper bound of theperformance of any coding-aware scheduling algorithmfor either the GA or LA model. Thus, if the performance ofour algorithm is within some ratio of OPT-NC, then it mustbe also within the same ratio of the optimum.

All our comparisons are based on the metrics of 1) long-term utility, which is defined as the sum of the logarithm ofthe downlink throughput and the uplink throughput of allMSs as given in (2), and 2) MS throughput, which is the sumof both downlink throughput and uplink throughput.

While some other coding-aware scheduling algorithmshave been proposed in the literature [24], [25], [26], they arenot included in this paper for performance comparison forthe following reasons. First, existing coding-aware schemesare proposed under an assumption that each schedulingunit can be continuously partitioned with no constraintimposed by the frame structure in OFDMA relay networksand, thus, cannot be applied for frame-based schedulingstudied in this paper. Second, it is unfair to compare withthem because they are designed for maximizing theaggregate throughput not proportional fairness. Finally,the result of a theoretical upper bound of the coding-aware

1734 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 9, SEPTEMBER 2013

Page 9: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

proportional-fair scheduling is included for performancecomparison. The fact that our proposed scheme performsclose to the upper bound, i.e., even closer to the optimalsolution, is sufficient to show its supreme performance.

6.2 Simulation Setup

We simulate a single cell OFDMA relay network with 1-kmcell radius. A number of RSs are randomly and uniformlylocated within the annulus with inner and outer radii of 350and 550 m, respectively. And a number of MSs aredistributed randomly over the whole cell coverage. Weadopt a simple distance-based metric for routing, in whicheach MS connects to the nearest RS or BS directly. To focuson the benefit of network coding, we only consider thoseMSs each of which connects to the BS via some RS.

We consider a bandwidth of 10 MHz at carrierfrequency of 5 GHz. The whole bandwidth is equally splitinto 1,080 subcarriers. The number of subchannels is 30,and each subchannel is made up of 36 contiguoussubcarriers. For links between BS and RS, the channelimpairment due to large scale fading is characterized by alarge-scale log-normal shadowing with a path loss expo-nent of 2.4 and a standard deviation of 5.4 dB [35]. For theRS-MS links, the small scale fading effects, caused by themovement of MS, is also incorporated using the Rayleighfading model. The inherent frequency selective fading iscaptured by an exponential power delay profile with adelay spread of 15 �s. Each subcarrier also has a Dopplerspread under the random waypoint model, where each MSis moving at a pedestrian speed of 5.0 km/h with 0 pauseperiod [36]. The combined complex gain is generated usingthe modified Jakes-like method [37]. We set the transmis-sion power of BS, RS, and MS at 41.7 dBm (15 watts),39.5 dBm (9 watts), and 37.8 dBm (6 watts), respectively.The noise power is set at �174 dBm/Hz. Once the SNR ofeach subcarrier is determined, the modulation and codingscheme, which decides the rate of the subcarrier, is chosenaccording to the SNR/modulation mapping from the IEEE802.16j standard [2]. Thus, the rate of a subchannel isobtained by summing up the rates of composite subcar-riers. In the experiments, we consider that each frame lasts10 ms with 60 time slots. For each network setting, i.e., agiven number of MSs and RSs, the simulation results areobtained from 100 randomly generated network topologiesby running over 1,000 frames on each. Important simula-tion parameters are summarized in Table 1.

6.3 Simulation Results

We first evaluate these algorithms under simulation settingswith fixed 5 RSs and 40 MSs over 100 randomly generatedinstances, which are classified into four categories accord-ing to the network coding gain in terms of mean throughput(the performance ratio of our algorithms to DIV1). Let IHAG

and IEMW be the instance sets sorted by the network codinggain obtained by HAG and EMW, respectively, both in adecreasing order. Category 1 consists of the instances thatare top 50 percent in both IHAG and IEMW, category 2consists of the instances that are top 50 percent in IEMW butnot top 50 percent in IHAG, category 3 consists of theinstances that are top 50 percent in IHAG but not top50 percent in IEMW, and category 4 consists of all otherinstances, i.e., the instances that are not top 50 percent ineither IEMW or IHAG. We then randomly select one instancefrom each category as a representative case. These four

concrete topologies are shown in Fig. 3, where Topology i,i ¼ 1; 2; 3; 4, is selected from category i.

The long-term utilities of evaluated algorithms on thefour topologies are plotted in Fig. 4. From this figure, wehave the following observations:

1. OPT-NC has higher utilities than OPT-NO-NC,showing that network coding does improve thenetwork performance in terms of throughput andfairness.

2. Both HAG and EMW perform better than DIV1.3. EMW also performs better than OPT-NO-NC in all

cases while HAG does in most cases including thetopologies given in Fig. 3. Note that even a lowerperformance than OPT-NO-NC cannot depreciateHAG, since OPT-NO-NC only provides an upperbound of the optimal solution of noncoding schedul-ing as explained in Section 6.1.

4. HAG achieves about 95.5 percent of OPT-NC, whileEMW achieves about 97 percent. Both of them arevery close to the optimum.

The cumulative distributions of MS throughput underthe four representative topologies are also plotted inFigs. 5a, 5b, 5c, and 5d, respectively. Compared to thenoncoding scheme DIV1, both HAG and EMW achievehigher mean throughput (the area between the correspond-ing curve and the y-axis), as well as median throughput(the corresponding x-coordinate when y-coordinate isequal to 0.5). In particular, HAG improves the meanthroughput by 15-20 percent and median throughput byabout 10-23 percent. Such improvement offered by EMWare both 15 and 30 percent.

We also counted the percentage of subchannels used fornetwork coding in the four representative topologies asshown in Table 2. By comparing the table and figures, wehave the following observations. 1) Both algorithms havesubstantially exploited network coding. For example, HAGused more than 80 percent of subchannels for networkcoding in all cases and EMW used more than 70 percent.2) HAG creates more coding opportunities than EMW. Thisis because HAG only focuses on exploiting the networkcoding gain, while EMW additionally exploits the multiuserdiversity, which would lead to some compromise on thecoding efficiency. 3) The coding chance correlates to theinstance category (i.e., the network coding gain). Forexample, both HAG and EMW use least subchannels for

TANG ET AL.: CODING-AWARE PROPORTIONAL-FAIR SCHEDULING IN OFDMA RELAY NETWORKS 1735

TABLE 1Simulation Parameters

Page 10: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

network coding in Topology 4, which belongs to category 4with low coding-gain instances. Here, is an intuitiveexplanation. By looking into the concrete topology ofTopology 4 as shown in Fig. 3d, we can find that thereare quite a few MSs located very closely to their associatedRSs. For these MSs, the rates over links (RS, BS) and (RS,MS) must vary greatly due to their significant fadingdifference. Thus, according to the broadcast rate constraintas shown in (1), there is little coding gain for these MSs, i.e.,few subchannels to be used for network coding in thistopology. However, in general, it is difficult for us todifferentiate the topologies such that HAG or EMW canachieve higher coding gain, as it is influenced by too manyfactors including network topologies, link rates, and so on.

The overall simulation results obtained from all these 100topologies are also analyzed. The cumulative distribution ofnetwork coding gains in terms of mean throughput andmedian throughput are plotted in Figs. 6a and 6b, respec-tively. As revealed by the dotted lines, HAG achieves anetwork coding gain of at least 1.17 for both mean and medianthroughput in half of all 100 network topologies, while EMWachieves about 1.23. Besides, HAG can achieve a networkcoding gain up to 1.22 for mean throughput, and up to 1.3 formedian throughput. Similarly, these coding gains achievedby EMW are up to 1.28 and 1.38, respectively.

Now, we evaluate the network coding gains achieved byour algorithms under network settings with variousnumbers of MSs and RSs. We first study the impact ofthe number of MSs M. For a fixed number of RSs (R ¼ 2;4; and 6), we vary M from 20 to 100 with an incrementalstep of 20. The results of mean throughput are shown in

1736 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 9, SEPTEMBER 2013

Fig. 3. Distribution of BS, RSs, and MSs over a plane in each of four representative topologies.

Fig. 4. Comparison of long-term utilities in four representativetopologies.

Page 11: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

Figs. 7a, 7b, and 7c, where data points represent theaverage network coding gains, and error bars indicatethe standard deviation (the results of median throughputare similar and, thus, omitted for the interest of space). Wenote the following facts:

1. The average network coding gain achieved by eitherEMW or HAG is little affected by the number ofMSs. It implies that the network coding opportu-nities do not increase much when more MSs areinvolved under a fixed number of RSs.

2. The error bars of both HAG and EMW becomeshorter when the number of MSs increases, indicat-ing that HAG and EMW perform more steadilywhen more MSs are included in the network.

3. HAG is more steady than EMW in all cases.4. Another interesting phenomenon is that the error

bars of EMW are quite long for any number of MSswhen R ¼ 2.

In the case of only two RSs (R ¼ 2) available as relays, thedistance of each MS to its associated RS varies a lot. Suchdistance variance (i.e., link rate variance) correlates with theerror bars of EMW because in each scheduling step

whether a subchannel is used for coding or not isdetermined by the rates of all participating links (RS, BS)and (RS, MS). When more RSs are in the network, thedistance variance will reduce, resulting in shorter errorbars. On the other hand, the distance variance makesshorter error bars of HAG because the coding rate isconstrained by the lower rate of only a pair of links (RS, BS)and (RS, MS) at each scheduling step.

Then, we study the impact of the number of RSs R. For afixed number of MSs (M ¼ 20; 60; and 100), we vary R from3 to 9 with an incremental step of 2. The correspondingresults of mean throughput are plotted in Figs. 8a, 8b, and8c. We have the following observations: 1) With the increaseof the number of RSs, the network coding gain achieved byEMW improves a lot. This is because more network codingopportunities are introduced. However, the performance ofHAG varies little due to its local scheme, where onlythe traffic through one RS will be scheduled each cycle.2) When the number of RSs is very small, HAG performsbetter than EMW; on the other hand, EMW has a betterperformance when more RSs are introduced in the network.3) Both HAG and EMW perform more steadily when thenumber of RSs becomes larger.

In summary, both HAG and EMW can improve themean as well as the median throughput over the noncodingscheme significantly. EMW has a slightly better perfor-mance than HAG. On the other hand, we recall that HAGhas lower overheads than EMW. They are both practicaland effective proposals, and can be used in differentapplications according to various tradeoff requirements.

TANG ET AL.: CODING-AWARE PROPORTIONAL-FAIR SCHEDULING IN OFDMA RELAY NETWORKS 1737

Fig. 5. Cumulative distribution of MS throughput in four representative topologies.

TABLE 2Percentage of Subchannels Used for Network

Coding-Based Broadcast

Page 12: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

7 CONCLUSION

In this paper, we study the network coding-aware schedul-

ing problem in OFDMA relay networks under the propor-

tional fair scheduling policy. We propose two approaches,

GA and LA, to solve the problem under a tradeoff

consideration between performance and overheads. For

each model, we establish its hardness and propose efficient

algorithms with low time complexity. The theoretical

performance of our proposal is also studied. To highlight

the efficiency of our algorithms, as well as the benefit of

network coding, extensive simulations have been conducted.

The experimental results show that our proposals outper-form one of the best existing schemes in terms of bothfairness and throughput.

ACKNOWLEDGMENTS

This work was partially supported by the National BasicResearch Program of China under Grant No.2009CB320705; the National Natural Science Foundationof China under Grant No. 61170069, 61073028, 61021062,91218302, and 60903025. Baoliu Ye and Sanglu Lu are thecorresponding authors.

1738 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 9, SEPTEMBER 2013

Fig. 6. Cumulative distribution of network coding gains in terms of mean throughput and median throughput in one hundred topologies.

Fig. 7. The network coding gain of the mean throughput under various numbers of MSs. Data points represent the average coding gain and the errorbars represent the standard deviation.

Fig. 8. Network coding gain of the mean throughput under various numbers of RSs.

Page 13: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

REFERENCES

[1] H. Yin and S. Alamouti, “OFDMA: A Broadband Wireless AccessTechnology,” Proc. IEEE Sarnoff Symp., pp. 1-4, 2006.

[2] 802.16: Air Interface for Broadband Wireless Access Systems, IEEEStandard, 2009.

[3] 3GPP, LTE Release 10 and Beyond (LTE Advanced), RP-090939,http://www.3gpp.org/LTE-Advanced, 2013.

[4] K. Sundaresan and S. Rangarajan, “On Exploiting Diversity andSpatial Reuse in Relay-Enabled Wireless Networks,” Proc. ACMMobiHoc, 2008.

[5] R. Ahlswede, N. Cai, S.Y. Li, and R.W. Yeung, “NetworkInformation Flow,” IEEE Trans. Information Theory, vol. 46, no. 4,pp. 1204-1216, July 2000.

[6] S. Katti, H. Rahul, W. Hu, D. Katabi, M. Medard, and J. Crowcroft,“XORs in the Air: Practical Wireless Network Coding,” Proc. ACMSIGCOMM, 2006.

[7] H.J. Kushner and P.A. Whiting, “Convergence of Proportional-Fair Sharing Algorithms under General Conditions,” IEEE Trans.Wireless Comm., vol. 2, no. 6, pp. 1150-1158, Nov. 2003.

[8] O. Oyman, “OFDMA2A: A Centralized Resource Allocation Policyfor Cellular Multi-Hop Networks,” Proc. IEEE Asilomar Conf.Signals, Systems and Computers, 2006.

[9] Y.W. Cheong, R.S. Cheng, K.B. Latief, and R.D. Murch, “MultiuserOFDM with Adaptive Subcarrier, Bit, and Power Allocation,”IEEE J. Selected Areas in Comm., vol. 17, no. 10, pp. 1747-1758, Oct.1999.

[10] D. Kivane, G. Li, and H. Liu, “Computationally EfficientBandwidth Allocation and Power Control for OFDMA,” IEEETrans. Wireless Comm., vol. 2, no. 6, pp. 1150-1158, Nov. 2003.

[11] M. Ergen, S. Coleri, and P. Varaiya, “QoS Aware AdaptiveResource Allocation Techniques for Fair Scheduling in OFDMABased Broadband Wireless Access Systems,” IEEE Trans. Broad-casting, vol. 49, no. 4, pp. 362-370, Dec. 2003.

[12] M. Andrews and L. Zhang, “Scheduling Algorithms for Multi-Carrier Wireless Data Systems,” Proc. ACM MobiCom, 2007.

[13] S. Bodas, S. Shakkottai, L. Ying, and R. Srikant, “Low-complexityScheduling Algorithms for Multi-Channel Downlink WirelessNetworks,” Proc. IEEE INFOCOM, 2010.

[14] S. Mengesha and H. Karl, “Relay Routing and Scheduling forCapacity Improvement in Cellular WLANs,” Proc. Modeling andOptimization in Mobile, Ad Hoc and Wireless Networks (WiOpt), 2003.

[15] A. So and B. Liang, “Effect of Relaying on Capacity Improvementin Wireless Local Area Networks,” Proc. IEEE Wireless Comm.Networking Conf. (WCNC), 2005.

[16] K. Sundaresan, X. Wang, and M. Madihian, “Low-overheadScheduling Algorithms for OFDMA Relay Networks,” Proc. FourthAnn. Int’l Conf. Wireless Internet (WiCON), 2008.

[17] S. Deb, V. Mhatre, and V. Ramaiyan, “WiMAX Relay Networks:Opportunistic Scheduling to Exploit Multiuser Diversity andFrequency Selectivity,” Proc. ACM MobiCom, 2008.

[18] C.Y. Hong and A.C. Pang, “Link Scheduling with QoS Guaranteefor Wireless Relay Networks,” Proc. IEEE INFOCOM, 2009.

[19] S. Rayanchu, S. Sen, J. Wu, S. Banerjee, and S. Sengupta, “Loss-Aware Network Coding for Unicast Wireless Sessions: Design,Implementation, and Performance Evaluation,” Proc. ACM SIG-METRICS Int’l Conf. Measurement and Modeling of ComputerSystems, 2008.

[20] S. Sengupta, S. Rayanchu, and S. Banerjee, “An Analysis ofWireless Network Coding for Unicast Sessions: The Case forCoding-Aware Routing,” Proc. IEEE INFOCOM, 2007.

[21] J.L. Le, J.C.S. Lui, and D.M. Chiu, “DCAR: Distributed Coding-Aware Routing in Wireless Networks,” Proc. IEEE 28th Int’l Conf.Distributed Computing Systems (ICDCS), 2008.

[22] H. Xu and B. Li, “XOR-Assisted Cooperative Diversity in OFDMAWireless Networks: Optimization Framework and ApproximationAlgorithms,” Proc. IEEE INFOCOM, 2009.

[23] X. Zhang and B. Li, “Network Coding Aware Dynamic SubcarrierAssignment in OFDMA Wireless Networks,” Proc. IEEE Int’l Conf.(ICC), 2008.

[24] Y. Xu, J.C.S. Lui, and D.M. Chiu, “Analysis and Scheduling ofPractical Network Coding in OFDMA Relay Networks,” ComputerNetworks, vol. 53, pp. 2120-2139, 2009.

[25] Y. Liu, M. Tao, B. Li, and H. Shen, “Optimization Framework andGraph-Based Approach for Relay-Assisted Bidirectional OFMDACellular Networks,” IEEE Trans. Wireless Comm., vol. 9, no. 11,pp. 3490-3500, Nov. 2010.

[26] B.G. Kim and J.W. Lee, “Opportunistic Subchannel Scheduling forOFDMA Networks with Network Coding at Relay Stations,” Proc.IEEE GlobeCom, 2010.

[27] J. Padhye, R. Draves, and B. Zill, “Routing in Multi-Radio, Multi-Hop Wireless Mesh Network,” Proc. ACM MobiCom, 2004.

[28] T. Nguyen and Y. Han, “A Proportional Fairness Algorithm withQoS Provision in Downlink OFDMA Systems,” IEEE Comm.Letters, vol. 10, no. 11, pp. 760-762, Nov. 2006.

[29] Y. Ma, “Rate Maximization for Downlink OFDMA with Propor-tional Fairness,” IEEE Trans. Vehicular Technology, vol. 57, no. 5,pp. 3267-3274, Sept. 2008.

[30] V.V. Vazirani, Approximation Algorithms. Springer, 2001.[31] L. Tassiulas and A. Ephremides, “Stability Properties of Con-

strained Queueing Systems and Scheduling Policies for MaximumThroughput in Multihop Radio Networks,” IEEE Trans. AutomaticControl, vol. 37, no. 12, pp. 1936-1948, Dec. 1992.

[32] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, R. Vijayaku-mar, and P. Whiting, “Providing Quality of Service over a SharedWireless Link,” IEEE Comm. Magazine, vol. 39, no. 2, pp. 150-154,Feb. 2001.

[33] M. Neely, E. Modiano, and C. Rohrs, “Power and ServerAllocation in a Multi-Beam Satellite with Time Varying Chan-nels,” Proc. IEEE INFOCOM, 2002.

[34] M. Fisher, G. Nemhauser, and L. Wolsey, “An Analysis ofApproximations for Maximizing Submodular Set Functions-II,”Math. Programming Study, vol. 14, pp. 265-294, 1978.

[35] J. Gross, H. Geerdes, H. Karl, and A. Wolisz, “PerformanceAnalysis of Dynamic OFDMA Systems with Inband Signaling,”IEEE J. Selected Areas in Comm., vol. 24, no. 3, pp. 427-436, Mar.2006.

[36] J. Broch, D.A. Maltz, D.B. Johnson, Y.-C. Ju, and J. Jetcheva, “APerformance Comparison of Multi-Hop Wireless Ad Hoc Net-work Routing Protocols,” Proc. ACM MobiCom, 1998.

[37] J.K. Cavers, Mobile Channel Characteristics. Kluwer AcademicPublishers, 2000.

[38] GLPK (GNU Linear Programming Kit), version 4.8, http://www.gnu.org/s/glpk/, 2013.

Bin Tang received the BS degree in computerscience from Nanjing University, Nanjing, China,in 2007, where he is currently working towardthe PhD degree with the Department of Com-puter Science and Technology. His researchinterests lie in the area of communications,network coding, and distributed computing, witha focus on the application of network coding tofile distribution in various networking environ-ments. He is a student member of the IEEE.

Baoliu Ye received the PhD degree in computerscience from Nanjing University, China, in 2004.He is currently an associate professor at theDepartment of Computer Science and Technol-ogy, Nanjing University, China. He served as avisiting researcher of the University of Aizu,Japan from March 2005 to July 2006. His currentresearch interests include peer-to-peer (P2P)computing, online/mobile social networking, andwireless network. He has published more than

40 technical papers in the above areas. He served as the TPC cochair ofHotPOST’12, HotPOST’11, and P2PNet’10. He is the regent of CCF anda member of the IEEE, ACM.

TANG ET AL.: CODING-AWARE PROPORTIONAL-FAIR SCHEDULING IN OFDMA RELAY NETWORKS 1739

Page 14: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

Sanglu Lu received the BS, MS, and PhDdegrees from Nanjing University in 1992, 1995,and 1997, respectively, all in computer science.She is currently a professor in the Department ofComputer Science & Technology and the StateKey Laboratory for Novel Software Technology.Her research interests include distributed com-puting, wireless networks and pervasive com-puting. She has published more than 80 papersin referred journals and conferences in the

above areas. She is a member of the IEEE.

Song Guo received the PhD degree in compu-ter science from University of Ottawa, Canada.He is currently an associate professor with theSchool of Computer Science and Engineering,the University of Aizu, Japan. His researchinterests are mainly in the areas of protocoldesign and performance analysis for computerand telecommunication networks. He has pub-lished more than 150 papers in referred journalsand conferences in these areas. He is currently

the associate editor of IEEE Transactions on Parallel and DistributedSystems, Wiley Wireless Communications and Mobile Computing, AdHoc & Sensor Wireless Networks, and so on. He is a senior member ofthe IEEE.

. For more information on this or any other computing topic,please visit our Digital Library at www.computer.org/publications/dlib.

1740 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 9, SEPTEMBER 2013

Page 15: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

1

APPENDIX APROOF OF THEOREM 4

We prove this theorem by explicitly constructing aPTAS for the CA-LA problem. In the following, wefirst introduce another integer linear programming(CA-LA-c) which is correlated to CA-LA. Then wepresent a PTAS for CA-LA-c. Based on this result, wefinally derive a PTAS for CA-LA.

A.1 Correlated Integer Linear Programming

By restricting Qd (Qu) as the maximum downlink (up-link) data amount that can be transmitted by the sub-channels in the relay sub-frames, CA-LA-c is definedas follows:

CA-LA-c:

max∑c∈C3

3∑k=1

βkr3(c, k)I3(c, k) (1)

s.t.∑c∈C3

∑k∈{1,3}

r3(c, k)I3(c, k) ≤ Qd (2)

∑c∈C3

∑k∈{2,3}

r3(c, k)I3(c, k) ≤ Qu (3)

3∑k=1

I3(c, k) ≤ 1,∀c ∈ C3 (4)

I3(c, k) ∈ {0, 1},∀c ∈ C3, k ∈ {1, 2, 3} (5)

where β1 = 1D , β2 = 1

U , and β3 = 1D + 1

U .Before building the relationship between CA-LA

and CA-LA-c, we have to introduce some usefulconcepts first. We denote the set of sub-channels usedin mode k as Ck

3 , and for each c ∈ Ck3 , we define

βkr3(c, k) as its virtual profit v(c) in CA-LA, and asits profit p(c) otherwise. As the names suggest, theobjective value of CA-LA may be smaller than thesum of v(c), while the objective value of CA-LA-c isexact the sum of p(c).

Denote the optimal values of CA-LA and CA-LA-cby OPT and OPT1, respectively. The following lemmashows that there is only a small gap between OPTand OPT1.

Lemma A.1: For any optimal assignment strategyof CA-LA, there must be two sub-channels c1 and c2,such that OPT ≤ OPT1 + v(c1) + v(c2).

Proof: Consider an arbitrary optimal assignmentstrategy C1

3 ∪C23 ∪C3

3 of CA-LA. We will adjust it to ob-tain a feasible solution of CA-LA-c with performanceguarantee.

We number the sub-channels in C3 as 1, 2, . . . , C3.Without loss of generality, we assume that

∑k∈{1,3}

∑c∈Ck

3

r3(c, k) > Qd and∑

k∈{2,3}

∑c∈Ck

3

r3(c, k) > Qu.

For other cases, it can be shown in a similar way. Thenthere must be some maximum positive integers i1 andi2 (1 ≤ i1, i2 ≤ C3), such that∑

k∈{1,3}

∑c∈Ck

3 :c<i1

r3(c, k) ≤ Qd,

and ∑k∈{2,3}

∑c∈Ck

3 :c<i2

r3(c, k) ≤ Qu.

Without loss of generality, we assume i1 ≤ i2. For allsub-channels c ≥ i1, we assign them for uplink only,i.e.,

Ck3 ← Ck

3 ∩ {c : c < i1}, k = 1, 3,

andC23 ← C2

3 ∪ {c : c ≥ i1}.Because r3(c, 3) ≤ r3(c, 2) holds for any sub-channel c,the above sub-channel reassignment never decreasesthe uplink data amount that can be transmitted.Hence, there must be some maximum integer i3 ≥i2 ≥ i1, such that∑

k∈{2,3}

∑c∈Ck

3 :c<i3

r3(c, k) > Qu.

After removing all sub-channels c ≥ i3, i.e.,

C23 ← C2

3 − {c : c ≥ i3},the new strategy C1

3 ∪ C23 ∪ C3

3 is feasible for theinstance of CA-LA-c. It is easily seen that this solutionhas a value of at least OPT − v(i1) − v(i3), whichimplies OPT1 ≥ OPT − v(i1) − v(i3). The proof isaccomplished.

A.2 PTAS for CA-LA-cBy extending the basic ideas in [1] for the m-dimensional 0-1 knapsack problem (m is fixed), wepropose a PTAS PLA-c(ε) for CA-LA-c. The outlinecan be described as follows:

• Guess the most profitable assignment with atmost η = min{C3, 3

ε } sub-channels;• Generate an integer linear programming (ILP) for

the remaining sub-channels, ensuring that a sub-channel can be allocated in some mode only if itsprofit is not more than the minimum profit of theguessed sub-channels;

• Solve the ILP by LP-relaxation and rounding. Aspecial rounding technique is employed to guar-antee that at most 3 sub-channels are removed forensuring the feasibility.

Now we describe PLA-c(ε) (see Algorithm 1) inmore details.

[Guessing most profitable sub-channels] We de-note any partial assignment with at most η sub-channels as S1 ∪ S2 ∪ S3, where Sk is the set of theassigned sub-channels in mode k. Let pmin be theminimum profit of these sub-channels. Then do the

Page 16: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

2

Algorithm 1 Algorithm PLA-c(ε) for CA-LA-c

1: η ← min{C3, 3ε }.

2: zA ← 0.3: for any disjoint subsets S1, S2 and S3 of C3 with

|S1| + |S2| + |S3| ≤ η do4: Qd ← Qd − ∑

k∈{1,3}∑

c∈Skr3(c, k).

5: Qu ← Qu − ∑k∈{2,3}

∑c∈Sk

r3(c, k).

6: if Qd ≥ 0 and Qu ≥ 0 then7: z ← (Qd − Qd)/D + (Qu − Qu)/U ;8: Construct CA-LA-c2, and obtain its LP-

relaxed solution I∗3 (c, k).9: Construct CA-LA-c3 using I∗3 (c, k).

10: Round I∗3 (c, k) to I3(c, k) by the methodproposed in [2].

11: for each violated constraint of (14)-(16)do

12: Let c be the sub-channel whose re-moval ensures feasibility.

13: I3(c, 0) ← 1; I3(c, k) ← 0, where k ∈{1, 2, 3} is the mode correspondingto the violated constraint.

14: end for15: z ← z +

∑c∈U

∑3k=1 βkr3(c, k)I3(c, k).

16: if z ≥ zA then17: zA ← z.18: end if19: end if20: end for21: return zA.

following steps. At the end of the algorithm thatchecks all possible combinations of S1, S2 and S3, itreturns the solution with the largest objective value.

[Constructing an ILP for remaining sub-channels]Let U denote the set of unassigned sub-channels. Wedefine

Qd = Qd −∑

k∈{1,3}

∑c∈Sk

r3(c, k), (6)

and

Qu = Qu −∑

k∈{2,3}

∑c∈Sk

r3(c, k). (7)

If Qd < 0 or Qu < 0, then the combination is infeasibleand is skipped. Otherwise, we construct an IP forremaining sub-channels. To make sure all remainingsub-channels can be assigned, we introduce an addi-tional mode 0 with r3(c, 0) = 0 for c ∈ U . By restrictingthat each sub-channel can be assigned in some non-zero mode only if the profit obtained is not larger thanpmin, the optimal assignment strategy for remainingsub-channels can be decided by solving the followingprogramming CA-LA-c2, where γk

c = pmin−βkr3(c, k)

.

CA-LA-c2:

min∑c∈U

3∑k=0

γkc I3(c, k) (8)

s.t.∑c∈U

∑k∈{1,3}

r3(c, k)I3(c, k) ≤ Qd (9)

∑c∈U

∑k∈{2,3}

r3(c, k)I3(c, k) ≤ Qu (10)

3∑k=0

I3(c, k) ≤ 1,∀c ∈ U (11)

I3(c, k) ∈ {0, 1},∀c ∈ U , 0 ≤ k ≤ 3 (12)I3(c, k) = 0, if βkr3(c, k) > pmin (13)

The benefit of representing the ILP in a minimizationform will be clarified in the next step.

[Solving the ILP by LP-relaxation and rounding]Let I∗3 (c, k) be the optimal fractional solution of CA-LA-c2. We further define Qb =

∑c∈U r3(c, 3)I∗3 (c, 3). A

new programming CA-LA-c3 is thus constructed byreplacing (9) and (10) in CA-LA-c2 with the followingconstraints: ∑

c∈Ur3(c, 1)I3(c, 1) ≤ Qd − Qb (14)

∑c∈U

r3(c, 2)I3(c, 2) ≤ Qu − Qb (15)

∑c∈U

r3(c, 3)I3(c, 3) ≤ Qb (16)

Note that I∗3 (c, k) is also the optimal fractional solu-tion of CA-LA-c3, and each feasible integer solutionof CA-LA-c3 is feasible to CA-LA-c2 as well. SinceCA-LA-c3 is exactly a special case of the general-ized minimum assignment problem, we can roundI∗3 (c, k) to integers using the algorithm of Shmoysand Tardos [2]. This rounded solution may still beinfeasible since constraints (14)-(16) might be violated.However, due to the analysis in [2], if some constraintis violated, then there must be some single sub-channel whose removal can guarantee the feasibility.By setting these sub-channels in mode 0, we obtain afeasible solution.

We can show that such solution of CA-LA-c2 (de-noted by SOL2) exceeds the optimum (denoted byOPT2) by at most 3pmin.

Lemma A.2: SOL2 ≤ OPT2 + 3pmin.Proof: Denote the obtained feasible solution and

the optimal fractional solution of CA-LA-c3 as SOL3

and OPT f3 , respectively. Similarly the optimal frac-

tional value of CA-LA-2 is defined as OPT f2 . Ac-

cording to the construction of CA-LA-c3, we haveOPT f

2 = OPT f3 and SOL2 = SOL3. On the other

hand, we have SOL3 ≤ OPT f3 +3pmin, since at most 3

constraints would be violated as analyzed by Shmoys

Page 17: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

3

and Tardos in [2]. Combining these equations withOPT f

2 ≤ OPT2, we have SOL2 ≤ OPT2 + 3pmin.Now we are in a position to show that PLA-c(ε) is

a PTAS for CA-LA-c exactly.Theorem A.3: The algorithm PLA-c(ε) is a PTAS for

CA-LA-c with an approximation factor of at least 1−εand a running time of O(C4

3 (3C3)3ε ).

Proof: Consider an optimal solution OPT1 of CA-LA-c. If no more than η sub-channels need to be as-signed, obviously, PLA-c(ε) returns the optimal value.Otherwise, without loss of generality, we assumep(1) ≥ p(2) ≥ · · · ≥ p(C3). When the enumeratedcombination consists of sub-channels from 1 to η withpmin = p(η), the minimization formulation of CA-LA-c2 guarantees

OPT1 =∑c≤η

p(c) + (C3 − η)pmin − OPT2.

On the other hand, according to Lemma A.2, thesolution obtained by PLA-c(ε) SOL1 is at least

SOL1 ≥∑c≤η

p(c) + (C3 − η)pmin − (OPT2 + 3pmin)

=∑c≤η

p(c) + (C3 − η)pmin − OPT2 − 3pmin.

Therefore, the approximation factor of PLA-c(ε) is atleast

SOL1

OPT1≥

∑c≤η p(c) − 3pmin∑

c≤η p(c)

≥ (η − 3)pmin

ηpmin

≥ 1 − ε.

Here, the basic inequality x+ax+b ≥ a

b for any b ≥ a > 0and x ≥ 0 is repeatedly used.

Now we analyze the time complexity of PLA-c(ε).There are

∑ηi=1 3i

(C3i

) ≤ ∑ηi=1 3iCi

3 = O(3ηCη3 ) combi-

nations, and for each combination, the running timeis dominated by solving an LP with 4C3 variables.By using the ellipsoid method [3], the LP can besolved in O(C4

3 ) time. Thereby, the running time ofPLA-c(ε) is O(C4

33ηCη3 ) = O(C4

3 (3C3)3ε ). The proof is

accomplished.

A.3 PTAS for CA-LAAs Lemma A.1 implies, the gap between the solutionsof CA-LA and CA-LA-c is at most the sum of virtualprofits of two sub-channels. Because the virtual profitscould be extremely large compared to pmin, PLA-c(ε)may not be a PTAS for CA-LA by just adjusting theparameter η. Fortunately, we can derive a PTAS forCA-LA using a similar approach and the theoreticalresults in previous sections. At the beginning, themost virtually profitable assignment with η = 4

ε sub-channels is obtained by guessing. Rather than

solving a CA-LA problem for the remaining sub-channels, we solve a corresponding CA-LA-c probleminstead by PLA-c to obtain an almost optimal solution.In this way, we can eventually get a solution forCA-LA that is arbitrarily close to the optimum inpolynomial time. The details of the PTAS PLA(ε) forCA-LA are shown in Algorithm 2.

Algorithm 2 Algorithm PLA(ε) for CA-LA

1: η ← min{C3, 4ε }.

2: zA ← 0.3: for any disjoint subsets S1, S2 and S3 of C3 with

|S1| + |S2| + |S3| = η do4: Qd

A ← Qd − ∑k∈{1,3}

∑c∈Sk

r3(c, k).5: Qu

A ← Qu − ∑k∈{2,3}

∑c∈Sk

r3(c, k).6: if Qd

A ≤ 0 then7: Set remaining sub-channels in mode 2.8: x ← ∑

c∈S3r3(c, 3) +∑

c∈C3−S1−S3r3(c, 2).

9: z ← Qd

D + min{Qu,x}U .

10: else if QuA ≤ 0 then

11: Set remaining sub-channels in mode 1.12: x ← ∑

c∈S3r3(c, 3) +∑

c∈C3−S2−S3r3(c, 1).

13: z ← min{Qd,x}D + Qu

U .14: else15: Construct an instance of CA-LA-c by re-

placing Qd, Qu and C3 with QdA, Qu

A andC3 − ∪3

i=1Si, respectively.16: Solve the instance using PLA-c( ε

2 ) withsolution z′.

17: z ← z′ + Qd−QdA

D + Qu−QuA

U .18: end if19: if z ≥ zA then20: zA ← z.21: end if22: end for23: return zA.

Theorem A.4: The algorithm PLA(ε) is a PTAS forCA-LA with an approximation factor of at least 1 − εand a running time of O(C4

3 (3C3)10ε ).

Proof: Consider an optimal solution of CA-LA,where the set of sub-channels working in mode k isdenoted by Ck

3 . If∑

k∈{1,3}∑

c∈Ck3 :c≤η r3(c, k) ≥ Qd or∑

k∈{2,3}∑

c∈Ck3 :c≤η r3(c, k) ≥ Qu, then PLA(ε) returns

an optimal solution (steps 6-13). Otherwise, withoutloss of generality, we assume v(1) ≥ v(2) ≥ . . . ≥v(C3). When the enumerated combination consists ofsub-channels from 1 to η, PLA(ε) solves an instance ofCA-LA-c by executing PLA-c( ε

2 ) with sub-channel set{c ∈ C3 : c > η}, and data amount bounds Qd

A and QuA

(steps 15-17). Let z∗ and z∗1 be the optimal solutionsof the CA-LA and CA-LA-c problems on the remain-ing sub-channel allocation, respectively. According toLemma A.1, there must be some sub-channels i1 > η

Page 18: IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED …cssongguo/papers/ofdma13.pdf · the multiuser diversity and frequency selectivity in multi-hop OFDMA relay networks, with an emphasis

4

and i2 > η, such that

z∗ ≤ z∗1 + v(i1) + v(i2). (17)

It is easily seen that the optimal solution OPT of theoriginal CA-LA is

OPT =∑c≤η

v(c) + z∗. (18)

On the other hand, due to the fact that PLA(ε) tra-verses all subsets of η sub-channels in C3, the solutionreturned by PLA(ε) is at least

SOL ≥∑c≤η

v(c) + (1 − ε

2)z∗1 . (19)

By combining (17), (18) and (19), the approximationfactor is

SOL

OPT≥

∑c≤η v(c) + (1 − ε

2 )z∗1∑c≤η v(c) + z∗

≥∑

c≤η v(c) + (1 − ε2 )z∗1∑

c≤η v(c) + z∗1 + v(i1) + v(i2)

≥∑

c≤η v(c) + z∗1∑c≤η v(c) + z∗1 + v(i1) + v(i2)

− ε

2

≥∑

c≤η v(c)∑c≤η v(c) + v(i1) + v(i2)

− ε

2.

Recalling that v(1) ≥ v(2) ≥ · · · v(n), we finally have

SOL

OPT≥ ηv(η)

ηv(η) + v(i1) + v(i2)− ε

2

≥ ηv(η)ηv(η) + v(η) + v(η)

− ε

2

η + 2− ε

2≥ 1 − ε.

Based on the time complexity of PLA-c(ε), it isstraightforward to show that the time complexity ofPLA(ε) is O(3η

(C3η

)C4

3 (3C3)3

ε/2 ) = O(C43 (3C3)

10ε ).

REFERENCES[1] A.M. Frieze, and M.R.B. Clarke, Approximation Algorithms for

the m-dimensional 0-1 Knapsack Problem: Worst-case and Probabilis-tic Analyses, European Journal of Operational Research, 1(15),pp.100-109, 1984.

[2] D.B. Shmoys, and E. Tardos, An Approximation Algorithm forthe Generalized Assignment Problem, Mathematical Programming,62(1993), pp.461-474.

[3] D. Alevras, and M.W. Padberg, Linear Optimization and Exten-sions: Problems and Extensions, Springer-Verlag, 2001.