A Strategy-Proof Combinatorial Heterogeneous Channel Auction Framework in Noncooperative Wireless Networks

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1 Ths s the author s verson of an artcle that has been publshed n ths journal. Changes were made to ths verson by the publsher pror to publcaton. The fnal verson of record s avalable at 1 A Strategy-Proof Combnatoral Heterogeneous Channel Aucton Framework n Noncooperatve Wreless Networks Zhenzhe Zheng, Student Member, IEEE, Fan Wu, Member, IEEE, and Guha Chen, Member, IEEE Abstract Aucton s beleved to be an effectve way to solve or releve the problem of rado spectrum shortage, by dynamcally redstrbutng dle wreless channels of prmary users to secondary users. However, to desgn a practcal channel aucton mechansm, we have to consder fve challenges, ncludng strategy-proofness, channel spatal reusablty, channel heterogenety, bd dversty, and socal welfare maxmzaton. Unfortunately, none of the exstng works fully consdered the fve desgn challenges. In ths paper, we present the frst n-depth study on the problem of dynamc channel redstrbuton jontly consderng the fve desgn challenges, and present SMASHER, whch s a famly of Strategy-proof combnatoral Aucton mechansms for HEterogeneous channel Redstrbuton. SMASHER contans two strategy-proof aucton mechansms, namely SMASHER-AP and SMASHER-GR. SMASHER-AP s a strategy-proof, approxmately effcent combnatoral aucton mechansm for ndvsble channel redstrbuton. We further consder the case, n whch channels can be shared by the users n a paradgm of tme-dvson multplexng and propose SMASHER-GR, whch s a strategy-proof channel allocaton and schedulng mechansm. We have extensvely evaluated our desgns. The evaluaton results show that our desgns acheve much better performance than exstng works. Index Terms Wreless Network, Channel Allocaton, Combnatoral Aucton. 1 INTRODUCTION THE last two decades have wtnessed a rapd development of wreless communcaton technology. Unfortunately, naturally lmted rado spectrum s becomng a more and more serous bottleneck of the ongong growth of wreless applcatons and servces. Most of the countres have specfc departments to regulate spectrum usage, e.g., Federal Communcatons Commsson (FCC) [1] n the US and Rado Admnstraton Bureau (RAB) n Chna [2]. They statcally allocate spectrum to wreless applcaton servce provders on a long term bass for large geographcal regons. Such statc management leads to low spectrum utlzaton n the spatal and temporal dmensons. Large chunks of rado spectrum are left dle most of the tme at a lot of places, whle new wreless applcatons are starvng for the rado spectrum. F. Wu s the correspondng author. Z. Zheng, F. Wu, G. Chen are wth the Department of Computer Scence and Engneerng, Shangha Key Laboratory of Scalable Computng and Systems, Shangha Jao Tong Unversty, Chna. (E-mal: zhengzhenzhe@sjtu.edu.cn {fwu,gchen}@cs.sjtu.edu.cn). Ths work was supported n part by the State Key Development Program for Basc Research of Chna (973 project 214CB3433 and 212CB31621), n part by Chna NSF grant and , n part by Shangha Scence and Technology fund 12PJ1449 and 12ZR14149, and n part by Program for Changjang Scholars and Innovatve Research Team n Unversty (IRT1158, PCSIRT) Chna. The opnons, fndngs, conclusons, and recommendatons expressed n ths paper are those of the authors and do not necessarly reflect the vews of the fundng agences or the government. Therefore, an open and market-based framework s hghly needed to dynamcally redstrbute the rado spectrum, and thus mprove the utlzaton of the rado spectrum [3]. Auctons are the most well-known market-based mechansms to redstrbute resources [4], [5]. Snce 1994, FCC has conducted a seres of auctons for the lcenses of rado spectrum. Whle FCC auctons target only at large wreless servce provders, our focus s on small wreless applcatons, such as communty wreless networks or home wreless networks. There exst many challenges n desgnng a practcal channel aucton mechansm [11], [12]. We lst fve major challenges: Strategy-Proofness: In strategy-proof aucton mechansms (please refer to Secton 2.1 for the defnton), smply submttng truthful channel demands (e.g., valuaton of the channels) maxmzes each partcpant s utlty. Snce the partcpants are normally ratonal and selfsh, they always tend to strategcally manpulate the aucton, f dong so can ncrease ther utltes. Such selfsh behavor nevtably hurts the other partcpants utltes. Therefore, t dscourages truthfully behavng partcpants from jonng the aucton, f strategy-proofness s not guaranteed. Spatal Reusablty: Spatal reusablty dfferentates the wreless channels from conventonal goods. Two wreless users can use the same wreless channel smultaneously, f they are wellseparated (.e., out of the nterference range of Copyrght (c) 214 IEEE. Personal use s permtted. For any other purposes, permsson must be obtaned from the IEEE by emalng pubs-permssons@eee.org.

2 Ths s the author s verson of an artcle that has been publshed n ths journal. Changes were made to ths verson by the publsher pror to publcaton. The fnal verson of record s avalable at 2 TABLE 1 Comparson wth Exstng Channel Aucton Mechansms. Exstng Works Strategy- Proofness Spatal Reusablty Channel Heterogenety Bd Dversty Socal Welfare VERITAS [6] No Guarantee TRUST [7] No Guarantee SMALL [8] No Guarantee TAHES [9] No Guarantee CRWDP [1] Approxmately Effcent SMASHER-AP Approxmately Effcent SMASHER-GR No Guarantee each other). Explotng spatal reusablty can hghly mprove spectrum utlzaton. Channel Heterogenety: The nature of wreless channels makes the goods n the channel aucton heterogeneous. The channel heterogenety comes from both spatal heterogenety and frequency heterogenety. On one hand, the avalablty and qualty of a channel vary at dfferent locatons. On the other hand, channels wth dfferent central frequency may have dfferent propagaton and penetraton characterstcs. Bd Dversty: Wreless devces may be equpped wth multple rados, each of whch can work on a dfferent channel at the same tme. Consequently, a wreless user may request multple channels, accordng to her qualty of servce requrement. Buyers have hgher opportuntes to obtan channels by submttng multple channel bundles, whch makes the channel redstrbuton more flexble. Therefore, t s necessary to allow users to express dverse demands for channels. Socal Welfare: The basc and common objectve of auctons s to maxmze socal welfare, whch s the sum of the aucton wnners valuatons of the allocated goods (please refer to Secton 2.1 for the defnton). A number of related works (e.g., [6] [1]) exst n the lterature. Unfortunately, none of these works fully consder the fve desgn challenges (as shown n Table 1). Some of strategy-proof channel aucton mechansms (e.g., VERITAS [6], TRUST [7], SMALL [8]) consder channel spatal reusablty, but only work when the tradng channels are homogenous. Two recent works TAHES [9] and CRWDP [1] consder the heterogenety of channels, but TAHES restrcts each buyer to bd for a sngle channel whle CRWDP gnores the spatal reusablty of channels. In ths paper, we conduct an n-depth study on the problem of dynamc channel redstrbuton jontly consderng the fve desgn challenges, and present SMASHER, whch s a famly of Strategyproof combnatoral Aucton mechansms for HEterogeneous channel Redstrbuton. SMASHER contans two dstnct aucton mechansms, namely SMASHER-AP and SMASHER-GR. Specfcally, SMASHER-AP s a novel combnatoral aucton mechansm for ndvsble heterogeneous channel redstrbuton, and acheves both strategy-proofness and approxmately effcent socal welfare. SMASHER-GR jontly consders channel allocaton and schedulng when channels can be shared n a paradgm of tme-dvson multplexng. We use Table 1 to show the comparson of our desgns wth closely related works. We make the followng contrbutons n ths paper: Frst, we present a general model of combnatoral aucton for heterogeneous channel redstrbuton. The aucton model s powerful enough to express channel spatal reusablty and heterogenety, as well as bd dversty. Second, we ntroduce the concept of vrtual channel to capture the conflcts of channel usage among dfferent aucton partcpants. By usng vrtual channels, we transform the problem of heterogeneous channel allocaton to a classc combnatoral aucton. Thrd, we propose SMASHER-AP, whch s a combnatoral aucton mechansm for heterogeneous channel redstrbuton, achevng both strategy-proofness and approxmately effcent socal welfare. Fourth, we further consder the case, n whch channels can be shared n a paradgm of tmedvson multplexng, and propose SMASHER- GR, whch s a strategy-proof combnatoral aucton mechansm for channel allocaton and schedulng. Fnally, we evaluate the performance of our desgns. Our smulaton results show that our desgns acheve much better performance than closely related works, n terms of socal welfare, buyer satsfacton rato, and channel utlzaton. The rest of ths paper s organzed as follows. In Secton 2, we present the model of combnatoral aucton for heterogeneous channel redstrbuton. In Secton 3, we ntroduce the concept of vrtual channel and convert the problem of heterogeneous channel allocaton to a classc combnatoral aucton. In Secton 4, we present the desgn of SMASHER-AP. In secton 5, we propose SMASHER-GR. In Secton 6, Copyrght (c) 214 IEEE. Personal use s permtted. For any other purposes, permsson must be obtaned from the IEEE by emalng pubs-permssons@eee.org.

3 Ths s the author s verson of an artcle that has been publshed n ths journal. Changes were made to ths verson by the publsher pror to publcaton. The fnal verson of record s avalable at 3 we report evaluaton results. In Secton 7, we revew related works. In Secton 8, we conclude the paper and dscuss future works. 2 PRELIMINARIES AND PROBLEM FORMU- LATION In ths secton, we present the aucton model for the problem of heterogeneous channel allocaton, and revew some mportant soluton concepts. 2.1 Aucton Model We consder a statc scenaro, n whch there s a prmary spectrum user, called seller, who wants to lease out her temporarly unused wreless channels, and some secondary users (e.g., WF access ponts), called buyers, who want to lease channels to provde servces to ther customers at certan qualty of servce (QoS). We consder that the channels for leasng are heterogeneous, and thus the buyers have ther own preference over the channels due to spatal varance (e.g., background nose, temperature, and landform). Snce wreless devces can be equpped wth multple rados, the buyers may request more than one channel accordng to ther requrements of QoS. Consderng the dversty of QoS demand and the heterogenety of channels, we allow the buyers to submt multple channel requests, among whch one of the requests can be granted. 1 We assume that buyer have unform valuaton over any of her channel requests, because the buyer s requrement of QoS can be satsfed f one of her requested bundles s allocated. Dfferent from the allocaton of tradtonal goods, wreless channels can be spatally reused, meanng that well-separated buyers can work on the same channel smultaneously, f they do not have nterference between each other. We model the process of heterogeneous channel redstrbuton as a sealed-bd combnatoral aucton, n whch buyers smultaneously submt ther demands for channels to a trustworthy auctoneer, such that no buyer can know other partcpants nformaton. The auctoneer makes the decson on channel allocaton and the charge to each wnner. We denote the set of orthogonal and heterogeneous channels for leasng by C {c 1, c 2,..., c m }, and the set of buyers by N {1, 2,..., n}. We lst useful notatons n our model of combnatoral channel aucton as follows: Channel Request R : Each buyer N submts a vector of requested channel bundles ( R S 1, S 2,..., S φ to the auctoneer. Any channel bundle S l C, 1 l φ can satsfy her QoS. We assume that the request 1. We dscuss ths model n Secton 4 and extend to the scenaro, n whch each buyer can be allocated multple bundles to reach her QoS n Secton 5. ) s strct, meanng that the buyer s only nterested n wnnng a whole bundle S l n her request vector. Although the buyer can submt a request vector R wth more than one channel bundle, only one channel bundle can be granted by the auctoneer. We call buyer, who submts a request vector of φ channel bundles, and s nterested n wnnng one of the bundles, as φ -mnded buyer. If φ = 1, then the buyer s sngle-mnded. Note that our aucton model s a generalzaton of exstng models wth snglemnded buyers (e.g., [9], [1]). The maxmum number of submtted channel bundles among all buyers s denoted by Φ max N φ. We denote the channel request vector R of all the buyers as R (R 1, R 2,..., R n ). Valuaton v : Each buyer N has a unform valuaton v over any requested channel bundles n R. Here, v s the prvate nformaton of the buyer. Ths s also known as type n mechansm desgn. The buyer valuaton has two propertes: Free Dsposal and Normalzaton. Free dsposal means that for any two subsets of channels S and T, f S T, then v (S) v (T ); whle normalzaton means that v ( ) =. We denote the valuaton vector V of all the buyers as V (v 1, v 2,..., v n ). Bd b : Each buyer N submts a bd b to the auctoneer, meanng that f she wns any channel bundle S l, she would lke to pay no more than b for t. Here, the bd b may not necessarly be equal to her valuaton v. Let vector B represent the bds of all the buyers B (b 1, b 2,..., b n ). Clearng prce p : The auctoneer charges each wnnng buyer N a clearng prce p. The loser n the aucton s free of any charge. We use vector P (p1, p 2,..., p n ) to represent the clearng prces of all the buyers. Utlty u : The utlty of a buyer N s defned as the dfference between her valuaton on the bundle of wnnng channels and her clearng prce p u v p. (1) We consder that the buyers are ratonal and selfsh, thus ther goals are to maxmze ther own utltes. In contrast to the buyers, the auctoneer s objectve s to maxmze socal welfare. Here socal welfare s defned as follows. Defnton 1 (Socal Welfare): The socal welfare n a channel aucton s the sum of wnnng buyers valuatons on ther allocated bundles of channels. where W s the set of wnners. SW W v, (2) Copyrght (c) 214 IEEE. Personal use s permtted. For any other purposes, permsson must be obtaned from the IEEE by emalng pubs-permssons@eee.org.

4 Ths s the author s verson of an artcle that has been publshed n ths journal. Changes were made to ths verson by the publsher pror to publcaton. The fnal verson of record s avalable at 4 In ths paper, we assume that buyers do not collude wth each other and do not cheat about ther channel bundles, 2 whle leavng these problems to our future works. 2.2 Soluton Concepts We brefly revew the soluton concepts used n ths paper. A strong soluton concept from game theory s domnant strategy. Defnton 2 (Domnant Strategy [2] [21]): Strategy s s player s domnant strategy, f for any strategy s s and any other player s strategy profle s : u (s, s ) u (s, s ). Intutvely, a domnant strategy of a player s a strategy that maxmzes her utlty, regardless of what strategy profle the other players choose. The concept of domnant strategy s the bass of ncentve-compatblty, whch means that there s no ncentve for any player to le about her prvate nformaton, and thus revealng truthful nformaton s the domnant strategy for every player. An accompanyng concept s ndvdual-ratonalty, whch means that every player partcpatng n the game expects to gan no less utlty than stayng outsde. We now can ntroduce the defnton of Strategy-Proof Mechansm. Defnton 3 (Strategy-Proof Mechansm [22] [23]): A mechansm s strategy-proof when t satsfes both ncentve-compatblty and ndvdual-ratonalty. The objectve of ths work s to desgn strategyproof combnatoral aucton mechansms for heterogeneous channel redstrbuton. 3 COMBINATORIAL CHANNEL AUCTION Dfferent from exstng works on strategy-proof channel allocaton, we ntroduce a novel concept of vrtual channel to represent the conflcts of channel usage among the buyers. By ntroducng vrtual channels, we transform the problem of heterogeneous channel allocaton to a classc combnatoral aucton, whch s computatonally ntractable. Therefore, we propose strategy-proof and approxmately effcent combnatoral aucton mechansms for heterogeneous channel redstrbuton n the followng sectons. 2. When both valuatons and channel bundles are prvate nformaton, buyers wll have more power to manpulate the aucton market,.e., they can further mprove ther utltes by cheatng on channel bundles, and our aucton model falls nto the general combnatoral auctons wth mult-parameter doman, whch s stll an open problem n algorthmc mechansm desgn [13]. Papers [14], [15] have characterzed the truthfulness for mechansms n multple parameter doman, and some negatve results are demonstrated [16] [19]. 3.1 Vrtual Channel We ntroduce vrtual channel to capture the nterference among the buyers on dfferent channels. Specfcally, a vrtual channel vc k,j denotes that the buyer and the buyer j may cause nterference between each other on channel c k, and thus they cannot work on channel c k smultaneously. Snce vrtual channel vc k,j represents the exclusve usage of channel c k between the buyer and j, ts quantty s set to 1. When vrtual channel vc k,j s added to the requested bundle(s) that contans channel c k from the buyer and j, at most one of the requests contanng channel c k from the two buyers can be granted. Consequently, the exclusve usage of channel c k between the buyer and j s guaranteed. The heterogeneous channel redstrbuton problem can be converted to the problem of exclusve vrtual channels allocaton. We present the defnton of vrtual channel as follows. Defnton 4 (Vrtual Channel): There s a vrtual channel vc k,j, f the buyer and buyer j are wthn the nterference range of each other on channel c k. In most of exstng works on channel aucton, a sngle conflct graph s used to represent the nterference among buyers [6], [7]. However, n case of heterogeneous channels, each channel may have a dstnctve conflct graph. Let G k (O k, E k ) denote the conflct graph on channel c k, where O k N s the set of buyers who can access channel c k, and each edge (, j) E k represents the nterference between the buyer and j on channel c k. Let G {G k c k C} denote the set of conflct graphs. We also denote the maxmum degree of all the conflct graphs as δ. These conflct graphs can be bult by the auctoneer through some measurement methods, e.g., measurement calbrated method [24]. We note that the conflct graphs used n ths paper belong to bnary nterference model, such as the protocol model. The problem of channel redstrbuton under physcal nterference model s totally dfferent, and please refer to papers [25], [26] for more dscusson. Snce the conflct graph s commonly assumed to be avalable n wreless networks, we construct the vrtual channel from the conflct graph. The process of convertng the edges n the conflct graphs to vrtual channels wth unt quantty s shown by Algorthm 1. We create a vrtual channel vc k,j (Lne 4), f there s an edge between the buyer and j n conflct graph G k, and append vc k,j to the requested bundle(s) contanng channel c k from the buyer and j, whle remanng the correspondng bd(s) unchanged (Lnes 6-11). After addng vrtual channels nto the channel bundles, we remove the orgnal channels from all updated channel bundles (Lne 14). Let VC be the set of vrtual channels (Lne 5). Let S l be the lth updated channel bundle of buyer. Snce the maxmum degree of conflct graphs s δ and there are at most m tradng channels, we have S l δ m, N, 1 l φ. Copyrght (c) 214 IEEE. Personal use s permtted. For any other purposes, permsson must be obtaned from the IEEE by emalng pubs-permssons@eee.org.

5 Ths s the author s verson of an artcle that has been publshed n ths journal. Changes were made to ths verson by the publsher pror to publcaton. The fnal verson of record s avalable at 5 Algorthm 1: Vrtual Channel Generaton Input: A set of conflct graph G, a vector of channel requests R. Output: A set of vrtual channels VC, a vector of updated requests R. 1 VC ; R R; 2 foreach G k = (O k, E k ) G do 3 foreach (, j) E k do 4 Create vrtual channel vc k,j ; 5 VC VC { vc,j} k ; 6 foreach S l R s.t. c k S l ( S j t R j, c ) k S j t do 7 S l S l { } vc k,j ; 8 end 9 foreach S j l R j s.t. c k S j l ( S t R, c k S t) do 1 S j l { } S l j vc k,j ; 11 end 12 end 13 end 14 Remove the orgnal channels C from updated channel bundles R ; 15 return VC and R ; We use a smple example n Fgure 1 to explan the concept of vrtual channel. In Fgure 1, there are 2 channels and 4 buyers. The two conflct graphs show the nterference among buyers on two heterogeneous channels c 1 and c 2. The upper rght table shows the buyers channel demands. Both sngle-mnded and mult-mnded buyers exst n ths example. Here, the buyer 2 s a sngle-mnded buyer, and only bds a bundle of channels ({c 1, c 2 }) for 15; the buyer 3 s a mult-mnded buyer, and submts three requests,.e., ({c 1 }, {c 2 }, {c 1, c 2 }), and a unform valuaton 13. After runnng Algorthm 1, the updated request vectors wth vrtual channels are shown n the lower rght table. Let s see buyer 2 s updated request as an example. Snce both buyer 1 and buyer 2 bd for channel c 1 and they nterfere wth each other on ths channel, we add a vrtual channel vc 1 1,2 wth unt quantty to buyer 2 s requested bundle. 3.2 Problem Formulaton Gven the vrtual channel ntroduced n the last secton, we are ready to transform the problem of heterogenous channel allocaton to a classc combnatoral aucton. The outcome of the aucton s the set of wnnng buyers and ther assgned channel bundles. The goods n the combnatoral channel aucton are the vrtual channels. The quantty of each vrtual channel vc k,j VC s 1. Gven the vector of requests wth vrtual channels R and the bd vector B, the auctoneer determnes the wnners and whch channel bundles to grant. Let x ( ), S l = 1 denote that the Conflct Graph on c Conflct Graph on c 2 Buyers (Channel Requests, Bd) 1 ({ },7) 2 ({, },15) 3 (({ },{ }, {, }),13) 4 ({ },1) Buyers Vrtual Channels (Updated Channel Requests, Bd) 1 ({, },7) 2 ({,, },15) 3 (({ },{ },{, }),13) 4 ({ },1) Fg. 1. An example showng the generaton of vrtual channels. channel set S l s granted to the buyer ; otherwse, x ( ), S l =. The process of wnner determnaton can be modeled as a bnary program. The objectve s to maxmze the socal welfare. We use b, nstead of v, because the strategy-proof mechansms shown n later sectons wll guarantee that bddng truthfully s the domnant strategy of each buyer N. Objectve: Maxmze Subject to: N S l R,S l vc k φ l=1 φ x(, S l ) b N l=1 x (, S l ) 1 vck VC (3) x (, S l ) 1 N (4) x (, S l ) {, 1} N, 1 l φ (5) Here constrant (3) ndcates the quantty lmtaton of vrtual channel. As the orgnal channels have been removed from the updated channel bundles, we do not have quantty constrants on the orgnal channels. Constrant (4) ndcates that each buyer can wn at most one bundle of channels out of her submtted requests. Constrant (5) ndcates the bnary value of the auctoneer s decson of allocaton. If the optmal socal welfare can be acheved by solvng the above bnary program, then the celebrated VCG mechansm (named after Vckrey [27], Clark [28], and Groves [29]) can be appled to calculate clearng prces that can ensure the strategy-proofness of the aucton mechansm. Unfortunately, the above wnner determnaton problem can be proven to NP-hard by reducng from the exact cover problem [3] n polynomal tme. Consderng the computatonal ntractablty of the wnner determnaton problem, we present an alternatve soluton wth greedy channel allocaton to acheve approxmately effcent socal welfare n Copyrght (c) 214 IEEE. Personal use s permtted. For any other purposes, permsson must be obtaned from the IEEE by emalng pubs-permssons@eee.org.

6 Ths s the author s verson of an artcle that has been publshed n ths journal. Changes were made to ths verson by the publsher pror to publcaton. The fnal verson of record s avalable at 6 next secton. Furthermore, we ntegrate the greedy allocaton algorthm wth a novel prcng mechansm to provde a strategy-proof and approxmately effcent combnatoral aucton mechansm for heterogeneous channel redstrbuton. 4 EXCLUSIVE CHANNEL REDISTRIBUTION We consder the case of ndvsble channels, whch can only be allocated exclusvely to non-nterferng buyers, n ths secton. As shown n Secton 3.2, fndng the optmal aucton decson s computatonally ntractable. Furthermore, exstng works [17], [18] show that t s mpossble to desgn a strategy-proof approxmaton combnatoral aucton mechansm n the general case, even f the goods are not spatally reusable. We assume that buyers have unform valuaton on ther multple channel requests, and present SMASHER-AP, whch s a strategy-proof and approxmately effcent combnatoral aucton mechansm for heterogeneous channel redstrbuton. 4.1 Desgn of SMASHER-AP SMASHER-AP conssts of the followng three major components: vrtual channel generaton, wnner determnaton, and clearng prce calculaton. We brefly descrbe the desgn ratonale of SMASHER-AP. We frst generate vrtual channels to capture the nterference of channel usage among buyers, and transform the problem of channel redstrbuton nto the exclusve vrtual channel allocaton. After that, we propose a greedy channel allocaton algorthm to determne wnnng buyers, whch leads to a good approxmaton rato. Fnally, a clearng prce calculaton scheme based on crtcal vrtual bd s desgned to guarantee the economc propertes of SMASHER-AP Vrtual Channel Generaton The process of vrtual channel generaton s the same as that of Algorthm 1 shown n Secton 3.1, except that we add one more vrtual channel vc wth unt quantty to each requested bundle of buyer N. Vrtual channel vc s used to ensure that at most one of the requested bundles from the buyer can be granted. S l = S l {vc }, N, 1 l φ, where S l s updated bundle wth vrtual channels. The set of vrtual channels s also updated VC = VC {vc N} Wnner Determnaton Before presentng the approxmaton algorthm for wnner determnaton, we ntroduce vrtual bd. The Algorthm 2: Approxmaton Algorthm for Wnner Determnaton Input: Vector of updated channel requests R, vector of bds B. Output: A par of sets of wnnng buyers and allocated bundles of channels (W, S). 1 (W, S) (, ); V ; 2 foreach N do ( 3 b b / max 1 l φ S l ) ; 4 end 5 Sort b n non-ncreasng order: L 1 : b 1 b 2... b n ; 6 for = 1 to n do 7 Sort S l n non-decreasng order of bundle sze: L 2 : S 1 S 2 S... φ ; 8 for l = 1 to φ do 9 f S l V = then 1 V V S l; 11 (W, S) ( W {}, S { }) S l ; 12 break; 13 end 14 end 15 end 16 return (W, S); unform vrtual bd b over any of requested bundles from the buyer s defned as b b ( S ). (6) max l SMASHER-AP sorts all the buyers accordng to ther vrtual bds n non-ncreasng order: L 1 : b 1 b 2... b n. In case of a te, SMASHER-AP breaks the te followng a bd-ndependent rule, such as lexcographc order of buyers IDs or channel number. Followng the order n L 1, SMASHER-AP greedly grants the smallest channel bundle, n whch no vrtual channel has already been allocated, to each buyer. 3 Algorthm 2 shows the pseudo-code of above wnner determnaton process. In practce, the number of buyers n s much larger than Φ, thus the tme complexty of Algorthm 2 s O(n log n) Clearng Prce Calculaton The clearng prce s calculated based on crtcal vrtual bd. Defnton 5 (Crtcal Vrtual Bd): The crtcal vrtual bd cr() L 1 of buyer N s the mnmum vrtual 3. Actually, we allocate the orgnal channel bundle S l to the wnnng buyer, when she s granted the updated channel bundle S l n Algorthm 2. Copyrght (c) 214 IEEE. Personal use s permtted. For any other purposes, permsson must be obtaned from the IEEE by emalng pubs-permssons@eee.org.

7 Ths s the author s verson of an artcle that has been publshed n ths journal. Changes were made to ths verson by the publsher pror to publcaton. The fnal verson of record s avalable at 7 bd that the buyer must exceed to be allocated one of her channel bundles,.e., f the vrtual bd of the buyer s hgher than cr(), she wns the aucton; otherwse, she loses. We note that accordng to the defnton of crtcal vrtual bd, no matter whch channel bundles of buyer s granted n the aucton, the crtcal vrtual bd cr() s always the same. The crtcal vrtual bd of buyer N can be calculated by the followng ( procedure. Gven other buyers requests and bds R, B ), we greedly select vrtual bds by rerunnng Algorthm 2 untl none of buyer s requests can be satsfed. The threshold vrtual bd cr() we select fnally s regarded as the crtcal vrtual bd of the buyer. We now show the method of calculatng the clearng prce of the buyer by dstngushng two cases: 1) If the buyer loses n the aucton or cr() does not exst (denoted by cr() = ), then her clearng prce s. 2) If the buyer s granted channel bundle S l and there exsts a crtcal vrtual bd cr(), the clearng prce p of buyer s set to 4.2 Analyss p cr() max ( S l ). (7) We prove the strategy-proofness and analyze the approxmaton rato of SMASHER-AP n ths secton Strategy-Proofness Theorem 1: SMASHER-AP s a strategy-proof combnatoral aucton mechansm for heterogeneous ndvsble channel redstrbuton. Proof: We frst show that buyer N cannot obtan hgher utlty by bddng untruthfully. We dscuss the problem n the followng two cases: The buyer wns bundle Ŝ l and gets utlty u when bddng truthfully,.e., b = v. Let Ŝ t Ŝ l be the bundle won by the buyer, when she cheats the bd,.e., b v. The utlty of the buyer remans the same: u = v p ( ) S = v cr() max l = u. If the buyer loses the aucton when she cheats the bd, her utlty s, whch s not better than that ganed when bddng truthfully. The buyer loses n the aucton when bddng truthfully. Then, her utlty u =. If she stll loses when bddng untruthfully, her utlty cannot be changed. We consder the case, n whch she cheats the bd b v and wns a bundle Ŝ t. We denote vrtual bd b and b for channel bundle when the buyer bds truthfully and untruthfully, respectvely. Then, we have b cr() b, because otherwse, she stll cannot wn any bundle. Her utlty now becomes non-postve: Ŝ t u = v p = v cr() max v b max v ( S l ) ( S l ) = v ( S max l ) max = v v =. ( S l ) From the above analyss of two cases, we can see that the buyer cannot ncrease her utlty by bddng any other value than v, and thus bddng truthfully s a domnant strategy for each buyer. Therefore, SMASHER-AP satsfes ncentve compatblty. We now prove that SMASHER-AP also satsfes ndvdual ratonalty. On one hand, buyer s utlty s zero f she loses n the aucton. On the other hand, wnnng buyer gets utlty: u = v p ( ) S = v cr() max l = = v max ( S ) cr() max l ( ) S ( b cr()) max l, ( ) S l where b s the vrtual bd of buyer, Snce the buyer s a wnner, we have b cr(), and thus u. Buyer utlty s always non-negatve, whch s not worse than stayng outsde the aucton (.e., the utlty s ). Therefore, SMASHER-AP satsfes ndvdual ratonalty. Snce SMASHER-AP satsfes both ncentve compatblty and ndvdual ratonalty, accordng to Defnton 3, SMASHER-AP s a strategy-proof mechansm. Our clam holds. Snce our mechansm belongs to sngle-parameter mechansm, we can also obtan the property of strategy-proofness by usng Myerson s well known characterzaton [31] Approxmaton Rato We now present the approxmaton rato of SMASHER-AP. Theorem 2: The approxmaton rato of SMASHER- AP s O(δm), where δ s the maxmum degree of conflct graphs and m s the number of channels. Proof: Let (W OP T, S OP T ) be the optmal channel allocaton, and (W AP P, S AP P ) be the allocaton Copyrght (c) 214 IEEE. Personal use s permtted. For any other purposes, permsson must be obtaned from the IEEE by emalng pubs-permssons@eee.org.

8 Ths s the author s verson of an artcle that has been publshed n ths journal. Changes were made to ths verson by the publsher pror to publcaton. The fnal verson of record s avalable at 8 acheved by SMASHER-AP. The socal welfare of the optmal soluton and SMASHER-AP s W OP T v and W AP P v, respectvely. For each buyer W AP P, we defne W OP T { b j j W OP T ( S max l ) b ( S j max l ), 1 l φ j ( ) } S j OP T S AP P VC to represent the buyers n W OP T, whose bundles n S OP T cannot be granted n SMASHER-AP because of the exstence of. Snce every j W OP T appears after n the ordered lst L 1, we have ( S v max l ) j 1 l φ j v j ( max S l ). Summng over all j W OP T, we can get v v j ( S j W max l ) ( ) S max l j. 1 l φ j OP T j W OP T (8) Usng the Cauchy-Schwarz nequalty, we can bound j W OP T W OP T max 1 l φ j ( S l j j W OP T ) ( max S l ) j. (9) 1 l φ j By ntegratng nequatons (8) and (9), we get v W ( OP T j W max S l ) j OP T 1 l φ j v j ( S j W max l ). OP T (1) Snce (W OP T, S OP T ) s the optmal channel allocaton, the channel bundles allocated to any par of buyers, j W OP T cannot overlap on any vrtual channel: S OP T Sj OP T VC =. Every bundle allocated to j W OP T n the optmal allocaton ntersects wth S AP P at least one vrtual ( channel. Consequently, there are at most max S l ) buyers n W OP T Snce ( W OP T max S l ( ) W OP T max S l ( max S l j δm + 1, we also have 1 l φ j (11) ( max S l ) j δm(δm + 1). 1 l φ j (12) j W OP T By ntegratng nequatons (1), (11) and (12), we get v j δm(δm + 1)v. (13) j W OP T Snce W OP T = W AP P W OP T, we fnally get W OP T v W AP P j W OP T δm(δm + 1) v j W AP P v. (14) Therefore, the approxmaton rato of SMASHER- AP s O(δm). 5 CHANNEL REDISTRIBUTION WITH TIME SCHEDULING In ths secton, we consder the scenaro, n whch the clocks of buyers are synchronzed [32] and the rados on buyers devces can swtch among dfferent channels wthn very short tme [33]. Therefore, a channel can be shared by wreless devces n a paradgm of tme-dvson multplexng, whch s smlar to the tme-frequency model n [1]. We extend SMASHER- AP to the channel redstrbuton wth tme flexblty, and desgn SMASHER-GR, whch s a strategy-proof combnatoral aucton mechansm for heterogeneous channel redstrbuton, jontly consderng spatal and temporal channel reusablty. SMASHER-GR dvdes the tme nto a seres of slots wth a fxed length of duraton τ. A tme slot of a channel can be scheduled to multple buyers usng tme-dvson multplexng, by whch each buyer uses a certan fracton of the slot. In other words, the channel s consdered as a knd of dvsble goods. Durng a tme slot, buyer N has a requested data throughput Q, whch can be derved from her subscrbers QoS. We use vector Q to denote the data throughput of all the buyers Q (Q 1, Q 2,..., Q n ). Due to the heterogenety of channels, dfferent channel bundles may provde dfferent data rates to dfferent buyers. The channel requests from the buyer can now be expressed as ( (S 1 R, d 1 ) (, S 2, d 2 ) ),..., (S φ, d φ ), where d l denotes the data rate acheved by the buyer f she operates on the channel bundle S l. In ths model, a mult-mnded buyer can work on multple channel bundles sequentally to reach her data throughput requrement Q n each tme slot. Here, we assume that buyers do not cheat on data throughput and data rate, and we wll relax these assumptons n our future works Smlarly, f we relax ths assumpton, the aucton model wll fall nto the general combnatoral aucton wth multple parameter doman, and the determnstc strategy-proof combnatoral aucton mechansms are stll unknown for multple parameters scenaros. Copyrght (c) 214 IEEE. Personal use s permtted. For any other purposes, permsson must be obtaned from the IEEE by emalng pubs-permssons@eee.org.

9 Ths s the author s verson of an artcle that has been publshed n ths journal. Changes were made to ths verson by the publsher pror to publcaton. The fnal verson of record s avalable at 9 Let bnary varable x = 1 denote that buyer can obtan throughput Q by workng on requred channel bundles n tme duraton τ; otherwse, x =. We use h l and tl to denote the startng tme and lastng tme of channel bundle S l from buyer, respectvely. Buyer can get throughput d l tl f she operates on bundle S l for tl tme. We jontly consder allocaton and schedulng algorthm to determne whch channel bundles to grant, and when and how long wnnng buyers can access allocated channel bundles. Let R be the updated channel requests wth vrtual channels for buyer. We denote the updated request vector R of all the buyers as R = (R 1, R 2,..., R n). We formalze the process of channel allocaton and schedulng as the followng mxed-nteger nonlnear program (MINLP). Objectve: Maxmze (x b ) Subject to: N (h l h l ) (hl + x x t l h l ) >, S l, S l R, S l S l VC (15) (h l hl ) (h l + x x t l hl ) >, φ l=1 S l, S l R, S l S l VC (16) ( d l t l ) = x Q N (17) h l N, 1 l φ (18) h l + x t l τ N, 1 l φ (19) x {, 1} N (2) Same as before, the objectve s to maxmze the socal welfare. Constrants (15) and (16) ndcate that any two wnnng channel bundles contanng the same vrtual channel should be carefully scheduled to avod nterference n the tme dmenson. Specfcally, we use ntervals [h l, hl +tl ] and [hl, hl +tl ] to denote the workng tme duraton of two allocated channel bundles S l and Sl, respectvely. Constrants (15) and (16) guarantee that these two ntervals are nonoverlappng. Constrant (17) ndcates that the sum of throughput obtaned from multple channel bundles should be equal to the requested throughput of each wnnng buyer. Constrants (18) and (19) guarantee that the startng tme should be larger than, and the endng tme should be less than τ. Constrant (2) shows the bnary value of x. The above allocaton and schedulng problem s NP-hard [34], and thus s computatonal ntractable. So we follow the desgn ratonale of SMASHER-AP, and desgn SMASHER-GR, ncludng greedy channel allocaton, schedulng and prcng calculaton, to adapt to channel redstrbuton wth tme flexblty. Algorthm 3: Greedy Algorthm for Wnner Determnaton and Schedulng. Input: Vector of updated channel requests R, vector of data throughput Q, vector of bds B, < vrtual channel, tme > space F. Output: Sets of wnnng buyers W, allocated bundle of channels S, schedulng matrx (H, T). 1 W ; S ; (H, T) ( n,φ, n,φ ); 2 foreach N do ( ) 3 b b / max 1 l φ S l Q /d l ; 4 end 5 Sort b n non-ncreasng order: L 3 : b 1 b 2... b n ; 6 for = 1 to n do 7 f Satsfable(Q, R, F) s true then 8 W W {}; 9 l 1; Q Q ; 1 whle Q > do 11 (Schedulable, H(, l), T(, l)) Schedulng(S l, F); 12 f Schedulable s true then 13 S S { } S l ; 14 Q ( Q dl T(, l)) ; 15 end 16 l l + 1; 17 end 18 end 19 end 2 return W, S, (H, T); 5.1 Desgn of SMASHER-GR The desgn ratonale of SMASHER-GR s brefly descrbed as follows. SMASHER-GR frst generates vrtual channels accordng to the conflct graphs to represent the spatal nterference of heterogeneous channels. Then t greedly selects the avalable and non-overlappng tme ntervals for wnnng buyers, consderng the channel nterference n the tme dmenson. Fnally, SMASHER-GR apples a prcng mechansm to guarantee economc propertes. SMASHER-GR conssts of three parts: vrtual channel generaton, wnner determnaton and clearng prce calculaton Vrtual Channel Generaton SMASHER-GR generates the vrtual channels by runnng Algorthm 1, descrbed n Secton 3.1. Dfferent from SMASHER-AP, SMASHER-GR does not generate vrtual channel vc for each buyer N, because buyer may be allocated multple channel bundles to acheve her throughput durng a tme slot. Copyrght (c) 214 IEEE. Personal use s permtted. For any other purposes, permsson must be obtaned from the IEEE by emalng pubs-permssons@eee.org.

10 Ths s the author s verson of an artcle that has been publshed n ths journal. Changes were made to ths verson by the publsher pror to publcaton. The fnal verson of record s avalable at Wnner Determnaton The wnner determnaton algorthm conssts of two parts,.e., channel allocaton and channel bundle schedulng. Before presentng the algorthm, we redefne the vrtual bd, consderng the dfferent data rates of channel bundles. The vrtual bd of the buyer s b b ( S ). (21) max l Q /d l Intutvely, a channel bundle wth larger sze ( S l ) and longer occupaton tme (Q /d l ), has lower prorty to be granted, because t may lead to more spatal and temporal conflcts wth other requests. SMASHER-GR sorts all the buyers by ther vrtual bds n non-ncreasng order: L 3 : b 1 b 2... b n, and breaks the te by usng a bd-ndependent rule. We use F to denote the < vrtual channel, tme > dmensonal space, whch records the avalable workng tme ntervals of vrtual channels. Followng the order n lst L 3, SMASHER-GR frst checks whether the buyer can fulfll her clamed throughput Q by beng allocated the remanng tme of vrtual channels n F. If the buyer s a wnner, SMASHER-GR then greedly selects the avalable and non-overlappng tme ntervals for channel bundle S l, and packs these ntervals nto space F. SMASHER-GR teratvely allocates tme ntervals to each bundle S l of buyer untl achevng her requested data throughput Q. Algorthm 3 shows the pseudo-code of greedy algorthm for wnner determnaton, ncludng channel allocaton and schedulng. The functon Satsfable(Q, R, F) checks whether the buyer can satsfy her data throughput Q by beng allocated the non-overlappng tme ntervals n current space F. Functon Schedulng(S l, F) returns trple tuple (Schedulable, H(, l), T(, l)), n whch the bnary varable Schedulable ndcates whether there exst non-overlappng tme ntervals n F for bundle S l. If the varable Schedulable s true, Schedulng also returns the startng tme H(, l) and lastng tme T(, l) for bundle S l.5 Functon Schedulng fnally packs the tme nterval [H(, l), H(, l) + T(, l)] nto space F. The process of schedulng can be done lnearly, then the complexty of Algorthm 3 s O(n log n) Clearng Prce Calculaton The prcng mechansm s also based on crtcal vrtual bd. Defnton 6 (Crtcal Vrtual Bd): The crtcal vrtual bd cr() L 3 of wnnng buyer W s the mnmum vrtual bd that buyer must exceed n order to fulfl 5. Smlar to Algorthm 2, the updated channel bundle S l s schedulable means that the correspondng orgnal channel bundle S l s also schedulable n Algorthm 3. her requested data throughput,.e., f the vrtual bd of the buyer s hgher than cr(), her requested data throughput would be satsfed; otherwse, she would lose the aucton. The crtcal vrtual bd cr() of buyer can be obtaned by the followng ( steps. Gven the other buyers channel demands R, Q, B ), we greedly select the vrtual bd from L 3 by runnng Algorthm 3 untl buyer s requested data throughput cannot be fulflled. The last vrtual bd we selected s consdered as the crtcal vrtual bd of buyer. Now we can calculate the clearng prce of buyer by dstngushng the followng two cases: If the buyer loses n the aucton or there exst no crtcal vrtual bds for her, then her clearng prce s. If the buyer s a wnner and her crtcal bd s cr(), we can calculate her clearng prce p : 5.2 Analyss p cr() max ( S l Q /d l ). (22) By combnng the channel allocaton, schedulng and prcng mechansms together, SMASHER-GR acheves the followng property. Theorem 3: SMASHER-GR s a jont allocaton and schedulng strategy-proof combnatoral aucton mechansm for heterogeneous channel redstrbuton wth tme flexblty. Proof: We frst prove that buyer N cannot ncrease her utlty by bddng untruthfully,.e., reportng true valuaton s a domnant strategy for the buyer. We dstngush two cases: Buyer acheves her throughput Q and gets her utlty u when bddng truthfully,.e., b = v. She gans channel bundle set S S and schedulng matrx (H, T ) (H, T). Buyer wns another channel bundle set S S, S S and schedulng matrx (H, T ) (H, T) when she reports another bd b v. Her utlty s not changed: u = v p ( ) S = v cr() max l Q /d l = u. If buyer loses n the aucton when she cheats the bd, her utlty s, whch s no more than when buyer bds truthfully. We consder the other case, n whch the buyer cannot fulfll her throughput Q when bddng truthfully. Then her utlty u =. The only way to mprove her utlty s to cheat the bd b v and become a wnner. We denote the buyer s Copyrght (c) 214 IEEE. Personal use s permtted. For any other purposes, permsson must be obtaned from the IEEE by emalng pubs-permssons@eee.org.

11 Ths s the author s verson of an artcle that has been publshed n ths journal. Changes were made to ths verson by the publsher pror to publcaton. The fnal verson of record s avalable at 11 wnnng channel bundle set S S and the correspondng schedulng matrx (H, T ) (H, T) when she cheats on bd. Let b and b denote the vrtual bd of buyer when she bds truthfully and untruthfully, respectvely. Then, we have b cr() b, because otherwse, she stll cannot fulfl her throughput. Her utlty now becomes non-postve: u = v p = v cr() max ( ) S l Q /d l ) ( S l v b max Q /d l ( S ) v max l Q /d l = v ( S ) max l Q /d l = v v =. Therefore, bddng the true valuaton s a domnant strategy for each buyer. We can conclude that SMASHER-GR satsfes ncentve compatblty. We now prove that SMASHER-GR also satsfes ndvdual ratonalty. On one hand, the buyers losng n the aucton get zero utlty. On the other hand, the wnnng buyer s utlty: u = v p ( ) S = v cr() max l Q /d l = = v ( ) cr() max S l Q /d l ( ) S max l Q /d l ( ) S ( b cr()) max l Q /d l where b s the vrtual bd of buyer. From the defnton of crtcal vrtual bd, we have b cr() for wnnng buyer. Then, wnnng buyer gets non-negatve utlty. Buyer utlty s always non-negatve, whch s not worse than stayng outsde the aucton (.e., utlty s equal to zero). Therefore, we can conclude that SMASHER-GR satsfes ndvdual ratonalty. Snce SMASHER-GR satsfes both ncentve compatblty and ndvdual ratonalty, SMASHER-GR s a strategy-proof mechansm, and then our clam holds. Smlar to SMASHER-AP, the property of strategy-proofness can also be analyzed by applyng Myerson s well known characterzaton [31]. We note that the performance of SMASHER-GR can be arbtrarly bad n some specal cases. But the evaluaton results show that SMASHER-GR performs qute well n most of cases. 6 EVALUATION RESULTS In ths secton, we show our evaluaton results. 6.1 Methodology We mplement SMASHER-AP and compare ts performance wth TAHES [9] and CRWDP [1]. We also show the performance of SMASHER-GR. Buyers are randomly dstrbuted n a terran area of 2 meters 2 meters. The number of buyers vares from 2 to 4 wth ncrement of 2. The number of leasng channels can be one of the three values: 6, 12 and 24. The heterogeneous channels have dfferent nterference ranges, spannng from 25 meters to 45 meters. We allow buyers to be equpped wth dfferent number of rados n our auctons, but lmt the maxmum sze of requested channel bundle to 3. We assume that the buyers valuatons are randomly dstrbuted over (, 1]. We consder the case of sngle-mnded buyers (.e., Φ = 1), and the case of mult-mnded buyers who can submt up to 3 channel bundles (.e., Φ = 3). In SMASHER-GR, we normalze the length of tme slot to 1. We assume the throughput of buyers and data rate of channel bundle are unformly dstrbuted n the nterval (, 1]. All the results of performance are averaged over 2 runs. 6 Metrcs: We evaluate three metrcs: Socal Welfare: Socal welfare s the sum of wnnng buyers valuatons on ther allocated bundles of channels. Satsfactory Rato: Satsfactory rato s the percentage of buyers who obtan one of ther demanded channel bundles n SMASHER-AP or acheve ther throughput n SMASHER-GR. Channel Utlzaton: Channel utlzaton s the average number of rados allocated to each channel. 6.2 Performance of SMASHER-AP We compare the performance of SMASHER-AP wth two strategy-proof heterogeneous channel aucton mechansms, TAHES and CRWDP. We evaluate the outcome of SMASHER-AP when the buyers are sngle-mnded (Φ = 1) and mult-mnded (Φ = 3). We also show the optmal results wth tolerance 1 4, denoted by IP-OPT, computed by solvng the bnary nteger program n Secton 3.2, as references of upper bound. Fgure 2 shows the evaluaton results when there are 12 channels and dfferent number of buyers. We can see that SMASHER-AP always outperforms the other two aucton mechansms, and ts performance approaches the optmum, especally when Φ = 1. When the number of nodes s smaller than 6, TAHES 6. All parameters can be dfferent from the ones used here. However, the evaluaton results of usng dfferent parameters are dentcal. Therefore, we only show the results for these parameters n ths paper. Copyrght (c) 214 IEEE. Personal use s permtted. For any other purposes, permsson must be obtaned from the IEEE by emalng pubs-permssons@eee.org.

12 Ths s the author s verson of an artcle that has been publshed n ths journal. Changes were made to ths verson by the publsher pror to publcaton. The fnal verson of record s avalable at 12 Socal Welfare IP-OPT Φ=3 IP-OPT Φ=1 SMASHER-AP Φ=3 SMASHER-AP Φ=1 TAHES CRWDP Number of Buyers (a) Socal Welfare Satsfacton Rato IP-OPT Φ=3 IP-OPT Φ=1 SMASHER-AP Φ=3 SMASHER-AP Φ=1 TAHES CRWDP Number of Buyers (b) Satsfacton Rato Chanel Utlzaton IP-OPT Φ=3 IP-OPT Φ=1 SMASHER-AP Φ=3 SMASHER-AP Φ=1 TAHES CRWDP Number of Buyers (c) Channel Utlzaton Fg. 2. Performance of SMASHER-AP, TAHES, CRWDP and IP-OPT, when there are 12 channels. Socal Welfare IP-OPT Φ=3 IP-OPT Φ=1 SMASHER-AP Φ=3 SMASHER-AP Φ=1 TAHES CRWDP Number of Channels Satsfacton Rato IP-OPT Φ=3 IP-OPT Φ=1 SMASHER-AP Φ=3 SMASHER-AP Φ=1 TAHES CRWDP Number of Channels Channel Utlzaton IP-OPT Φ=3 IP-OPT Φ=1 SMASHER-AP Φ=3 SMASHER-AP Φ=1 TAHES CRWDP Number of Channels (a) Socal Welfare (b) Satsfacton Rato (c) Channel Utlzaton Fg. 3. Performance of SMASHER-AP, TAHES, CRWDP and IP-OPT, when there are 2 buyers. Socal Welfare SMASHER-GR Φ=3 m=24 SMASHER-GR Φ=1 m=24 SMASHER-GR Φ=3 m=12 SMASHER-GR Φ=1 m= Number of Buyers (a) Socal Welfare Satsfacton Rato SMASHER-GR Φ=3 m=24.2 SMASHER-GR Φ=1 m=24 SMASHER-GR Φ=3 m=12 SMASHER-GR Φ=1 m= Number of Buyers (b) Satsfacton Rato Chanel Utlzaton SMASHER-GR Φ=3 m=24 SMASHER-GR Φ=1 m=24 SMASHER-GR Φ=3 m=12 SMASHER-GR Φ=1 m= Number of Buyers (c) Channel Utlzaton Fg. 4. Performance of SMASHER-GR n dfferent network scenaros. cannot form suffcent buyer groups wth a large number of bds, and thus does not perform well n ths case. When the number of buyer s larger than 6, CRWDP s performance s not good because CRWDP does not consder channel spatal reusablty (.e., the channel utlzaton of CRWDP s equal to 1 n all cases). Fgure 2 also shows that when the buyer number ncreases, the socal welfare and channel utlzaton ncrease, but the satsfacton rato decreases. On one hand, the larger number of buyers leads to more ntense competton on lmted channels, thus decreasng the satsfacton rato. On the other hand, SMASHER-AP can allocate channels more effcently among more buyers, hence the socal welfare and channel utlzaton ncrease. Fgure 3 shows the evaluaton results when there are 2 buyers and the number of channels s 6, 12, 24. Agan, SMASHER-AP always acheves better performance than TAHES and CRWDP, whenever Φ = 1 or Φ = 3. Fgure 3 also shows that when the number of channels ncreases, the socal welfare and satsfacton rato ncrease and the channel utlzaton decreases. The reason s that larger supply of leasng channels leads to more trades n the aucton, thus the socal welfare and satsfacton rato ncrease when there exsts a fxed number of buyers. The channel utlzaton decreases because buyers rados can be allocated to more channels when the number of channels ncreases. From Fgure 2 and Fgure 3, we can see that SMASHER-AP sacrfces lmted system performance to acheve economc robustness. Although IP-OPT acheves near optmal socal welfare, we cannot apply t to channel redstrbuton problem, because IP-OPT has not any guarantee on economc propertes. We observe that SMASHER-AP wth mult-mnded buyers (.e., Φ = 3) always performs better than SMASHER- AP wth sngle-mnded buyers (.e., Φ = 1), on all the three metrcs. Ths s because mult-mnded buyers have hgher chance to obtan channel bundles than Copyrght (c) 214 IEEE. Personal use s permtted. For any other purposes, permsson must be obtaned from the IEEE by emalng pubs-permssons@eee.org.

13 Ths s the author s verson of an artcle that has been publshed n ths journal. Changes were made to ths verson by the publsher pror to publcaton. The fnal verson of record s avalable at 13 sngle-mnded buyers. Ths leads to more trades n the aucton. Therefore, allowng buyers to submt multple spectrum requests ndeed mproves the aucton performance. 6.3 Performance of SMASHER-GR Solvng the mxed nteger nonlnear program, shown n Secton 5, s computatonal ntractable even n small scale network scenaros. CRWDP only allocates tme-frequency blocks, wthout consderng the channel schedulng problem whle TAHES doest not take tme multplexng of channels nto account, regardng channels as ndvsble goods. Therefore, we only show the performance results of SMASHER-GR n Fgure 4. Smlarly, when the buyer number ncreases, the socal welfare and channel utlzaton ncrease and the satsfacton rato decreases. We can also observe that when the number of leasng channels ncreases from 12 to 24, the socal welfare and satsfacton rato ncrease whle the channel utlzaton decreases. The reasons of the relaton between system performance and the number of buyers or the number of channels s smlar to those analyzed n SMASHER-AP. Fgure 4 also shows that SMASHER-GR wth mult-mnded buyers (.e., Φ = 3) always outperforms SMASHER- GR wth sngle-mnded buyers (.e., Φ = 1), on all the three metrcs. Ths result verfes that dverse spectrum requests lead to hgher aucton performance. We can draw a concluson that bd dversty s an effectve strategy to mprove the performance of channel redstrbuton system. Compared wth SMASHER-AP, SMASHER-GR performs better on all the three metrcs. Ths s because channels can be shared among buyers n a paradgm of tme-dvson multplexng n SMASHER-GR. SMASHER-GR s an effectve aucton mechansm to channel redstrbuton, consderng both spatal and temporal channel reusablty. 7 RELATED WORKS In ths secton, we brefly revew related works on channel aucton and aucton mechansm desgn. 7.1 Channel Allocaton Wth Selfsh Partcpants A number of works model the problem of channel allocaton by game theory. Felegyhaz et al. [35] studed Nash Equlbra n a statc mult-rado mult-channel allocaton game. Later, Wu et al. [36] desgned an ncentve scheme for the mult-rado mult-channel allocaton game, makng the system converge to a much stronger equlbrum state. Gao et al. studed the mult-rado channel allocaton problem n multhop wreless networks, and proposed the mn-max coalton-proof Nash Equlbrum channel allocaton scheme n the cooperatve game [37]. Yang et al. consdered the channel allocaton n mult-rado multchannel wreless networks wth multple collson domans [38]. Chen et al. proposed dstrbuted spectrum sharng schemes to coverage Nash equlbrum n spectrum access game [39], [4]. In cogntve rado networks, Kasbekar et al. analyzed spectrum prcng game and computed Nash Equlbrum n dfferent scenaros [41] [43]. Byun et al., computed a market equlbrum, whch s defned n context of extended Fsher model, for spectrum sharng n cogntve rado networks [44]. The paper [45] proposed a Qualty of Experence drven channel allocaton scheme for secondary users n cogntve rado networks. Resource allocaton among selfsh partcpants has been studed n dfferent network scenaros, such as wreless mesh networks [46], OFDMA femtocell networks [47], and LTE networks [48]. The most closely related works are VERITAS [6], TRUST [7], and SMALL [8], all of whch are auctonbased strategy-proof channel allocaton mechansms. VERITAS and SMALL are sngle-sded auctons both supportng multple channel requests. In contrast, TRUST elegantly extends double aucton to consder both channel sellers and buyers ncentves. Recently, TAHES [9] was proposed to solve the problem of heterogeneous channel allocaton. Besdes, there are some other related works on channel aucton, such as onlne channel auctons [49] [51], colluson-resstant channel aucton [52], revenue generaton for spectrum aucton [53], and approxmate algorthms for dfferent models of nterference and dfferent formats of valuatons [25], [26]. 7.2 Mechansm Desgn A large number of works on combnatoral auctons have been proposed durng the last decades. Dobznsk [17], Buchfuhrer et al. [54], and Papadmtrou et al. [16] proved that gettng optmal socal welfare and ensurng strategy-proofness cannot be acheved smultaneously n general combnatoral auctons. Lehmann et al. [18] even asserted that there s no payment scheme to make greedy allocaton algorthm strategy-proof n general combnatoral auctons wth mult-mned buyers. Consderng the ntractablty of combnatoral aucton, a number of strategy-proof aucton mechansms wth well bounded approxmaton ratos were proposed [55] [58]. In [1], the author modeled the tme-frequency allocaton problem as a combnatoral aucton wth sngle-mnded buyers, and proposed a greedy allocaton algorthm to acheve approxmately effcent socal welfare n a sngle collson doman. However, none of the above combnatoral aucton consders the spectrum spatal reusablty. Aucton mechansms have been proposed to address dfferent knds of resource allocaton problems, such as resource management n cloud computng [59], [6], sensng tasks dstrbuton n moble Copyrght (c) 214 IEEE. Personal use s permtted. For any other purposes, permsson must be obtaned from the IEEE by emalng pubs-permssons@eee.org.

14 Ths s the author s verson of an artcle that has been publshed n ths journal. Changes were made to ths verson by the publsher pror to publcaton. The fnal verson of record s avalable at 14 crowdsensng [61], [62], and cooperaton dynamcs on collaboratve socal networks [63]. Schedulng theory has receved a growng nterest snce ts orgns, and there are varous types of schedulng problems [64]. Recently, some works have studed the schedulng problems n the mechansm desgn context. Nsan and Ronen [65] were the frst to consder makespan-mnmzaton on unrelated machnes. Later, Archer and Tardos [66] consdered the related-machne problem and gave a 3-approxmaton truthful-n-expectaton mechansm. However, the objectve of these works s to mnmze the makespan, whle our objectve s to maxmze socal welfare and the schedulng model we consdered s more complcated. 8 CONCLUSION AND FUTURE WORKS In ths paper, we have made an n-depth study on channel redstrbuton problem by jontly consderng the fve desgn challenges. We have presented two closely related strategy-proof combnatoral aucton mechansms for dynamc heterogeneous channel redstrbuton, namely SMASHER-AP and SMASHER- GR. SMASHER-AP s a combnatoral aucton mechansm for ndvsble channel redstrbuton, achevng strategy-proofness and approxmately effcent socal welfare. SMASHER-GR s a strategy-proof combnatoral aucton for jont channel allocaton and schedulng for the scenaros, n whch channels can be shared n a tme-multplexng way. We have also evaluated the performance of our desgns. The smulaton results have shown that our desgns acheve good performance, n terms of socal welfare, buyer satsfacton rato, and channel utlzaton. As for future works, we are nterested n desgnng combnatoral channel aucton mechansms that can prevent colluson among multple buyers. Desgnng a strategy-proof combnatoral channel aucton mechansm to prevent buyers from cheatng on channel bundle s also an nterestng problem. 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For any other purposes, permsson must be obtaned from the IEEE by emalng pubs-permssons@eee.org.

16 Ths s the author s verson of an artcle that has been publshed n ths journal. Changes were made to ths verson by the publsher pror to publcaton. The fnal verson of record s avalable at 16 on Foudatons of Computer Scence (FOCS), Las Vegas, Nevada, Oct. 21. Zhenzhe Zheng s a master student from the Department of Computer Scence and Engneerng at Shangha Jao Tong Unversty, P. R. Chna. Hs research nterests le n algorthmc game theory, wreless networkng and moble computng. He s a student member of ACM, CCF and IEEE. Fan Wu s an assocate professor n the Department of Computer Scence and Engneerng at Shangha Jao Tong Unversty, P. R. Chna. He receved hs B.S. n Computer Scence from Nanjng Unversty n 24, and Ph.D. n Computer Scence and Engneerng from the State Unversty of New York at Buffalo n 29. He has vsted the Unversty of Illnos at Urbana-Champagn (UIUC) as a Post Doc Research Assocate. Hs current research nterests nclude wreless networkng, algorthmc mechansm desgn, and prvacy preservaton. He receved Excellent Young Scholar award of Shangha Jao Tong Unversty n 211, and Pujang Scholar award n 212. He s a member of ACM, CCF, and IEEE. For more nformaton, please vst Guha Chen earned hs B.S. degree from Nanjng Unversty n 1984, M.E. degree from Southeast Unversty n 1987, and Ph.D. degree from the Unversty of Hong Kong n He s a dstngushed professor of Shangha Jaotong Unversty, Chna. He had been nvted as a vstng professor by many unverstes ncludng Kyushu Insttute of Technology, Japan n 1998, Unversty of Queensland, Australa n 2, and Wayne State Unversty, USA durng September 21 to August 23. He has a wde range of research nterests wth focus on sensor networks, peer-to-peer computng, hgh-performance computer archtecture and combnatorcs. He has publshed more than 2 peer-revewed papers, and more than 12 of them are n well-archved nternatonal journals such as IEEE Transactons on Parallel and Dstrbuted Systems, Journal of Parallel and Dstrbuted Computng, Wreless Networks, The Computer Journal, Internatonal Journal of Foundatons of Computer Scence, and Performance Evaluaton, and also n well-known conference proceedngs such as HPCA, MOBIHOC, INFOCOM, ICNP, ICPP, IPDPS and ICDCS. Copyrght (c) 214 IEEE. Personal use s permtted. For any other purposes, permsson must be obtaned from the IEEE by emalng pubs-permssons@eee.org.