Truthful Spectrum Auction Design for Secondary Networks

Transcription

1 Truthful Spectrum Aucton Desgn for Secondary Networks Yuefe Zhu, Baochun L Department of Electrcal and Computer Engneerng Unversty of Toronto Zongpeng L Department of Computer Scence Unversty of Calgary Abstract Opportunstc wreless channel access by nonlcensed users has emerged as a promsng soluton for addressng the bandwdth scarcty challenge. Auctons represent a natural mechansm for allocatng the spectrum, generatng an economc ncentve for the lcensed user to relnqush channels. A severe lmtaton of exstng spectrum aucton desgns les n the oversmplfyng assumpton that every non-lcensed user s a snglenode or sngle-lnk secondary user. Whle such an assumpton makes the aucton desgn easer, t does not capture practcal scenaros where users have multhop routng deman. For the frst tme n the lterature, we propose to model non-lcensed users as secondary networks (SNs), each of whch comprses of a multhop network wth end-to-end routng deman. We am to desgn truthful auctons for allocatng channels to SNs n a coordnated fashon that maxmzes socal welfare of the system. We use smple examples to show that such auctons among SNs dffer drastcally from smple auctons among sngle-hop users, and prevous solutons suffer severely from local, per-hop decson makng. We frst desgn a smple, heurstc aucton that takes nter-sn nterference nto consderaton, and s truthful. We then desgn a randomzed aucton based on prmal-dual lnear optmzaton, wth a proven performance guarantee for approachng optmal socal welfare. A key technque n our soluton s to decompose a lnear program (LP) soluton for channel assgnment nto a set of nteger program (IP) solutons, then applyng a par of talored prmal and dual LPs for computng probabltes of choosng each IP soluton. We prove the truthfulness and performance bound of our soluton, and verfy ts effectveness through smulaton studes. I. INTRODUCTION Recent years have wtnessed substantal growth n wreless technology and applcatons, whch rely crucally on the avalablty of bandwdth spectrum. Tradtonal spectrum allocaton s statc, and s prone to neffcent spectrum utlzaton n both temporal and spatal domans: large spectrum chunks reman dlng whle new users are unable to access them. Such an observaton has prompted research nterest n desgnng a secondary spectrum market, where new users can access a lcensed channel when not n use by ts owner, wth approprate remuneraton transferred to the latter. In a secondary spectrum market, a spectrum owner or prmary user (PU) leases ts dle spectrum chunks (channels) to secondary users (SUs) through auctons [], [2]. SUs submt b for channels, and pay the PU a prce to access a channel f ther b are successful. A natural goal of spectrum aucton desgn s truthfulness, under whch an SU s best strategy s to bd ts true valuaton of a channel, wth no ncentve to le. A truthful aucton smplfes decson makng at SUs, and lays a foundaton for good decson makng at the PU. Another mportant goal n spectrum aucton desgn s socal welfare maxmzaton,.e., maxmzng the aggregated happness of everyone n the system. Such an aucton ten to allocate channels to SUs who value them the most. A unque feature of spectrum aucton desgn s the need of approprate consderaton for wreless nterference and spatal reuse of channels. A channel can be allocated to multple SUs provded that they are far apart, wth no mutual nterference. Optmal channel assgnment for socal welfare maxmzaton s equvalent to the graph colourng problem, and s NP-hard [3], even assumng truthful b are gven for free. Exstng works on spectrum auctons mostly focus on resolvng such a challenge (e.g., [4], [5]) whle assumng the smplest model of a SU: a sngle node, or a sngle lnk, smlar to a sngle hop transmsson n cellular networks [2], [4], [5]. SN 3 SN3 2 SN2 Fg. : A secondary spectrum market wth 3 SNs and 2 channels. After extensve research durng the past fve years, aucton desgn for sngle-hop users, each requestng a sngle channel, has been relatvely well understood. However, a practcal SU may very well comprse of multple nodes formng a multhop network, whch we refer to as a secondary network (SN). These nclude scenaros such as users wth multhop access to base statons, or users wth ther own moble ad hoc networks. SNs requre coordnated end-to-end channel assgnment, and n general beneft from mult-channel dversty along ts path. The SN model subsumes the SU model as the smplest specal case. Fg. depcts three co-located SNs, SN, SN2 and SN3, whch have nterference wth one another, because ther network regons overlap. The prmary network (PN) has two channels, Ch and Ch2, whch have been allocated to SN and SN2, respectvely. Now SN3 wshes to route along a two-hop path 2 3. Under exstng sngle-channel

2 auctons for SUs, SN3 cannot obtan a channel, because each channel nterferes wth ether SN or SN2. However, a soluton exsts by relaxng the one channel per user assumpton, and assgnng Ch to lnk 2 and Ch2 to the lnk 2 3. In general, takng multchannel, multhop transmssons by SNs nto consderaton can apparently mprove channel utlzaton and socal welfare. Note here that the model n whch an SN b for multple channels s napplcable, because due to the unawareness of other SNs nformaton, an SN cannot know the number of channels to bd for, to form a feasble path. Desgnng truthful auctons for SNs s an nterestng problem, but by no means an easy one. We note that t s hard for an SN to decde by tself an optmal or good path to bd for. Such decson makng requres global nformaton on other SNs as well, and s naturally best made by the auctoneer,.e., the PN. Consequently, a bd from an SN ncludes just a prce t wshes to pay, wth two nodes t wshes to connect usng a path. Furthermore, SNs now nterfere wth each other n a more complex manner. Not only that they transmt along multhop paths, but each path can be assgned wth dstnct channels at dfferent lnks. The PN, after recevng b, nee to make judcous jont routng and channel assgnment decsons. In ths work, we frst desgn a smple heurstc aucton for spectrum allocaton to SNs, whch guarantees both truthfulness and nterference-free channel allocaton, provdng wnnng SNs wth end-to-end multhop paths, wth a channel assgned to each hop. The heurstc aucton enables mult-channel assgnment along a path, thereby reducng the possblty that a path s blocked due to nterference. To acheve truthfulness, we employ a greedy, monotonc allocaton rule and desgn an accompanyng payment scheme, by referrng to Myerson s characterzaton of truthful auctons [6]. The heurstc aucton provdes no hard guarantee on socal welfare. We next desgn a randomzed aucton, whch s truthful n expectaton, and s provably approxmate optmal n socal welfare. We note that absolute optmal socal welfare s mpossble, snce the jont routng-channel assgnment problem s already NP-hard wth truthful b gven for free. We relax an nteger program (IP) formulaton to the socal welfare maxmzaton problem nto a lnear program (LP), and prove an upper-bound on the ntegralty gap. We then employ the decomposton technque due to Lav and Swamy [7] to decompose an LP soluton nto a set of feasble IP solutons (allocatons). A par of prmal-dual LPs are formulated, for computng a probablty dstrbuton over the allocaton set. Based on the set and the probabltes, an approxmaton algorthm s fnally desgned, for computng a feasble soluton to the orgnal IP wth a provable performance guarantee. Such a soluton assgns channels to paths constructed for wnnng SNs. We then apply the classc VCG [8] [] payment scheme, to conclude the desgn of a truthful aucton that approxmately maxmzes socal welfare for spectrum allocaton to SNs. The remander of the paper s organzed as follows. We dscuss related work n Sec. II, and present prelmnares n Sec. III. A heurstc truthful aucton s desgned n Sec. IV. In Sec. V, we propose and analyze a randomzed aucton wth performance bound. Smulaton studes are presented n Sec. VI. Sec. VII concludes the paper. II. RELATED WORK Auctons serve as an effcent mechansm for dstrbutng scarce resources to competng partcpants n a market. To smplfy the strategcal behavour of agents and hence encourage partcpaton, truthfulness s desred. A celebrated work s the VCG mechansm due to Vckrey [], Clarke [8], and Groves [9]. However, the VCG mechansm s only sutable when optmal solutons are computatonally feasble, and s not drectly applcable for secondary spectrum auctons, because nterference-free channel allocaton s NP-Hard. Allocatng spectrum usng auctons has receved consderable research attenton recently. Early solutons nclude auctons that allocate power [] and allocate a channel to each wnnng user [2]. These auctons are unfortunately not truthful. VERITAS [] s the frst truthful spectrum aucton based on the monotonc allocaton rule. Zhou and Zheng propose TRUST [3], whch s a double aucton wth multple sellers (lcensed users). Ja et al. [4] desgn a spectrum aucton mechansm that not only encourages truthful behavour but also computes approxmately maxmum revenue, whch s an alternatve goal to maxmum socal welfare. For spectrum auctons that take nterference among secondary users nto consderaton, Wu et al. [2] develop a semdefnte programmng based mechansm, whch s truthful and resstant to bdder colluson. Gopnathan [5] et al. propose auctons that ncorporate farness consderatons nto channel allocaton. Ther goal s to maxmze socal welfare, whle ensurng a noton of farness among bdders when the aucton s repeatedly held. A truthful and scalable spectrum aucton enablng both sharng and exclusve access s proposed by Kash et al. [4]. Ths aucton handles heterogeneous agent types wth dfferent transmsson powers and spectrum nee. We note that, all the works mentoned above focus on sngle-hop users bddng for a sngle channel only. Our work essentally generalzes the problem to mult-hop users, who may enjoy mult-channel paths. III. PRELIMINARIES In ths secton, we frst ntroduce some background n truthful aucton desgn n Sec. III-A, then descrbe our system model n Sec. III-B. A. Truthful Aucton Desgn Aucton theory s a branch of economcs that studes how people act n an aucton and analyzes the propertes of aucton markets. We frst ntroduce some basc and most related concepts, defntons and theorems from aucton desgn. An aucton allocates tems or goo (channels n our case) to compettve agents wth b and prvate valuatons. We adopt w as nonnegatve valuatons of each agent, whch s often prvate nformaton known only to the agent tself. Besdes determnng an allocaton, an aucton also computes

3 payments/charges for wnnng bdders. We denote by p() and b the payment and bd of agent, respectvely. Then the utlty of s a functon of all the b: { w p() f agent wth bd b gets an tem u (b, b ) = otherwse where b s the vector of all the b except b. We frst adopt some conventonal assumptons n economcs here. We assume that each agent s selfsh and ratonal. A selfsh agent s one that acts strategcally to maxmze ts utlty. An agent s sad to be ratonal n that t always prefers the outcome that brngs tself a larger utlty. Hence, an agent may le about ts valuaton, and bd b w f dong so yel a hgher utlty. Truthfulness s a desrable property of an aucton, where reportng true valuaton n the bd s optmal for each agent, regardless of other agents b. If agents have ncentves to le, other agents are forced to strategcally respond to these les, makng the aucton and ts analyss complex. A key advantage of a truthful aucton s that t smplfes agent strateges. Formally, an aucton s truthful f for any agent wth any b w, any b, we have u (w, b ) u (b, b ) () An aucton s randomzed f ts allocaton decson makng nvolves flppng a (based) con. The payment and utlty of an agent are then random varables. A randomzed aucton s truthful n expectaton f () hol n expectaton. Besdes, we also prefer an aucton to be ndvdually ratonal, n whch agents pay no more than ther gan (valuatons). As dscussed, the classc VCG mechansm for truthful aucton desgn requres the optmal allocaton to be effcently computable, and s not practcal for spectrum auctons, snce optmal channel allocaton s NP-hard. If we am to desgn a talored, heurstc truthful aucton, then we may rely on the characterzaton of truthful auctons by Myerson [6]. Theorem. Let P (b ) be the probablty of agent wth bd b wnnng an aucton. An aucton s truthful f and only f the followngs hold for a fxed b : P (b ) s monotoncally non-decreasng n b ; Agent bddng b s charged b P (b ) b P (b)db. Gven Theorem, we see that once the allocaton rule P( ) = {P (b )} N s fxed (N s the set of bdders), the payment rule s also fxed. For the case where the aucton s determnstc, there are two equvalent ways to nterpret Theorem : () there exsts a mnmum bd b, such that wll wn only f agent b at least b,.e., the monotoncty of P (b ) mples that, there s some crtcal bd b, such that P (b ) s for all b > b and for all b < b ; () the payment charged to agent for a fxed b should be ndependent of b (formally, p (b ) = b b db = b b ). B. System Model We assume there s a set of SNs, N. Each SN has deployed a set of nodes n a geographcal regon, and has a demand for multhop transmsson from a source to a destnaton. A PN TABLE I: Lst of notatons w valuaton of agent b bd of agent b bd of all agents except b φ() vrtual bd of agent p() payment of agent u a node n SN luv lnk from u to v fuv flow rate on lnk luv O(w) objectve functon of IP (5) S(w) objectve functon of the LPR I s() the set of SNs that nterfere wth SN along ts path x(c, luv) bnary var: whether channel c s allocated to lnk luv G (E, V ) connectvty graph of SN wth lnk set E, node set V H(E H, V H ) conflct graph of lnks of all the SNs wth edge set E H and vertex set V H has a set of channels, C, avalable for auctonng n the regon. We refer to SNs as agents and the PN as the auctoneer. Each node wthn an SN s equpped wth a rado that s capable of swtchng between dfferent channels. SNs do not collaborate wth each other, and nodes from dfferent SNs are not requred to forward traffc for each other. We assume nodes from each SN form a connected graph G (E, V ), whch also contans node locatons. We use node and lnk for the connectvty graphs and vertex and edge for the conflct graph ntroduced later. To better formulate the jont routng-channel assgnment problem, we ncorporate the concept of network flows. Let u be a node n SN and s, d be the source and the destnaton n SN. We use luv to denote the lnk from node u to node v belongng to SN, and fuv to denote the amount of flow on lnk luv. Later we connect d back to s wth a vrtual feedback lnk l, for a compact formulaton of the jont optmzaton IP. We defne a conflct graph H(E H, V H ), whose vertces correspond to lnks from all the connectvty graphs. We use (luv, lpq) j to denote an edge n E H, ndcatng that lnk luv and lnk lpq j nterfere f allocated a common channel. Before the aucton starts, each SN submts to the auctoneer a compound bd, defned as B = (G (E, V ), s, d, b ). Then the conflct graph can be centrally obtaned by the auctoneer. We denote by w the prvate valuaton of SN for a feasble path between s and d, and p() ts payment. b, w and p() all represent monetary amounts. Note that we assume agents only have ncentves to le about ther valuatons. We denote by R T (u ) and R I (u ) the transmsson range and nterference range of node u, respectvely. We assume that R I (u ) R T (u ) = and R T (u ) R max for any node u where. Snce no nter-sn collaboraton s assumed, lnks from dfferent SNs do not partcpate n jont MAC schedulng, and cannot be assgned the same channel f they nterfere. As a result, two lnks luv and lpq j nterfere f a node n {u, v} s wthn the nterference range of a node n {p, q}, and cannot be assgned the same channel f j. Formally, let a bnary varable x(c, luv) {, } denote whether channel c C s assgned to lnk luv for user. If for channel c C, x(c, luv) = x(c, lpq), j then (luv, lpq) j / E H. Hence, for the jont routng-channel assgnment problem we have the Channel Interference Constrants: x(c, l uv) + x(c, l j pq), (l uv, l j pq) E H, c C (2) We also need Flow Conservaton Constrants,.e., at any

4 node n V, the total ncomng and outgong flows equal (recall the vrtual feedback lnk): fuv = fvu, v V (3) u V u V Assumng each channel has the same unt capacty, we have the Capacty Constrants: x(c, luv) (4) uv u V \{d }f c C whch also ensures that a lnk can be assgned a sngle channel only. An agent nee an end-to-end path between ts source and destnaton. Ths correspon to a network flow of rate. Note that the lnk flow on the feedback lnk f equals the end-toend flow for SN. We formulate the jont routng-channel assgnment problem for SNs nto an IP: subject to maxmze O(w) = N w f (5) x(c, luv) + x(c, lpq) j, fuv = fvu, u V u V uv u V \{d }f x(c, luv), c C f uv, x(c, l uv) {, }. (l uv, l j pq) E H, c C v V v V where O(w) denotes the objectve functon of the IP. Solvng ths IP to optmal s an NP-hard problem. Therefore we frst ntroduce a heurstc aucton n Sec. IV, whch s based on the technque of monotonc allocaton and crtcal b, and s smple and truthful. However, t does not provde any bound on the socal welfare generated. A more sophstcated, randomzed aucton wth a proven bound s studed next, where the LP relaxaton of IP (5) s solved as a frst step. IV. A HEURISTIC TRUTHFUL AUCTION In ths secton, we desgn an aucton wth a greedy style allocaton and a payment scheme to ensure truthfulness. The aucton conssts of two phases: Algorthm determnes the channel assgnment and wnnng bdders, and Algorthm 2 computes the payments for wnnng agents. A. Channel Allocaton As dscussed n Sec. III, the key to desgnng a truthful aucton s to have a non-decreasng allocaton rule. Prces can then be calculated by the crtcal b to make the aucton truthful. A greedy allocaton s adopted n Algorthm. Assume channels are ndexed by, 2,... C. For a smple heurstc aucton, we frst compute the shortest path for each agent as ts end-to-end path. Let I s () be the set of SNs that nterfere wth along the path. We defne the vrtual bd of SN as φ() = b (6) I s () The ratonale behnd scalng the bd by I s () s to take s nterference wth other agents nto consderaton, for heurstcally maxmzng socal welfare. Then we greedly assgn mnmum ndexed avalable channels along the paths to each lnk, accordng to a non-ncreasng order of vrtual b φ(). Algorthm A greedy truthful aucton channel allocaton.. Input: Set of channels C, all the compound b B = (G (E, V ), s, d, b ), conflct graph H(E H, V H ) 2. for all N do 3. I s () ; 4. Compute the shortest path P from s to d ; 5. for all N do 6. for all luv along path P do 7. x(c, luv) c C; 8. f (luv, lpq) j E H then 9. I s () I s () {};. φ() b I ; s(). Wn() TRUE; 2. for N n non-ncreasng order of φ() do 3. for all luv along path P do 4. Let Tuv C; 5. for all c Tuv do 6. f x(c, lpq) j = wth (luv, lpq) j E H, p, q then 7. Tuv Tuv\{c}; 8. f Tuv = then 9. Wn() FALSE; 2. f Wn() = TRUE then for all luv along path P do Choose the mnmum ndexed channel c m n Tuv; 23. x(c m, luv) ; Fg. 2 shows an example to llustrate the channel assgnment procedure. There are four SNs, a, b, c and d, where φ(a) > φ(b) > φ(c) > φ(d). Two channels are avalable for allocaton. In the fgure, two ntersectng lnks also nterfere wth each other. If two lnks from two dfferent SNs ntersect, they cannot be allocated wth the same channel. The algorthm frst assgns Channel to SN a. As a result, t cannot assgn Channel to the frst lnk of SN b, whch receves Channel 2 nstead, as shown n Fg. 2b, leavng SN c wthout a channel t s mpossble to assgn ether channel to c s frst lnk. However, SN d wns, and receves a channel assgnment along ts path wthout ntroducng nterference to a or b. We now prove that the greedy aucton s monotone. Lemma. Algorthm s monotone. That s, the probablty of bdder wth bd b wnnng the aucton s non-decreasng n b, and crtcal b for wnnng agents exst. Proof: Bddng hgher can only ncrease an agent s vrtual bd, and therefore ncrease ts rank n Algorthm. Hence, the probablty of assgnng a channel to the agent s nondecreasng. Besdes, Algorthm s determnstc, so a crtcal bd b exsts for a wnnng bdder, such that agent always wns f t b b b.

5 d a c b d a c 2 b d a (a) Assgn channels to SN a (b) Assgn channels to SN b (c) Assgn channels to SN d Fg. 2: Procedure of channel assgnment. Dots and squares represent source and destnaton nodes respectvely. c 2 2 b B. Payment Calculaton Algorthm 2 computes payments for the wnnng agents. The payment scheme desgn s where we ensure the truthfulness of an aucton. Algorthm 2 ams to fnd a crtcal bdder wth crtcal bd b for a wnnng agent, such that s guaranteed to wn as long as s vrtual bd φ() φ (). Here φ () = b I s() s the crtcal vrtual bd for. If b s ndependent from b, then chargng agent b wll ensure that the aucton s truthful, whch we wll argue formally later. We now explan how Algorthm 2 works. It frst clears a wnnng agent s bd, and hence ts vrtual bd, to. Then we run Algorthm based on (, b ). In Algorthm, an agent loses only f a lnk along ts shortest path s unable to receve any channel. If we are unable to accommodate agent, there must exst at least one lnk along ts shortest path whose neghbourng lnks (neghbourng vertces n the conflct graph) have used all the channels. From all the agents that block lnks of agent, we fnd out an agent j wth the mnmum vrtual bd, set t as s crtcal bdder, and compute s payment. We clam that φ() φ(j), because otherwse agent would not be a wnnng agent among agents n I s () {}. Agent s payment can be computed as follows: p() = φ () I s () = φ(j) I s () (7) For the example n Fg. 2, we frst set SN a s bd to, and run Algorthm based on the new bd vector. After assgnng channels to agent c, we fnd that there are no avalable channels for the second lnk of agent a. Hence, agent c becomes the crtcal bdder of agent a, whch lea to a s payment p(a) = φ(c) I s (a). The rule apples to the other two wnnng agents b and d as well, where p(b) = φ(c) I s (b) and p(d) =. We next show that the aucton s ndvdually ratonal and truthful. Lemma 2. The aucton shown n Algorthms and 2 s ndvdually ratonal. Proof: Assume agent wns by bddng b, and let j be the crtcal bdder of. Then we have φ() φ(j), so p() = φ(j) I s () φ() I s () = b. Theorem 2. The aucton n Algorthms and 2 s truthful. Proof: Fx, b. Let w and w be agent s bd when beng truthful and not, respectvely. We need to show that for agent wth valuaton w, the utlty of bddng w s no less than the utlty of bddng w. We analyze the aucton case by Algorthm 2 A greedy truthful aucton payment calculaton.. Input: Set of channels C, all the compound b B = (G (E, V ), s, d, b ), conflct graph H(E H, V H ), all the routng paths P and channel assgnment from Algorthm. 2. for N n non-ncreasng order of φ() do 3. p() ; 4. f Wn() = then 5. Set b ; 6. Run Algorthm on (b, b ); 7. f Wn() = FALSE then 8. Let φ () + ; 9. for all luv along path P do. Let Tuv C;. for all c Tuv do 2. f x(c, lpq) j = wth (luv, lpq) j E H then 3. Tuv Tuv\{c}; 4. f T uv = then 5. A {j (luv, lpq) j E H, p, q; Wn(j) = TRUE}; 6. φ () mn(φ (), mn j A φ(j)); 7. p() φ () I s () ; case. Let { f agent wth bd b receves a channel; r(b ) = f agent wth bd b doesn t receve a channel. Frst, assume that w < w. Accordng to Lemma, t s mpossble for agent to have r(w ) = and r(w ) =. If r(w ) = and r(w ) =, there s no ncentve to le. If r(w ) = and r(w ) =, because the aucton s ndvdually ratonal, also has no ncentve to le. If r(w ) = and r(w ) =, then by observng that the crtcal bdder does not change, the utlty for agent remans the same. Next, assume that w > w. Agan t s mpossble for agent to have r(w ) = and r(w ) =. If r(w ) = and r(w ) =, the crtcal bdder does not change, so the utlty remans the same. If r(w ) = and r(w ) =, then has no ncentve to le. If r(w ) = and r(w ) =, then at least one lnk s unable to receve a channel (blocked), when agent b w. From all the agents that block lnks of, we fnd out the wnnng agent j wth mnmum vrtual bd. If b w > w and get assgned a channel, agent j would be the crtcal agent of. Then agent s payment would be p() = φ(j) I s () w, whch mples a nonpostve utlty of. Ths heurstc aucton, amng at achevng a monotonc

6 allocaton rule and fndng out a crtcal bdder for each wnnng agent, s drectly based on Theorem. V. A TRUTHFUL AUCTION FOR APPROXIMATELY MAXIMIZING SOCIAL WELFARE The greedy aucton just presented n Sec. IV, whle smple and truthful, attempts to maxmze socal welfare n a heurstc manner, wthout provdng any guarantee. In ths secton, we set out to desgn a more sophstcated, randomzed, lnear programmng based truthful aucton wth proven boun on approxmate socal welfare maxmzaton. We frst obtan the lnear programmng relaxaton (LPR) of IP (5), and solve the LPR optmally. We can turn the optmal LPR soluton, scaled down by, to a convex combnaton of nteger solutons, through a prmal-dual lnear program. Here s the upper bound of the ntegralty gap of IP (5) and the LPR. Vewng ths convex combnaton as specfyng a probablty dstrbuton over the nteger solutons, we arrve at a randomzed, truthful n expectaton aucton wth the VCG payment scheme. We prove that t approxmately maxmzes socal welfare. A. Decomposng the Fractonal Soluton Optmally and effcently solvng IP (5) s nfeasble, so we resort to an approxmaton approach. Frst, we obtan an LPR by allowng the nteger varables fuv, x(c, luv) to take fractonal values n [, ]. We use S(w) to denote the objectve functon of the LPR. Let f ( ) be the optmal flow vector from solvng the LPR, whch also contans agents wnnng/losng nformaton. Assume the ntegralty gap between IP (5) and ts LPR s at most, and there s a -approxmaton algorthm that verfes ths gap,.e., t acheves at least S(w), for any set of w. We employ the technque due to Lav and Swamy [7]. We utlze the -approxmaton algorthm and compute n polynomal tme a convex decomposton of f ( ) nto nteger solutons n polynomal sze. That s, we wll have ρ(l) values f ( ) such that = l I ρ(l)f (l), where {f (l)} l I s the set of all nteger solutons, I s ts ndex set and ρ(l), l I ρ(l) =. Now we can vew ths decomposton as specfyng a probablty dstrbuton over the nteger solutons. A soluton f (l) s selected wth probablty equal to ρ(l). If we set prces n such a way that the expected prces are VCG prces scaled down by, that wll make our randomzed aucton truthful n expectaton. Furthermore, the expected socal welfare s exactly the value of the LPR scaled by. We next provde detals on the decomposton and then prove the upper bound on the ntegralty gap. Let f ( ) be an optmal soluton to the LPR contanng all the f values. We use Z(F) = {f (l)} l I to denote the set of all nteger solutons to the LPR, where F denotes the feasble regon of the LPR, I s an ndex set of the nteger solutons. The crux of the method s to decde ρ(l) values such that f ( ) = l I ρ(l)f (l), where ρ(l), l I ρ(l) =. Now one can vew ths convex combnaton as specfyng a probablty dstrbuton over the nteger solutons, where a soluton f (l) s selected wth probablty equal to ρ(l). We solve the LP below to obtan the convex decomposton: subject to l I mnmze l I ρ(l)f (l) = f ( ) ρ(l) l I The dual of LP (8) s: subject to maxmze ρ(l) (8) N ρ(l) l I N η f(l) + λ N λ η unconstraned η f ( ) + λ (9) l I N To obtan the decomposton, an algorthm that approxmately computes the maxmum socal welfare s needed. We adapt a jont traffc routng and channel assgnment soluton framework due to Alcherry et al. [5] nto our settng, wth the objectve functon beng the weghted sum of throughput (socal welfare n our problem) and wth the number of rados at each node beng. Gven that ths algorthm may result n fractonal flows, we ntroduce the followng modfcaton: for an SN whose flow s fractonal, among SNs n {} I s () that also have fractonal flows, fnd out the SN wth maxmum w f, set ts flow to and others to. We denote ths modfed algorthm as A. We show later that Algorthm A verfes the upper bound of the ntegralty gap between (5) and the LPR. The prmal LP (8) has an exponental number of varables, whch wll take exponental tme to solve wth the smplex or nteror pont method [6]. Hence we resort to ts dual (9), whch has an exponental number of constrants. If we have a separaton oracle, we can apply the ellpsod method to solve the dual (9) and hence (8) n polynomal tme [6]. We show a separaton oracle for the dual later, so we can effcently solve ths par of prmal-dual LPs. One can vew η as a valuaton. A potental problem s that the η values could be negatve, whereas A s only for non-negatve valuatons. However, one can nstead use A wth the non-negatve valuatons η (+) gven by η (+) = max(η, ), and ths yel a separaton oracle [7]. Clam. Let η = {η } N be any weght vector. η (+) = max(η, ). Gven any nteger soluton f Z(F), one can obtan f (l) Z(F) such that N η f (l) = N η(+) f. Proof: We frst explot the packng property. That s, f a Z(F) and a 2 a s ntegral then a 2 Z(F). Now

7 we set f (l) = f f η and otherwse. Clearly, N η f (l) = N η(+) f. Snce f (l) f s ntegral, by the packng property f (l) Z(F). Now we are ready to show the followng lemma: Lemma 3. If the optmal soluton to (8) s ρ, then we have l I ρ (l) =. Proof: We show that the optmal value of (9) s, and hence the lemma follows by strong LP dualty. If we smply set λ =, η = for all N, t provdes a feasble soluton wth value. We then prove that the optmal value s at most by way of contradcton. Let (η ( ), λ ( ) ) denote the optmal soluton to (9). Suppose N η( ) f ( ) + λ ( ) >. Usng Algorthm A and Clam, we can compute a socal welfare maxmzng feasble soluton f (l), such that η ( ) f(l) S(η( )(+) ) N = N N η ( )(+) f ( ) η ( ) f ( ). () Now we have N η( ) f (l)+λ( ) N η( ) f ( ) + λ ( ) >, whch contradcts the frst set of nequaltes of (9), thereby contradctng the feasblty of (η ( ), λ ( ) ). The above lemma shows that wthout beng more restrctve, the nequalty N η f ( ) + λ can be added to the dual (9). We wll run the ellpsod method to solve ths dual LP. The frst set of nequaltes of (9) wll be the volated nequaltes returned by the separaton oracle durng the executon of the ellpsod method. The separaton oracle s, at a pont (η, λ), f N η f ( ) + λ >, then we can use Algorthm A and Clam to fnd an f (l) for whch the constrants of (9) s volated; otherwse, we use the half space N η f ( ) + λ to cut the current ellpsod. Snce the ellpsod method s guaranteed to take at most a polynomal number of steps, t wll return a set of solutons {f (l)} l I that s polynomal n sze. Then we can plug back these solutons to (8), leadng to a lnear program wth a polynomal number of varables and constrants, whch we can solve to recover ρ(l) s that sum to. B. Studyng the Integralty Gap Now we nvestgate the ntegralty gap of the IP (5) and the LPR. In the LPR, fractonal channel allocaton s drectly related to lnk flows, whch can be vewed as the fracton of tme a specfc lnk s actve. Smlarly, we can turn the constrant (2) nto the followng Lnk Schedulng Constrant [5]: fuv + fpq j. () l j pq:(l j pq,l uv ) E H where we assume there s only one channel, for ease of exposton. We wll argue later that the upper bound of the ntegralty gap between IP (5) and the LPR does not change when consderng multple channels. 2R max R max Fg. 3: The crcumstance formed by nterference regons of one lnk 2R max 2R max Fg. 4: The crcumstance formed by nterference regons of two lnks. We frst ntroduce the followng lemma for a sngle lnk, based on geometrc arguments: Lemma 4. For any channel c C, x(c, luv) + x(c, lpq) j α( ) (2) l j pq:(l j pq,l uv ) E H where α( ) s a constant that depen only on. Proof: For a gven lnk luv, we need to fnd the maxmum number of lnks that nterferes wth luv yet does not nterfere wth one another. For smplcty, we prove the followng for = 2; proofs for other values of can be derved smlarly. The worst case happens when R T (u ) = R T (v ) = R max and d(u, v ) = R max. We use U(u, v ) to denote the regon formed by the unon of two crcles wth radus 2R max and centre u and v, respectvely. Then the problem becomes placng a max number of ponts on the crcumference of U(u, v ) that are at least 2R max apart. The crcumference s composed of two major arcs wth length 4π 3, shown n Fg. 3. Among these lnks, every ndependent set s of sze at most 8. Hence n ths case α( ) = 9. We then nvestgate the ntegralty gap based on the worst case analyss of multhop transmssons. Lemma 5. For any channel c C and an SN wth a path that has L hops, there are at most g(l, ) nterference free SNs among I s () f all of them are assgned channel c, where g(l, ) s a functon that only depen on L and. Proof: Consderng the = 2 case as well, for an end-toend path, the worst case happens when the path s formed as a straght lne, and all the nterference-free lnks along the path belong to dfferent SNs. Addng one lnk nto a path ncreases the length of crcumstance by 4 arcsn 4, whch s due to the bold arcs shown n Fg. 4, where a two-hop path s formed. Then for an SN wth path length L, we have { L + 7 L s odd g(l, 2) = (3) L + 6 L s even We can loosen ths bound to L + 7, for all L. Smlar lnear functons of L can be derved wth other values of. Theorem 3. Assume that a SN s path s at most L max -hops. Then the ntegralty gap between the IP (5) and the LPR s at most = g(l max, ) +. Proof: For an SN and a sngle channel c, we know from Lemma 5 that there are at most g(l, ) nterference free SNs among I s (), ensurng every hop obeys (2). In the worst

8 case, the ntegral soluton pcks only one SN from at most g(l, ) + SNs. Snce ths s true for any SN and g(l, ) s an ncreasng functon of L, the lemma follows for a sngle channel case. If there are C channels, for an SN, we can magne that the maxmum ndependent set of a lnk s duplcated nto C copes, so that the ntegral soluton wll pck SNs from less than ( C )g(l, )L+( C )L+ = ( C )(g(l, )+ )L SNs. Snce the ntegral soluton pcks at least ( C )L+ SNs (pcks, and C SNs per lnk along s path), the ntegralty gap s at most ( C )(g(l max, ) + )L max + ( C )L max + ( C )(g(l max, ) + )L max ( C )L max g(l max, ) + (4) Snce Algorthm A modfes a fractonal flow to among I s (), t apparently verfes. C. A Randomzed Approxmaton Aucton We now present the desgn of our randomzed aucton, n Algorthm 3. It frst solves the LPR to obtan the optmal f ( ). Then a decomposton technque descrbed prevously s employed to compute a feasble set of allocatons and a probablty dstrbuton. Next, a soluton s chosen accordng to ts assocated probablty. Snce the ellpsod method and A both run n polynomal tme, our aucton s computatonally effcent. For each bdder that wns and acheves throughput f (l) wth probablty ρ(l), the payment s calculated as follows: p() = ( b f ( ) j y(j) b j f ( ) ) (5) j j where y s obtaned by recomputng the LPR wth b =. Algorthm 3 A truthful-n-expectaton aucton. Input: Set of channels C, all the compound b B = (G (E, V ), s, d, b ), conflct graph H(E H, V H ). 2. Solve the LPR, obtan optmal soluton f ( ) ; 3. Use the ellpsod method and Algorthm A on (9) wth f ( ), obtan a polynomally szed set of {f (l)}; 4. Solve (8) wth f ( ) and {f (l)}, fndng the ρ(l) values; 5. Pck some soluton f (l) wth probablty ρ(l); 6. Let p(), N ; 7. for all such that f (l) = do 8. Compute S(b =, b ) wth the LPR. Let y be the soluton; 9. p() ( j b jy(j) j b jf ( ) ); f ( ) Theorem 4. The aucton shown n Algorthm 3 s truthful n expectaton. Proof: Fx, b. Let w and w be agent s bd when beng truthful and not truthful, respectvely. Let a and a be the solutons to the LPR for b (w, b ) and (w, b ), respectvely. The expected utlty of when bddng truthfully s E[u (w )] = a() [w a() ( b j y(j) b j a(j))] j j = (w a() + j b j a(j) j b j y(j)) (6) Snce for every bdder, a s optmal for (w, w ). Assumng other bdders bd truthfully yel E[u (w )] (w a () + b j a (j) b j y(j)) j j = E[u (w )] (7) We now arrve at the followng theorem: Theorem 5. The aucton shown n Algorthm 3 acheves a -approxmate maxmum socal welfare n expectaton. VI. SIMULATION RESULTS Our auctons were evaluated through smulatons. We frst focus on the heurstc aucton. For each SN, we randomly and unformly dstrbute some nodes n a regon. Two nodes are connected f ther Eucldean dstance s at most.5. The largest connected component s used as the connected graph for the correspondng SN. All b are taken from a unform dstrbuton n the range of [4, ]. Snce the aucton has already been proven to be truthful and the optmal socal welfare s hard to obtan, we evaluate ts performance n terms of aucton effcency, whch s defned as N ϑ = w f N w (8) We then vary the number of channels n the smulatons to study the performance of the aucton, where all data are averaged over experments. We observe that, n general, as the number of channels ncreases, the aucton effcency ncreases as well, whch verfes the ntuton that the more channels, the hgher probablty for a bdder to wn. Frst we change the number of bdders (SNs), whle fxng = 4 and the number of nodes for each SN at 3. From Fg. 5a, we can see that our aucton n general effectvely explots the ncreasng number of channels avalable. Even n the extremely nterferng case where there are SNs n the regon, the effcency ncreases approxmately lnearly wth the number of channels. Fg. 5b (wth 5 bdders and 3 nodes for each SN) and Fg. 5c (wth 5 bdders and = 4) also show the performance of our aucton n terms of the severty of nterference, by changng and the sze of SNs. We can see that the change of s does not hurt the performance too much. However, large szes of SNs may ncrease nterference sgnfcantly, thereby decreasng the aucton effcency, where the connected graph for an SN wth 5 nodes dstrbuted can contan more than 5 nodes.

9 Effcency Bdders 2 Bdders.2 5 Bdders Bdders Number of Channels (a) Aucton effcency wth dfferent numbers of bdders enrolled. Effcency = = 2.2 = 3 = Number of Channels (b) Aucton effcency under dfferent nterference stuatons. Fg. 5: Performance evaluaton of the heurstc aucton. Effcency Nodes.2 3 Nodes 5 Nodes Number of Channels (c) Aucton effcency wth dfferent szes of networks. We then compare the performance of our aucton wth two other approaches. One s a greedy aucton that only assgns one channel to an SN, n whch the same channel cannot be assgned to SNs who nterfere wth one another. The other one s smply a mult-tem aucton that greedly assgns channels to lnks n each SN, wthout global vson of formng an end-toend path. We fx = 4, the number of potental nodes for each SN s 45, and the number of bdders s 5. We can see from Fg. 6 that our aucton and the sngle-channel aucton perform much better than the mult-tem aucton. Another observaton s that the effcency of our aucton ncreases faster than the sngle-channel one as the number of channels ncreases. Ths justfes the use of multchannel assgnment for each SN. Effcency Multchannel Enabled Sngle Channel Mult Item Number of Channels Fg. 6: Comparson of three dfferent aucton settngs. Number of SNs Amount of Flows Fg. 7: A hstogram of fractonal solutons of the LPR. Fg. 7 shows the dstrbuton of the fractonal soluton to the LPR for one smulaton nstance (wth 5 bdders, 3 nodes for each SN, one channel and = 4), where the sum of all flows equals The agents wth relatvely large non-zero flows n the soluton are shown n Table II. We can see that there s only one agent guaranteed to wn and most of them almost always lose (the amount of flows s approxmately ). In the worst case, our randomzed aucton wll select agents wth flow amount,.85,.54, and one of agents wth flow amount.5, and another agent wth fractonal flow. Note that two agents wth flow amount larger than.5 must not nterfere wth each other. Hence, n ths experment, algorthm A can actually acheve of the soluton to the LPR, hence rasng.3 the socal welfare wth our randomzed aucton. TABLE II: Agents wth non-zero fractonal flows Agent Flow VII. CONCLUSION Secondary spectrum auctons are emergng as a promsng approach to effcently dstrbutng and sharng scarce wreless spectrum. For the frst tme n the lterature, we propose the concept of a secondary network, relaxng the over-smplfyng assumpton on secondary users n exstng research. We desgned two auctons for spectrum allocaton among SNs. The frst s a smple, greedy style determnstc aucton that heurstcally maxmzes socal welfare. The heurstc aucton s truthful due to ts monotone allocaton rule. The second s a randomzed, lnear optmzaton based aucton that s not only truthful (n expectaton), but also provdes proven guarantees on socal welfare. In future work, we plan to further mprove the performance guarantee of the randomzed aucton, by provng a tghter bound on socal welfare approxmaton. REFERENCES [] X. Zhou, S. Gandh, S. Sur, and H. Zheng, ebay n the Sky: Strategy- Proof Wrelss Spectrum Auctons, n Proc. ACM MobCom, August 25. [2] Y. Wu, B. Wang, K. J. R. Lu, and T. C. Clancy, A Scalable Colluson-Resstant Mult-Wnner Cogntve Spectrum Aucton Game, IEEE Trans. Comm., vol. 57, no. 2, pp , 29. [3] M. Garey and D. Johnson, Computers and Intractablty: A Gude to the Theory of NP-Completeness. W.H. Freeman and Company, 99. [4] I. Kash, R. Murty, and D. C. Parkes, Enablng Spectrum Sharng n Secondary Market Auctons, n Proc. NetEcon, June 2. [5] A. Gopnathan, Z. L, and C. Wu, Strategyproof Auctons for Balancng Socal Welfare and Farness n Secondary Spectrum Markets, n Proc. IEEE INFOCOM, Aprl 2, pp [6] R. Myerson, Optmal Aucton Desgn, Mathematcs of Operatons Research, vol. 6, no., pp , 98. [7] R. Lav and C. Swamy, Truthful and Near-Optmal Mechansm Desgn va Lnear Programmng, n Proc. IEEE FOCS, October 25. [8] E. H. Clarke, Multpart Prcng of Publc Goo, Publc Choce, vol., pp. 7 33, 97. [9] T. Groves, Incentves n Teams, Econometrca: Journal of the Econometrc Socety, pp , 973. [] W. Vckrey, Counterspeculaton, Auctons, and Compettve Sealed Tenders, Journal of Fnance, pp. 8 37, 96. [] J. Huang, R. A. Berry, and M. L. Hong, Aucton Mechansms for Dstrbuted Spectrum Sharng, n Proc. Allerton, September 24. [2] M. M. Buddhkot and K. Ryan, Spectrum Management n Coordnated Dynamc Spectrum Access Based Cellular Networks, n Proc. IEEE DySPAN, November 25, pp [3] X. Zhou and H. Zheng, TRUST: A General Framework for Truthful Double Spectrum Auctons, n Proc. IEEE INFOCOM 29, Aprl 29. [4] J. Ja, Q. Zhang, Q. Zhang, and M. Lu, Revenue Generaton for Truthful Spectrum Aucton n Dynamc Spectrum Access, n Proc. ACM MobHoc, May 29, pp [5] M. Alcherry, R. Bhata, and L. L, Jont Channel Assgnment and Routng for Throughput Optmzaton n Mult-rado Wreless Mesh Networks, n Proc. ACM MobCom, August 25, pp [6] G. B. Dantzg and M. N. Thapa, Lnear Programmng 2: Theory and Extensons. Sprnger, 23.