Joint Optimization of Scheduling and Power Control in Wireless Networks: Multi-Dimensional Modeling and Decomposition

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1 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /TMC , IEEE Transactions on Mobie Computing Joint Optimization of Scheduing and Power Contro in Wireess Networks: Muti-Dimensiona Modeing and Decomposition Lu Liu, Student Member, IEEE, Yu Cheng, Senior Member, IEEE, Xianghui Cao, Senior Member, IEEE, Sheng Zhou, Member, IEEE, Zhisheng Niu, Feow, IEEE and Ping Wang, Senior Member, IEEE 1 Abstract The energy efficiency of future networks is becoming a significant and urgent issue, caing for greener network designs. However, the increasing compexity in network structure and resource space ead to growing probem scaes and couped resource dimensions, which bring great chaenges in obtaining a joint soution in optimizing the energy efficiency. In this paper, we deveop a muti-dimensiona network mode on the basis of tupe-inks associated with transmission patterns (TPs) and formuate the optimization probem as a TP based scheduing probem which jointy soves transmission scheduing, routing, power contro, radio and channe assignment. In order to tacke the compexity issues, we propose a nove agorithm by expoiting the deay coumn generation technique to decompose the couped probem into recursivey soving a master probem for scheduing and a sub-probem for power aocation. Further, we theoreticay prove that the performance gap between the proposed agorithm and the optimum is upper bounded by that for the sub-probem soution, where the atter is derived by soving a reaxed version of the sub-probem. Numerica resuts demonstrate the effectiveness of the muti-dimensiona framework and the benefit of the proposed joint optimization in improving network energy efficiency. Index Terms Muti-radio muti-channe networks, optimization, resource aocation, energy efficiency 1 INTRODUCTION Energy efficiency of next generation wireess networks is a critica and urgent issue. The key to improving energy efficiency of wireess networks reies on configuring and aocating various network resources in tempora, spatia, spectra and power dimensions in terms of routing, ink scheduing, channe aocation and power contro. Generay, different network resources are couped such that they cannot be determined indepenty for optima performance, which demands a joint optimization soution. What s more, in order to meet the rapidy growing traffic demands, wireess networks are evoving into more and more compex structures and hence arge scaes of joint optimization probems. Obtaining a joint optimization soution over wireess networks becomes a chaenging issue, which motivated us to deveop more efficient soutions. Many wireess networks can be abstracted as that each network node has mutipe radio interfaces operating on mutipe avaiabe wireess channes, yieding the generic muti-radio muti-channe (MR-MC) network mode with muti-dimensiona resource space [1], [2], [3]. With this mode for the joint optimization probem, the optimization variabe can be viewed as a compound of mutipe resource aocation strategies, incuding seection of transmitters and receivers for transmission inks, radio and channe assignment, Lu Liu and Yu Cheng are with the Department of Eectrica and Computer Engineering, Iinois Institute of Technoogy, Chicago, IL E-mai: iu41@hawk.iit.edu Xianghui Cao is with Schoo of Automation, Southeast University, Nanjing , China. Sheng Zhou and Zhisheng Niu are with Tsinghua Nationa Laboratory for Information Science and Technoogy, Tsinghua University, Beijing , China. Ping Wang is with Schoo of Computer Engineering, Nanyang Technoogica University, Singapore transmit power contro, routing and ink scheduing. The existing studies on energy-efficient networking in MR- MC wireess networks have addressed the joint optimization issues over different dimensions, but a generic joint optimization soution over the whoe muti-dimensiona space (especiay when power contro is invoved) is sti not avaiabe, to the best of our knowedge. Radio/channe assignment and transmission scheduing in MR-MC wireess networks have been studied with the objective to maximize network capacity [4], [5], [6], [7]. Specificay, protoco interference mode is widey adopted to characterize the interferences among inks as a confict graph, over which indepent set based scheduing is then used to faciitate a inear programming (LP) based formuation [8], [9], [10]. However, such a mode simpifies transmission inks to be either deactivated or activated with fixed transmit power, which can neither mode dynamic power assignment nor accuratey refect the practica interference magnitude. The more reaistic signa-to-interferencepus-noise ratio (SINR) based physica interference mode can mode transmission interferences under the power contro. Link scheduing for capacity optimization under the physica interference mode has been studied in [11], [12], [13], but is imited to singe-channe scenarios. How to incorporate physica interference mode based power contro into muti-dimensiona resource space so as to provide energy-efficient joint scheduing and power contro soution remains a chaenging issue. It has been known that the optima transmission scheduing probem in wireess networks is NP hard [14], [15], [16], [17], et aone the more compicated probem of energy-efficient joint scheduing and power contro. To this, we appy the tupe-ink based muti-dimensiona network mode [6], with which the joint aocation over muti-dimensiona resource space is reduced to the scheduing of tupe-inks. Further, we propose a new concept of

2 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /TMC , IEEE Transactions on Mobie Computing transmission pattern (TP) which integrates both scheduing and power contro to faciitate LP formuation of the joint optimization probem. A TP is defined as a vector of transmit power assignment of a the tupe-inks in the network. In a TP, the SINR at the receiver of each tupe-ink can be cacuated based on the power aocation of a the tupe-inks so that the transmission capacity of the tupe-ink can be determined according to the Shannon-Hartey equation. Therefore, a TP characterizes a possibe transmission state in the network, incuding the resource aocation information across a the dimensions. By considering discretized transmit power eves, the joint scheduing and power assignment probem in the muti-dimensiona resource space can be utimatey transformed into a scheduing probem of a finite number of TPs, which faciitates an LP formuation. This TP based scheduing probem is formuated in a simiar manner as that of indepent set based scheduing [3], [6], but is compatibe with physica interference mode and fexibe power aocation. The soution to the TP based optimization provides joint scheduing and power contro, as we as the resource aocation on a the other resource dimensions. The TP based scheduing transforms the NP hardness into an extremey arge probem scae due to exponentiay many possibe patterns, thus exponentiay many decision variabes. We then expoit a decomposition based approach by everaging deay coumn generation (DCG), which starts with an initia subset of TPs and then graduay adding new TPs that can improve the objective vaue. The DCG based method repeatedy soves a master probem and a sub-probem, where the master probem performs scheduing on existing TPs and the sub-probem searches for a new entering TP by soving a maximum utiity probem. We further revea that the sub-probem is indeed to find the most energy-efficient TP according to the information extracted from the existing TPs, and show that it is equivaent to finding the optima power aocation over tupe-inks. Thus the joint optimization probem is decomposed into an iterative procedure combining scheduing phase and power contro phase, whie optimaity remains intact during the decomposition. As soving the sub-probem sti incurs high computationa compexity in searching over the entire TP space, we further propose a greedy agorithm to sove the sub-probem efficienty. Moreover, through theoretica anaysis, we prove that the performance gap between the achieved and optima soutions is upper bounded by the gap achieved in soving the sub-probem, which can be derived by soving a reaxed sub-probem. Some preiminary resuts appeared in [18] focused on the simpe singe-hop scenario where per-ink traffic demand was expicity specified in the optimization formuation. This paper exts the framework to generic muti-hop scenarios with mutipe commodity fows. The obtained joint soution further provides routing information in a way that the source to destination paths for each commodity fow are impied by the obtained schedue of the inks. In addition, a new modeing method and the concept of interference coefficient is introduced in this paper which seamessy integrates radio confict and co-channe interference. Further, in soving the sub-probem, a greedy agorithm is proposed in this paper instead of the earning based agorithm in [18] for higher computationa efficiency. The main contributions of this paper are as foows: 1) We transform the joint scheduing and power contro probem for wireess network energy efficiency optimization in the muti-dimensiona resource space into a TP based scheduing probem and formuate it as an LP. 2) To sove the arge-scae TP scheduing probem, we expoit a decomposition approach by everaging the DCG technique that decomposes the optimization probem into scheduing (master probem) and energy-efficient TP seection (sub-probem), and revea the physica meaning of the decomposition. Further, we propose a greedy agorithm to efficienty sove the NP hard sub-probem. 3) We theoreticay prove that the performance gap of the origina probem s soution is bounded by that of the subprobem, and derive the atter by formuating and soving a reaxed version of the sub-probem. 4) We present numerica resuts to demonstrate the energy efficiency improvement of joint scheduing and power contro, and anayze how the aocations of mutidimensiona resources affect the energy efficiency in the network. The remainder of this paper is organized as foows. Section 2 reviews more reated work. Section 3 describes the system mode and probem formuation. Section 4 presents the decomposition framework and agorithm, with the performance bound of the proposed agorithm anayzed in Section 5. Section 6 presents numerica resuts, and Section 7 gives the concusion remarks. Notations: Throughout this paper, we use A to denote the size of set A. Bodfaced capita etters are used to denote matrices, whie bodfaced ower-case etters are used to denote vectors. A the vectors are coumn vectors by defaut, and the transpose of a matrix A is denoted as A. 2 RELATED WORK Energy-efficient wireess networking has gained great attention in the iterature, especiay for networks with muti-dimensiona resource space such as heterogeneous networks [19], cognitive radio networks [20], [21], [22] and networks with device-to-device communications [23], [24]. Resource aocation for heterogeneous cognitive radio network is studied in [19], where a Stackeberg game approach is adopted with gradient based iteration agorithm as a soution. Channe assignment and power contro are investigated in [20] which aims to maximize energy efficiency of cognitive radio networks and maps the optimization probem to a maximum matching probem. Simiary, a joint soution of channe and power aocation is proposed in [21], with the objective of maximizing overa network throughput. In that paper, physica interference mode is appied and the probem is soved by formuating a bargaining based cooperative game. The work in [22] investigates the joint optimization of spectrum and energy efficiency in cognitive networks with power and subchanne aocation, where the authors propose a tradeoff metric based probem transformation and expoiting convex probem structure. In [23], an energy efficiency maximization probem is formuated as a non-convex program, which is then transformed into a convex optimization probem with noninear fractiona programming. The authors in [24] consider joint radio and power aocation for energy efficiency optimization, and deveop an auction game based approach. The above works focus on specific network scenarios or configurations, which coud not be appied to generic MR-MC networks with muti-dimensiona resource spaces. Furthermore, as most of them focus on channe and power aocation, joint optimizations incorporating ink scheduing has not been we studied. In [25], the probem of energy efficiency optimization in MR- MC networks is considered to derive radio/channe assignment and 2

3 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /TMC , IEEE Transactions on Mobie Computing N C R,R v P λ q (λ) TABLE 1 Notations node set set of channes radio set, radio set of node v set of avaiabe power eves fow commodity demand of commodity λ, L tupe-ink, set of a tupe-inks SINR of tupe-ink γ g m generaized interference coefficient between and m α, A TP, set of a TPs t α portion of transmission time assigned to TP α p,α, r,α transmit power, capacity of ink achieved in TP α u, U, ũ ink utiity, system utiity, utopian utiity scheduing soutions for optima energy efficiency under the requirement of fu network capacity. A simiar approach is adopted in [26] to minimize energy consumption with guaranteed capacity requirement. The probem is soved with a decomposed approach due to the arge scae soution space. Whie these works take protoco interference mode to simpify the scheduing probem, the more reaistic physica interference mode is appied in [27] for a joint scheduing and radio configuration probem. However, power contro is not accounted, i.e., they a use fixed transmit power in the formuation. To take power aocation into resource aocation, the authors in [28] propose an agorithm to jointy aocate channe and power with a utiity based earning method in a decentraized manner. The utiity is characterized by the transmission rate achieved by inks and the soution can maximize the sum rate of inks, but without considering energy cost. For energy efficiency optimization, a joint ce seection and power aocation probem for heterogeneous networks is formuated in [29] and soved with a Lagrange dua based method, where the proposed mode does not appy to generic muti-dimensiona resource space. The work in [30] proposes a two-step approach which first fixes transmit power to sove for scheduing and then optimizes the transmit power on the soved scheduing soution. However, such a decomposition wi ead to sub-optimaity since scheduing and power contro are indeed soved separatey. A joint soution of scheduing, channe aocation and power contro is proposed in [31], but the achievabe data rate on inks is assumed constant. This mode cannot fuy refect the ink capabiity, since the atter is characterized by the rea-time SINR at the receiver. In sum, in the iterature, a joint optimization soution towards energy efficient networking over the muti-dimensiona resource space incuding routing, ink scheduing, radio/channe assignment and power aocation has not been fuy investigated, which is then to be studied in this paper. 3 PROBLEM FORMULATION 3.1 Network Mode Consider a generic MR-MC network with node set N. Each node v N is equipped with one or mutipe radio interfaces which are denoted as radio set R v. Define the set of a radios in the network as R, thus R = v N R v. For each radio, a the other nodes radios within its maximum transmission range are defined as its neighbors. For a non-isoated node, each of its radios can set up transmission inks to a its neighbors. Denote the maximum transmit power of a radio as p max, and assume that the transmit power of each radio takes vaue from a discrete set of power eves P. There is a number of non-overapping channes avaiabe to each radio. We denote a the channes as set C. We consider a sot-based mode, which means the network is static within a time sot such that a the network parameters remain unchanged during this sot. The objective is to minimize the tota energy consumption in the network under traffic demand requirement. Denote the set of mutipe commodity fows as {1,, λ,, Λ}. Each fow λ is specified by its corresponding source-destination node pair and fow demand. Therefore, it requires to jointy address: routing, ink scheduing, radio and channe assignments, and transmit power contro. In this optimization, the scheduing probem is to seect transmission inks and decide the transmission time for them. It can be seen that the joint optimization probem invoves both continuous and discrete decision variabes, making it a mixedinteger probem which is known of high compexity. In what foows, we present a tupe-ink based framework to remode the network, which faciitates an LP formuation and probem decomposition. A tupe-ink is defined as a combined resource aocation for a transmission indicating the transmitter radio, the receiver radio 1 and the operating channe [3]. Denote L as the set of a the tupeinks in the network. Tupe-ink ony exists when there exists a corresponding physica ink (between a radio and its neighbor); a physica ink can be mapped to mutipe tupe-inks. Fig. 1 gives an exampe of tupe-inks between two nodes, where Node 1 has two radios, Node 2 has one radio, and 2 channes are avaiabe. As shown by the dash ines, there exist 8 tupe-inks specified by different transmitters, receivers or channes. With this tupe- Fig. 1. Tupe-ink exampe. Node 1 Node 2 Channe 1 Channe 2 ink based framework, the above optimization probem becomes to jointy sove scheduing and power contro of the tupe-inks since radio and channe assignment is encapsuated into tupe-ink seection. In the rest of this paper, we use ink to stand for tupe-ink uness stated otherwise. In a wireess network, inks may suffer from interference from other concurrent transmitting inks. In this paper, we consider physica interference mode, in which the capacity of a ink can be characterized by the SINR at the receiver. For a ink L, the received SINR is defined as γ = g p I + σ 2 = g p g m p m + σ 2 (1) m L\ where g, p, I, σ 2 denote the ink gain, transmit power, received interference and the noise power, respectivey. Particuary, 1. Tupe-ink is directiona since the transmitter and receiver are specified. 3

4 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /TMC , IEEE Transactions on Mobie Computing g m is used to characterize the strength of interference from m to, to be discussed in detais in the foowing. The capacity (maximum achievabe transmission rate) of ink can be expressed as r = B og 2 (1 + γ ) (2) where B is the corresponding channe bandwidth of. The ink gain of is given as g = ρ(d ), where d is the distance between s transmitter and receiver and ρ( ) is a function of d (e.g., the path oss function). Simiary, g m = ρ(d m ) if and m transmit in the same channe with d m as the distance from m s transmitter to s receiver. Notice that two inks on different channes wi not generate co-channe interference to each other and in this case g m = 0. Besides co-channe interference, two inks may not work simutaneousy due to radio confict. For exampe, two inks cannot share the same transmitter radio for simutaneous transmissions. The radio confict is usuay expressed as integer constraints in optimization probems [27], [31] or considered separatey aside from other resource aocation [18]. In order to faciitate inear programming, we ext the definition of g m to cover the radio confict reationship, and specificay redefine it as interference coefficient. We appy a very arge interference coefficient between inks sharing the same radio. For exampe, if and m have the same transmitter radio, then we may set g m =. Assume a the radios in the network are haf-dupex, and at any time a radio interface can be occupied by at most one ink for transmission. In this case, the interference coefficient can be defined as ρ(d m ), if and m use different radios and are on the same channe; g m 0, if and m use different radios and are on different channes;, if and m share one or two radios. where stands for a significanty arge number. According to this definition, the SINR expression in (1) is abe to characterize both the radio confict and interference, thus the optimization probem can be formuated without additiona radio constraints Optimization Probem Formuation Considering the conficting objectives of throughput enhancement and energy saving, we wi take a muti-objective optimization approach, which is to keep one objective and transform the other one to constraint [26], [32], [33]. Particuary, the energy efficiency is optimized by minimizing the tota energy consumption in the network whie satisfying fow demands of mutipe commodities. Suppose the demand of commodity fow λ is q (λ). To specify the source and destination of each fow, define demand vector q (λ) = (q (λ) 1,..., q(λ) N ) as q (λ) i q (λ), if i is the source node of fow λ q (λ), if i is the destination node of fow λ 0, otherwise to nodes with an N by L node-fow incident matrix H, whose entries are defined as 1, if ink j carries outgoing fow from node i h ij 1, if ink j carries incoming fow to node i 0, otherwise. Based on the above definition, the fow baance constraints for each commodity can be expressed as Hf (λ) = q (λ), λ = 1,, Λ (3) Generay, a ink may use different transmit power at different time such that the mutua interference among inks can be dynamicay coordinated and the transmission rate can be adjusted. At a time instance, the transmit power eves of a the tupe-inks form a transmission pattern (TP). A TP impies the transmission state of a the inks in the network, incuding which radios and channes are being used as we as the corresponding transmit power and ink capacity. Reca that the scheduing probem we defined is to decide when and how ong the inks shoud transmit such that the fow demands can be satisfied with minimum energy consumption. Therefore, with the concept of TP introduced, the probem of joint scheduing and power contro becomes to seect TPs and decide transmission time for them. Since the sets of inks and transmit power eves are finite, the tota number of possibe TPs is finite. In each TP, if a ink is assigned a non-zero transmit power eve, the tupe-ink is considered to be active. Let A be the set of a TPs in the network. Denote the portion of transmission time assigned to pattern α as t α. Let the transmit power and the capacity of ink achieved in pattern α be p,α and r,α, respectivey. Since each TP defines the transmit power eves of a inks, r,α can be expressed as a function of p,α, which is g p,α r,α = B og 2 (1 + m L\ g mp m,α + σ 2 ) (4) Accordingy, the tota traffic rate (incuding a commodities) on ink is bounded as f = λ Λ f (λ) α A r,α t α, L (5) t α = 1 (6) α A Thus, the energy-efficient resource aocation probem can be formuated as a TP based scheduing probem to minimize power consumption whie satisfying fow demand, i.e., Probem 1 (Origina optimization probem): min {f (λ),t α} E = p,α t α (7) α A L s.t. constraints (3),(4),(5),(6) f (λ) 0, L, λ = 1,, Λ (8) t α 0, α A (9) 4 Denote f (λ) as fow rate for commodity λ on ink, then f (λ) = (f (λ) 1,..., f (λ) L ) is the fow vector of commodity λ on a the inks. Since fow rate is defined for each ink, it can be reated 2. The generaized radio confict mode can adapt to other network scenarios that have different types of radio constraints by correspondingy adjusting the vaues in the definition. The optimization variabes are fow variabes f (λ), as we as transmission time portion t α assigned to TPs. The objective function in (7) stands for the tota power consumption which is the summation of power consumption over a the inks in a TPs. Lemma 1. In the optima soution of Probem 1, constraint (5) wi reach equaity.

5 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /TMC , IEEE Transactions on Mobie Computing Proof: The emma can be proved by contradiction. Suppose with the optima soution, constraint (5) does not reach equaity, i.e., there exists a ink such that f < α A r,αt α. We ca such a ink over-schedued, which indicates some pattern is providing more than necessary capacity to ink. Since f is non-negative, there must exist a pattern α 1 with r,α1 t α1 > 0 (p,α1 > 0). Then, we ook for a pattern α 2 that has smaer capacity on but no ower rate on the other inks. In other words, α 2 shoud satisfy 0 r,α2 < r,α1 and r m,α2 r m,α1, m. It can be seen that any pattern with a ower power eve on and same eves on the other inks appies. Since p,α1 > 0, we can aways find such patterns. Notice that α 2 has ess power consumption than α 1. The equaity on ink can be achieved by removing the overschedued capacity on ink, which can be done by moving part of the traffic oad from α 1 to α 2. In other words, the equaity can be achieved by designing a new schedue that moves a portion of t α1 to t α2. Such a new schedue {t α} can be obtained by soving: t α 1 + t α 2 = t α1 + t α2 (10) t α = t α, α A \ {α 1, α 2 } (11) f = α A r,α t α. (12) Under the new schedue, the capacity of other inks wi not be reduced whie the inequaity on ink wi become equaity, which means (5) sti hods and the new schedue is a feasibe soution. Since part of the transmission time of pattern α 1 is reschedued to pattern α 2 whie the atter has ess power consumption, the new soution wi consume ess power compared to the origina one, which means the origina soution is not optima and contradicts the assumption. This competes the proof. 3 Then, with a the constraints in equaity, we can rewrite the optimization probem into standard matrix form as foows: Probem 1 (Origina probem in matrix form): min x s.t. c x Ax = b x 0 with x = (f (1),..., f (Λ), t 1,..., t A ) and H... A = H I L I L R 1 1 A b = (q (1),..., q (Λ), 0 1 L, 1) c = (0 1 ( L Λ), L p,1,, L p, A ) where R is the L A ink capacity matrix with entries r,α. Notice that the non-zero entries in c correspond to the energy consumption of TPs. Remark 1. Athough there are some standard agorithms for soving LPs, e.g., Eipsoid and simpex, to run such agorithms requires compete information of the constraint matrix A. In our case, such agorithms may become infeasibe due to the practica 3. The physica meaning of Lemma 1 is that whenever there is an overschedued ink, we can aways adjust the scheduing to remove the redundant capacity by averaging out the traffic oad from a current pattern to others with smaer capacity on. difficuty in preparing the matrix A in advance. It is impractica, in either computation time or storage space, to ist a those transmission patterns in advance. Without the constraint matrix A in advance, standard agorithms such as simpex, interior point, and eipsoid method cannot be appied to sove the probem. Remark 2. The number of patterns grows exponentiay with the number of inks (i.e., tupe-inks) in the network, eading to exponentiay many coumns in the constraint matrix A. In this sense, the LP formuation does not change the NP-hard nature of the wireess network scheduing probem, but can aow the convenience to design efficient approximation agorithms. 4 DECOMPOSITION FRAMEWORK In this section, we deveop a decomposed method to iterativey sove Probem 1. A the probems rered in the decomposition process are summarized in Tabe 2 and iustrated in Fig. 2. start Probem 1 (Agorithm 2) Probem 2 Converge? Yes Yes No Probem 3 (Agorithm 1) optiona Probem 3R (Agorihtm 3) Improve? Fig. 2. Fowchart of the proposed joint optimization framework. Intuitivey, not a the TPs wi contribute to fow deivery and there is no need to aocate transmission time to TPs with itte contribution. Our experiments in tupe-ink scheduing [6], [9] aso indicate that ony a subset of A wi be schedued. In other words, instead of considering a the patterns in A, we ony need to find and schedue the critica ones. To this, we deveop a decomposition technique based on deayed coumn generation (DCG) [9] to iterativey find such a subset of critica TPs. 4.1 DCG-Based Decomposition According to the matrix form of Probem 1, the size of the eft haf of constraint matrix A is determined by the network topoogy, whie each coumn in the right haf corresponds to a TP. Therefore the number of schedued TPs is equa to the number of coumns in the right haf of A. Starting from an initia feasibe soution obtained from a sma subset of A, the DCG method iterativey searches for new coumns (or equivaenty TPs) that are promising in improving the objective. Let A (k) denote the subset of TPs aready found at the beginning of Step k. In Step k, the optima soution with given A (k) can be obtained by soving the foowing master probem: Probem 2 (Master Probem): ( min {f (λ),t α} s.t. E (k) = α A (k) L f = f (λ) = λ Λ No p,α ) t α, (13) α A (k) r,α t α L (14) α A (k) t α = 1 (15) t α 0, α A (k) (16) constraints (3),(8) 5

6 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /TMC , IEEE Transactions on Mobie Computing TABLE 2 Summarization of optimization probems Probem 1 Probem 2 Probem 3 Probem 3R Description origina probem master probem sub-probem reaxed sub-probem Compexity (origina) P L P ( N 2 R v 2 C ) NA L 2 P L NA Optima soution E E (k) U Ũ Proposed soution Ê E (k) Û Ũ Agorithm Agorithm 2 any LP agorithm Agorithm 1 Agorithm 3 Compexity (proposed) iterative with LP LP with moderate size P L 3 R Bound ower bounded (Theorem 1) optimay soved upper bounded (Theorem 2 and 3 ) optimay soved 6 Or in matrix form, Probem 2 (Master probem in matrix form): min x (k) s.t. c (k) x (k) A (k) x (k) = b x (k) 0 where c = (0 1 ( L Λ), L p,1,, L p, A ) (k) and x = (f (1),..., f (Λ), t 1,..., t A ) (k). In the master probem, A (k) has the same number of rows as A but much fewer coumns than A. The above master probem can be easiy soved if the subset A (k) is of moderate size. The soution of the master probem provides the scheduing time t (k) α for each pattern α in A (k) aong with the dua variabe vector w (k) associated with the constraints (where w (k) is obtained by soving the dua probem of Probem 2). The next probem is to search for a new coumn A i to be added into A (k) to generate A (k+1), which can improve the objective of the optimization probem. In DCG agorithm, such an improvement is evauated by the reduced cost c i w (k) A i where i denotes the index of the new coumn [34]. If a coumn is associated with negative reduced cost, then adding this coumn wi improve the objective vaue. Since the added coumns ony correspond to the right haf of constraint matrix, it can be observed that adding a coumn is equivaent to adding a new pattern α, whose improvement 4 can be evauated as w (k) A α c α (17) = 0 + L = L w (k) r,α + w (k) 0 1 p,α L ( w (k) r,α p,α ) + w (k) 0 (18) where w (k) is the dua variabe corresponding to the th row of matrix R (k) in A (k) (entry r,α ) and w (k) 0 is the dua variabe associated with the ast row of A. Define the term w (k) r,α p,α as the utiity of ink in pattern α, and the utiity sum of a the inks as system utiity U (k). The utiity of each ink consists of the contribution to fow traffic and the power cost, where the fow contribution of a ink is further determined by both the ink capacity r,α and the dua variabe w (k). The expression of utiity function w (k) r,α p,α indicates 4. We use the additive inverse of reduced cost as a measurement of the performance improvement to keep consistency with the ater definition of utiity. that it shoud have the same unit as p,α, which is power, whie w (k) acts as a price factor to convert throughput into wefare. A new TP wi be added to A (k) if it maximizes the improvement in (18). Since w (k) 0 is a constant indepent of the TP to be added in Step k for a given A (k), it can be ignored during pattern seection. Then seecting a new TP is equivaent to soving the foowing probem: Probem 3 (Sub-Probem): ( ) U α (k) w (k) r,α p,α (19) max α A\A (k) = L Since energy efficiency is a compound of both the benefit in fow contribution and cost in power consumption, the expression in (19) naturay provides an evauation function of a TP with these considerations. Therefore, the sub-probem can be interpreted as to search for the most energy-efficient TP, which is evauated by the corresponding system utiity. As mentioned previousy, the system utiity shares the same unit as that of power, which indicates that the objectives in sub-probem and origina probem are consistent in unit. The new TP, if found, is then added to A (k) to form A (k+1). The master probem is then updated and soved to provide a new set of soutions. The process is repeated unti no improvement can be made (or no coumn can be added), i.e., w (k) A i c i 0, i. Then, the standard DCG theory shows that the current soution wi be the optima soution of Probem 1 [34]. The physica meaning behind the DCG decomposition can be expained as foows. We search for energy efficient TPs to perform scheduing, where the energy efficiency of TPs deps on the information obtained from current soution. Each time soving the master probem wi provide an updated evauation on a the inks regarding their capabiities in satisfying traffic demand based on their performance in existing TPs, and such an evauation is conveyed through dua variabes w (k). Then according to this evauation, the most energy efficient TP that can maximize the system utiity is searched and fed back to the master probem. With this new information, inks wi be re-evauated through soving the updated master probem. Repeating these steps wi provide more and more accurate evauations on the energy efficiency of TPs and therefore approach the optima soution. According to the definition of TP, finding a TP is equivaent to finding a power aocation over a the inks. In this sense, the proposed framework can be interpreted as decomposing the origina probem into scheduing phase (master probem) and power aocation phase (sub-probem). Optimaity remains intact during the decomposition process by iterativey soving the two phases. Thus, with the muti-dimensiona modeing, TP based scheduing and DCG based decomposition, the joint optima soution over a dimensions of network resources can be obtained.

7 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /TMC , IEEE Transactions on Mobie Computing 4.2 Initia Soution It is usuay difficut to find an initia subset of A that can yied feasibe soution. However, even if the initia soution is infeasibe, we can sti find new coumns based on the dua variabes, and the newy added coumns can potentiay drive the iteration to yied a feasibe soution. Thus, even starting from an infeasibe soution, the feasibiity wi be restored in severa rounds, providing that the origina Probem 1 is feasibe. Based on this, the initia subset can be constructed with randomy seected TPs. However, if a ink is not incuded in the initia subset, then the ink wi probaby never be evauated or invoved into the probem. Taken this issue into consideration, we need to cover every ink in the initia subset. In addition, the constraint matrix shoud have fu row rank [34]. Based on these, the initia subset can be set by choosing L TPs where each TP has exacty one unique ink activated. In this way, we can get a diagona matrix R and matrix A wi have fu row rank. Remark 3. Theoreticay, if the sub-probem can be soved optimay, the iterative process of DCG method wi finay converge to the optima soution of Probem 1 [34]. In our mode, the subprobem is to find a TP with maximum utiity, which is done by searching over a the unused patterns. Again, due to the arge size of TPs in the network ( A ), it becomes computationay impractica to search over a the patterns to obtain the optima soution. Therefore, we propose a ight-weight greedy agorithm to approximatey sove the sub-probem. 4.3 Greedy Agorithm for Soving the Sub-Probem As previousy mentioned, a TP is defined as a power aocation on a the inks, therefore it is equivaent to finding an optima power aocation on inks to maximize the system utiity (the superscript indicating number of rounds is omitted in this sub-section since the sub-probem is soved within one round): max {p } U = u = w r p (20) L L Power aocation on inks with the objective of maximizing system utiity is a chaenging probem since the utiities of inks are mutuay depent. Even if the utiity of each ink is fixed, the probem is sti NP hard (which can be viewed as a maximum weighted indepent set probem under physica interference mode as in [11]). To obtain a practicay feasibe soution, we deveop a greedy agorithm to find the optima power aocation. The greedy power aocation is done by starting with azero power aocation and graduay activating (assigning positive power eves to) inks unti the system utiity no onger increases. Whether an inactive ink can be activated deps on its contribution to the system utiity. Among a the inactive inks, the one with the argest contribution wi be activated. The detais of the greedy agorithm are shown in Agorithm 1. Denote the set of active inks and inactive inks as S a and S i, respectivey. At each step, the agorithm evauates a the inactive inks and seects one into the active set. According to Eq. (1) and the definition of utiity, the utiity of each ink can be written as a function of its transmit power p and active ink set S, u = u (p, S) = w r p = w B og 2 (1 + m S\{} g p g m p m + σ 2 ) p (21) Each inactive ink first cacuates a myopic optima power eve ˆp that maximizes its own utiity assuming that the power eves of a the other inks keep unchanged. ˆp can be obtained by cacuating the utiities at a possibe power eves and choosing the one that gives maximum utiity. Then it cacuates the change of system utiity U if it is activated by using power ˆp, U = u (ˆp, S a )+ u m (p m, S a {}) u m (p m, S a ) m S a m S a (22) i.e., U can be viewed as the contribution of if activated. For the ink with the argest contribution, if its contribution is arger than a pre-defined non-negative constant ɛ 5, it means activating this ink can increase the system utiity and this ink wi be activated at the cacuated power eve ˆp. Otherwise, the system utiity cannot be increased and the agorithm stops. At the of the agorithm, it outputs the power aocation to a inks. Agorithm 1: Greedy Agorithm for Probem 3 Input: dua variabes {w } =1,, L ; Initiaization: p = 0, L; S a = ; S i = L; Û = 0; whie S i do for S i do ˆp = arg max u (p, S a ); Cacuate U according to Eq. (22); = arg max U (If the soution is not unique, S i randomy seect one ink with max U ); S i if U > ɛ then move from S i to S a ; p = ˆp ; Û = Û + U ; ese Agorithm stops; Output: Û as the soution of Probem 3; The corresponding power aocation {p } =1,, L. 4.4 Compexity Anaysis Each utiity computation (as in Eq. (21)) incurs a compexity in the order of S a. Within each iteration of the greedy agorithm, each inactive ink in S i wi perform P utiity computations to find the optima power eve and at most 2 S a utiity computations (as in Eq. (22)) to cacuate the effect on other inks. As a resut, each iteration requires a tota number of ( P + 2 S a ) S i S a computations to evauate the contributions of a the inactive inks, pus L computations to perform sorting. The iteration wi be repeated by S a times, eading to a tota compexity of [( P + 2 S a ) S i S a + L ] S a. In practice, there can be at most R /2 inks actived simutaneousy due to radio confict. Therefore in the resut of the agorithm we wi have S a R /2. In addition, S i L. Based on this, the computation compexity is in the order of ( R + P ) R 2 L. Further, R is usuay ess than L. Therefore the greedy agorithm s compexity wi be in the order of P L 3. Since the number of TPs is P L and the compexity of cacuating system utiity of each TP is L 2, the compexity 5. ɛ can be set to 0, or positive if we want to terminate the agorithm earier when the contribution of adding a new ink is very sma. 7

8 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /TMC , IEEE Transactions on Mobie Computing of brute force searching over the entire space to find maximum utiity TP is L 2 P L, which is significanty higher than that by Agorithm Agorithm Design With the decomposition framework and the greedy agorithm, we can now design the decomposition agorithm for soving the origina probem, as shown in Agorithm 2. Agorithm 2: Decomposition Agorithm for Probem 1 Initia transmission pattern set A (0) ; k=0; whie E (k) < E (k 1) do //Master stage: Update master probem (Probem 2) with current TPs A (k) ; Sove master probem to obtain energy E (k) and dua variabes w (k) ; //Sub-probem stage: Sove the sub-probem (Probem 3) using Agorithm 1 to obtain a new TP α (with the power aocation) and Û (to cacuate performance bound); if w (k) A α c α > 0 then Add the new TP to A (k) and obtain A (k+1) ; k k + 1; Go to master stage; ese break; Ê = E (k) ; Output: Ê as the approximate soution of Probem 1. In the proposed framework, both the probem formuation and the proposed agorithms invove goba information of the network such as the ink states and power aocation. Therefore, a centra agent is required to coect the information and perform the optimization computation, and a centraized approach is taken to deveop agorithms in this paper. However, the ideas behind the proposed agorithms, such as the framework for jointy optimizing different types of network resources, the corresponding decomposition scheme to tacke the compexity issue, and the interpay between different network dimensions and the optimization, can be combined or incorporated with other approaches to efficienty sove optimization probems in various scenarios. For exampe, the concept of transmission patterns can be adopted in onine scheduing agorithms for joint scheduing and power aocation in a dynamic network, and the decomposition approach can be combined with distributed agorithms to further reduce the compexity. We eave these investigations for our future work. 5 PERFORMANCE ANALYSIS It is known that, theoreticay, the DCG-based iterative agorithm wi converge to an optima soution of the origina probem, providing that the sub-probem is optimay soved in every step [34]. Therefore, in our case, the optimaity of the obtained soution is determined by that of the sub-probem. Beow, we wi first show how the performance of the greedy agorithm affects that of the origina probem (Probem 1). 5.1 Performance of the Origina Probem Soution Denote the corresponding objectives achievaby optima soutions of the origina probem (Probem 1) and the sub-probem (Probem 3) as E and U, respectivey. When Agorithm 2 stops, et the soution of the sub-probem soved by the greedy agorithm be Û, and the corresponding soution to Agorithm 2 be Ê. Then we have the foowing reationship: Theorem 1. The performance gap of Agorithm 2 in soving the origina probem is upper bounded by that of Agorithm 1 in soving the sub-probem, i.e., E = Ê E U Û = U (23) For the origina probem, suppose the dua vector associated with Ê is ŵ, whose ast entry is ŵ 0. Before proving Theorem 1, we first present the foowing resut. Lemma 2. If ŵ s ast entry (ŵ 0 ) is repaced by ŵ 0 U, the resuting vector, denoted as w, wi sti be a feasibe soution to Probem 1 s dua probem. Proof: Denote the dua probem of Probem 1 as: Probem 1D (dua of Probem 1): max w w b s.t. w A c Denote A i as a coumn of A. For the coumns in the eft haf of A, repacing ŵ 0 with ŵ 0 U wi not affect the vaue of ŵ A i since the ast row of eft haf of A ony has zero entries. Therefore, for these coumns, w A i c i sti hods. Recaing that each coumn of the right haf of A is associated with a TP, we can write Eq. (18) as w A α c α = U α + ŵ 0, where U α is the system utiity achieved by pattern α. Since the decomposition agorithm stops at Û, we have Û + ŵ 0 = ŵ A α c α 0 ŵ 0 U U For w and every coumn in the right haf of A, we have w A α c α = U α + ŵ 0 U U α U 0 Above a, w is a feasibe soution to Probem 3. Then we continue to prove Theorem 1. Proof: Suppose x and w are the optima soutions of the origina probem (Probem 1) and its dua probem (Probem 1D), respectivey. From Lemma 2, w is a feasibe soution to Probem 1D. Therefore w b w b = ŵ b U = Ê U According to weak duaity, cx w b, which eads to Ê E = Ê cx Ê w b Ê (Ê U ) = U thus competing the proof of Theorem 1. As discussed in Section 4.1, it can be observed that both performance gaps of the soutions of origina probem and subprobem are in the unit of power, which shows the consistency of unit in Theorem 1. Moreover, Theorem 1 shows that the performance gap of the origina optimization probems soution is bounded by that of the sub-probem. Therefore, the performance of the decomposition agorithm can be evauated through investigating the performance gap of Agorithm 1 in soving the subprobem. 8

9 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /TMC , IEEE Transactions on Mobie Computing 5.2 Performance of the Sub-Probem Soution The objective of the sub-probem is the system utiity, whose maximum vaue is reated to how many inks can be activated simutaneousy. Due to radio confict, inks sharing the same radio wi not be activated at the same time, otherwise both of them wi resut in 0 utiity. Considering that the tupe-ink based mutidimensiona network mode can be abstracted as a graph with radios being vertices and inks being edges, the maximum number of concurrent inks with positive utiity can be characterized by the matching number of the graph associated with the network. Let M denote the matching number of the network. We have the foowing statement. Lemma 3. In the optima soution of the sub-probem, there can be at most M inks with positive utiity. The proof of Lemma 3 foows directy the definition of matching number of graph. Define ũ = u (ˆp, ) as the utopian utiity of a ink, which is its optima utiity when ignoring any mutua interference. Notice that utopian utiity is aso the utiity of each ink in the first round of the greedy agorithm (Agorithm 1), and wi not be smaer than the practicay achieved utiity when the corresponding ink is schedued. Without oss of generaity, suppose {ũ } is sorted in descing order, i.e., ũ 1 > ũ 2 >. Based on this, we can derive one performance bound of Agorithm 1 as Lemma 4. Û U ũ1 M =1 ũ 1 M (24) Proof: According to Agorithm 1, the system utiity wi be increased every time a new ink is added, therefore the fina system utiity of the greedy agorithm Û wi not be smaer than that in the first round, i.e., Û ũ 1. On the other hand, there can be at most M inks with positive utiity according to Lemma 3. Hence, U M ũ. Together we have =1 Û U ũ1 M =1 ũ ũ 1 M ũ 1 = 1 M Lemma 4 shows that the performance of the greedy agorithm is constant-bounded, and impies two ways of evauating performance gap U, which are shown in the foowing theorem. Theorem 2. The performance gap of Agorithm 1 in soving subprobem is upper bounded as U ũ (25) M =2 (M 1)Û (26) 5.3 Bound from Sub-Probem Reaxation For a given Û, estimating the upper bound of U is equivaent to estimating the upper bound of U. An upper bound of U* can be obtained by soving a reaxed version of Probem 3 as foows. The probem can be reaxed by ignoring some interference without changing the formuation. In other words, the reaxation of Probem 3 can be done by reducing the vaues of g m s. For exampe, one reaxation can be ignoring a the interference or radio confict in the network (i.e., g m = 0,, m L) but imiting the tota number of active inks to M. In this case, the utiity of each ink is indepent of other inks activities and the optima system utiity is M ũ, which is an interpretation of =1 the bound in Eq. (25). However, in this case, tupe-inks associated with the same physica ink usuay have the same dua vaues (w ), which means they t to be activated simutaneousy if mutua interference is ignored. As a resut, in the soution of this reaxed optimization probem, many inks share same radios, which is physicay infeasibe. Therefore, this reaxation may yied a oose bound. In order to formuate a proper reaxed probem to characterize an upper bound of U, we tighten the above reaxation by adding back part of the radio confict as constraints. This reaxation wi ignore co-channe interference and modify the radio constraint as foows: There coud be at most min{r u, R v } tupe-inks activated on any physica ink between node u and v, whie the tota number of active tupe-inks in the network is imited by M. Denote the reaxed probem as Probem 3R. The soution of Probem 3R can be obtained by greediy picking {ũ } as ong as the above constraint is not vioated, as summarized in Agorithm 3. Agorithm 3: Soving Probem 3R Input: {ũ } (sorted in descing order); Initiaization: Number of seected tupe-inks on each physica ink n uv = 0, u, v N ; Tota Number of active tupe-inks n = 0; Tota utiity Ũ = 0; = 1; whie n < M and ũ > 0 do Find s corresponding physica ink uv; if n uv < min{r u, R v } then Tupe-ink is activated; n = n + 1; n uv = n uv + 1; Ũ = Ũ + ũ ; = + 1; Output: Ũ as the (optima) soution of Probem 3R. Lemma 5. Agorithm 3 yieds the optima soution of Probem 3R. Proof: Since the constraint ony appies to each physica ink ocay and there is no co-channe interference, the behavior of each physica ink has no infuence on other physica inks. On the other hand, it can be observed that the greedy seection of the inks associated with one physica ink is ocay optima. As a resut, combining the oca optima soutions of indepent physica inks wi yied the goba optima soution. Define the ist of utopian utiities as utopian ist ({ũ }, = 1, 2,..., L). In Agorithm 3, at most M utopian utiities are added to the fina system utiity. Notice that these added utiities may not be the first M ones (M argest ones) in the utopian ist, since some might be excuded due to the constraint defined in Probem 3R. Then define the ist of added utopian utiities associated with the seected inks in Agorithm 3 as reduced utopian ist ({ũ R }, = 1, 2,..., M ). From the previous anaysis, it can be 9