Spectrum Auction Framework for Access Allocation in Cognitive Radio Networks

Transcription

1 Spectrum Aucton Framework for Access Allocaton n Cogntve Rado Networks ABSTRACT Gaurav S. Kasbekar Department of Electrcal and Systems Engneerng Unversty of Pennsylvana kgaurav@seas.upenn.edu Cogntve rado networks are emergng as a promsng technology for the effcent use of rado spectrum. In these networks, there are two categores of networks on dfferent channels: prmary networks and secondary networks. A prmary network on a channel has prortzed access to the channel and secondary networks can use the channel when the prmary network s not usng t. The access allocaton problem s to select the prmary and secondary networks on each channel. We develop an aucton-based framework that allows networks to bd for prmary and secondary access based on ther utltes and traffc demands, and uses the bds to solve the access allocaton problem. We develop algorthms for the access allocaton problem and show how they can be used ether to maxmze the auctoneer s revenue gven the bds, or to maxmze the socal welfare of the bddng networks, whle enforcng ncentve compatblty. We frst consder the case when the bds of a network depend on whch other networks t wll share channels wth. When there can be only one secondary network on a channel, we desgn an optmal polynomal-tme algorthm for the access allocaton problem based on reducton to a maxmum matchng problemnweghtedgraphs. Whentherecanbetwoormore secondary networks on a channel, we show that the optmal access allocaton problem s NP-Complete. Next, we consder the case when the bds of a network are ndependent of whch other networks t wll share channels wth. We desgn a polynomal-tme dynamc programmng algorthm to optmally solve the access allocaton problem when the number of possble cardnaltes of the set of secondary networks on a channel s upper-bounded. Fnally, we desgn a polynomal-tme algorthm whch approxmates the access allocaton problem wthn a factor of 2 when the above upper bound does not exst. Categores and Subject Descrptors C.2 [Computer-Communcaton Networks]: Wreless Communcaton Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. MobHoc 09, May 18 21, 2009, New Orleans, Lousana, USA. Copyrght 2009 ACM /09/05...$5.00. Saswat Sarkar Department of Electrcal and Systems Engneerng Unversty of Pennsylvana swat@seas.upenn.edu General Terms Algorthms, Desgn, Economcs, Theory Keywords Cogntve Rado Networks, Spectrum Auctons, Algorthms 1. INTRODUCTION Wth the prolferaton of dfferent wreless network technologes lke cellular networks, Wreless Local Area Networks, Wreless Meteropoltan Area networks etc., demand for rado spectrum s ncreasng. Currently, spectrum s regulated by a government agency lke the Federal Communcatons Commsson (FCC) and t allocates spectrum by assgnng exclusve lcenses to users to operate ther networks n dfferent geographcal regons. There s a wdespread belef that rado spectrum s becomng ncreasngly crowded. However, spectrum measurements ndcate that the allocated spectrum s under-utlzed,.e. at any gven tme and locaton, much of the spectrum s unused [2]. Cogntve rado networks are emergng as a promsng soluton to ths dlemma. In these networks, there are two levels of networks on a channel prmary networks and secondary networks. A prmary or secondary network s a network of multple wreless devces, whch we call prmary and secondary nodes respectvely. A prmary node has prortzed access to the channel,.e. t can transmt on the channel regardless of the transmssons of secondary nodes. On the other hand, a secondary node can transmt on the channel provded prmary nodes are not transmttng. So whenever a secondary node wants to transmt on the channel, t senses the channel to check for ongong transmssons. It ntates a transmsson only f a prmary node s not transmttng. Moreover, f a secondary node s transmttng and a prmary node wshes to transmt, then the secondary node suspends ts transmsson untl after the prmary node fnshes ts transmsson. Cogntve rado technology [4] allows secondary nodes to detect whch channel s not beng used by prmary nodes, share ths channel wth other secondary nodes and vacate the channel when a prmary node s detected. Surveys on cogntve rado networks can be found n [1] and [4]. An mportant queston faced by a spectrum regulator s how to allocate the rghts to be prmary and secondary networks on ts channels. Dfferent networks may attach dfferent value to beng prmary and secondary. A network may wsh to manly transmt delay-senstve traffc lke voce or vdeo. Such a network wll attach a hgh value to the

2 rghts to be prmary. On the other hand, a network may be manly nterested n transmttng delay-nsenstve traffc lke emal or fle transfer. Such a network would not need prmary rghts and would prefer secondary rghts snce the latter would be prced lower than the former. Also, a network whose traffc s a mxture of delay-senstve and delay-nsenstve traffc would want prmary rghts on some channels and secondary rghts on some channels. Auctons are sutable for sellng the rghts to be prmary and secondary on the channels. Snce the regulator need not know the values that bdders attach to prmary and secondary rghts, auctons provde a mechansm for the regulator to get a hgher revenue than that obtanable through statc prcng [9]. Auctons are also benefcal for the bdders snce n general they assgn goods to the bdders who value them most [9]. FCC has been conductng spectrum auctons snce 1994 to allocate lcenses for rado spectrum [3] (however, so far, no auctons have been conducted for cogntve rado networks). Spectrum auctons have been studed n [9], [6], [7], [8]. We now explan how our work dffers from prevous work. In some of the exstng work on spectrum auctons [9], [8], each channel s assgned to a sngle network,.e. there s no noton of prmary and secondary networks on a sngle channel. We consder the case when there s a prmary network and one or more secondary networks on each channel. Now, there are two possbltes [10] for allocatng secondary rghts on the channels. In one possblty, the regulator allocates channels to prmary networks and the prmary networks ndependently allocate unused portons on ther channels to the secondary networks. Auctons have been desgned for ths scenaro n [6] and [7]. In the other possblty, the regulator allocates the rghts to be the prmary and secondary networks on the channels n a sngle allocaton [10]. To the best of our knowledge, no work has been done n desgnng auctons for ths scenaro. In ths paper, we develop a comprehensve aucton framework usng whch a regulator can smultaneously allocate the rghts to be prmary and secondary on the channels. Ths scenaro may be more desrable than the frst possblty above n certan cases. For example, t gves a greater degree of control over the allocaton to the regulator than n the case when prmary networks allocate unused portons on ther channels to the secondary networks. We consder a scenaro n whch the regulator conducts an aucton to sell the rghts to be prmary and secondary networks on a set of channels. Networks can bd for these rghts based on ther utltes and traffc demands. The regulator uses these bds to solve the access allocaton problem,.e. the problem of decdng whch networks wll be the prmary and secondary networks on each channel. The goal of the regulator may be ether to maxmze ts revenue or to maxmze the socal welfare of the bddng networks. Now, networks can have utltes or valuatons that are functons of the number of channels on whch they get prmary and secondary rghts, on how many and whch other networks they share these channels wth etc. The number of valuatons of a network may be large and an exponental amount of space may be requred to express a bd for each valuaton. So we desgn bddng languages, that s, compact formats for networks to express bds for ther valuatons. For dfferent bddng languages, we desgn algorthms for the access allocaton problem. The paper s organzed as follows. We descrbe the system model n Secton 2. In Secton 3, we descrbe how the bddng languages and algorthms that we desgn n the paper can be used to maxmze the auctoneer s revenue or to maxmze socal welfare. In Secton 4, we descrbe a model n whch the bds of a network depend on whch other networks t wll share channels wth. In Secton 4.1, we desgn an optmal algorthm for the access allocaton problem for a smple case wth only one secondary network on each channel. We show the ntractablty (NP-Competeness of the access allocaton problem or exponental sze of bds) of the extensons of ths smple case n Secton 4.2. In Secton 5, we consder the case n whch the bds of a network are ndependent of whch networks t wll share channels wth and provde an optmal polynomal-tme algorthm for the access allocaton problem when the number of cardnaltes of the set of secondary networks on a channel s upper-bounded. In Secton 6, we descrbe a bddng language that can be used for the ndependent bds case when the above bound does not exst and provde a greedy 2-approxmaton algorthm for the access allocaton problem. Due to space constrants, we omt the proofs of several results and outlne the proofs for some others. 2. SYSTEM MODEL We consder a scenaro n whch there are M dentcal orthogonal channels n a regon. A regulator conducts an aucton to sell the rghts to be the prmary and secondary networks on the channels. N bdders partcpate n the aucton. Each bdder s an ndependent network of multple wreless nodes. Each bddng network submts bds to the regulator and based on the bds, the latter allocates the rghts to be the prmary and secondary networks on the channels. A prmary network on a channel must have prortzed access to the channel. If two or more ndependent networks were to be the prmary networks on a sngle channel, then the access of each one of them would be constraned by the transmssons of the other prmary networks, whch would transmt at the same prorty level. To avod ths, we assume that there s exactly one prmary network on each channel. However, we allow multple networks to have secondary rghts on a channel. We assume that all the secondary networks on a channel have equal rghts on the channel. Ths s because complcated multple access protocols [5] would be requred to grant access at dfferent prorty levels to dfferent secondary networks on a channel (wth all of them gettng lower prorty access than the prmary network). On the other hand, smple multple access protocols would suffce f all secondary networks have equal rghts on the channel. Now, snce a prmary network has prortzed access on a channel, the average delay of ts traffc s low. On the other hand, the average delay of a secondary network s traffc s hgh. Hence, prmary rghts (respectvely secondary rghts) are sutable for communcatng delay-senstve (respectvely delay-nsenstve) traffc. We assume that each network has two knds of traffc: (a) delay-senstve traffc lke voce, vdeo etc. and (b) delay-nsenstve or elastc traffc lke emal, fle-transfer etc. A network uses ts prmary rghts to transmt ts delay-senstve traffc and ts secondary rghts to transmt ts elastc traffc. We allow a sngle network to be both the prmary network and one of the secondary networks on a channel. In ths

3 case, we assume that t transmts ts delay-senstve traffc as a prmary network,.e. wth hgh-prorty and elastc traffc as a secondary network,.e. when t has no delay-senstve traffc to transmt. Also, the other secondary networks on the channel can transmt whenever network s not transmttng ts delay-senstve traffc. Let K be the set of all possble ways n whch the M channels can be allocated to the N bdders. For example, consder the smple case n whch M =3,N =9andthere can be at most four secondary networks on a channel. An example of an allocaton of the channels s one n whch network 1 becomes the prmary network on channels 1 and 2, network 2 becomes prmary on channel 3, network 3 becomes the sole secondary network on channel 1, networks 4 and 5 become secondary networks on channel 2, networks 1, 4, 6 and 7 become secondary networks on channel 3 and networks 8 and 9 do not become prmary or secondary networks on any channel. Let x (k) benetwork s valuaton or utlty from the channel allocaton k K,.e., the value that t conjectures or expects to derve from the allocaton k. Note that snce network wll share channels wth other networks n the allocaton k, theactual utlty that network wll derve from an allocaton k depends on the transmsson patterns of the other networks that are not completely known to network. Hence, each network bds for access based on ts conjectures about ts actual utlty. Henceforth, we use the terms valuaton or utlty for x (k), but they should be understood to mean the conjectured utlty or valuaton of network for the channel allocaton k. The valuatons x (.) ofnetwork for the allocatons n K depend on ts traffc demands,.e. the volumes of delaysenstve and elastc traffc that t wants to transmt. Now, for gven traffc demands, the valuaton of a network for a channel allocaton k K may depend upon the number of channels on whch network has prmary and secondary rghts n the allocaton k, how many and whch other networks have rghts on each of the channels on whch network has prmary or secondary rghts etc. Note that network may have the same valuaton for dfferent allocatons k K. Network s net utlty s of the form: u (k, τ,x )=x (k) τ (1) where τ s the payment that network makes to the auctoneer. The auctoneer determnes the channel allocaton and the payment τ that each network makes to the auctoneer. The socal welfare of an allocaton k s defned to be the quantty: N x (k) =1 Thus, the socal welfare s the sum of utltes of all bdders from the allocaton k. Now, there could be two goals for desgnng the aucton: revenue maxmzaton and maxmzng socal welfare. In the frst goal, based on ts valuatons, each network submts a set of bds to the auctoneer. Let z (k) be the bd of network for the allocaton k K,.e. the amount of money t s wllng to pay f the allocaton k K s chosen. Let k be the channel allocaton that maxmzes the revenue of the auctoneer, gven the bds z (.) for bdders 1,...,N. That s, k satsfes: N N z (k ) z (k) k K (2) =1 =1 As we wll see n Secton 3, when z (.) are not the bds of the networks, but have a dfferent nterpretaton, the channel allocaton that maxmzes the socal welfare of the N networks can be found by fndng the k satsfyng the above equaton. The access allocaton problem s to determne the channel allocaton k satsfyng (2). Dependng on the nterpretaton of z (.), ths allocaton k ether maxmzes the auctoneer s revenue or the socal welfare of the N networks. Now, the set K of possble channel allocatons may be exponental n sze. As noted earler, a bdder may have the same valuaton for two or more allocatons n K and hence t need not specfy a bd for each k K. The total number of dfferent valuatons of network may stll be exponental. However, t s not feasble to communcate a bd for each valuaton n ths large set. So we ntroduce bddng languages for the aucton models that we consder. A bddng language [11] s a format to compactly encode the bd nformaton of a bdder. When there are an exponental number of valuatons, a bddng language expresses the bds approxmately, not exactly. 3. SOLUTION FRAMEWORK As stated earler, an aucton could be desgned for two dfferent objectves. In our context, the frst objectve s to choose the channel allocaton that maxmzes the regulator s revenue for a gven set of bds of the bdders. Ths can be done by choosng the allocaton k satsfyng (2) when z (k) s the bd of network for the channel allocaton k. The second possble objectve for the aucton could be to acheve effcency, that s, to choose the allocaton that maxmzes socal welfare. To ths end, each bdder s asked to declare ts valuaton functon x (.). Wth an abuse of notaton, let z (k) denotethedeclared valuaton of network for the allocaton k, whch may be dfferent from x (k) f bdder beleves that falsely declarng ts valuatons wll mprove ts net utlty. Truth-tellng s sad to be a weakly-domnant strategy [17] for network, f for any possble declaratons of networks other than, the net utlty of network s maxmzedwhentsetsz (k) =x (k) k K. It follows from the revelaton prncple [17] that to maxmze socal welfare, t s suffcent to consder mechansms n whch the payments τ are chosen such that for each bdder, truth-tellng s a weakly domnant strategy. Such a mechansm s called ncentve compatble. To date, the Vckrey-Clarke-Groves (VCG) mechansm [17] s the only known general ncentve compatble mechansm that can be used to maxmze socal welfare. Under ths mechansm, gven the declared valuaton functons z (.) of the bdders, the allocaton k satsfyng (2) s chosen and the payments are chosen so as to enforce ncentve compatblty [17]. It can be shown that the VCG mechansm can be mplemented by runnng an algorthm for the access allocaton problem (n equaton (2)) (N + 1) tmes once wth all bdders and once each for the set of bdders {1,...,N}\ for =1,...,N. Now, n general, the set of dfferent valuatons of a bddng network s exponental n sze. Frst we consder the specal case when the number of dfferent valuatons of each bddng

4 network s of poynomal space complexty (but K can stll be exponental n sze). Even n ths case, t s sometmes computatonally ntractable to devse an algorthm to fnd the optmal allocaton k satsfyng (2), possbly because ths s NP-hard, but nstead an approxmaton algorthm for the access allocaton problem can be devsed. In ths case, the VCG mechansm cannot be used to enforce ncentve compatblty. To address ths problem, Nsan and Ronen [19] devsed the second-chance mechansm whch enforces ncentve compatblty under the assumpton that there s a lmt on the computatonal resources of each bdder. Moreover, the socal welfare attaned by the second-chance mechansm s at least as good as the socal welfare of the approxmaton algorthm used for the access allocaton problem. Now, n some cases, the set of valuatons of a bdder takes an exponental amount of space and hence bdders have to use ncomplete bddng languages (see Secton 2) to convey ther valuatons. In ths case as well, the VCG mechansm cannot be used to enforce ncentve compatblty. As a soluton to ths problem, Ronen [20] devsed the extended secondchance mechansm that, under reasonable assumptons [20], can be used to enforce ncentve compatblty and acheve a socal welfare at least as good as that of the approxmaton algorthm used for the access allocaton problem. In ths paper, we propose several spectrum aucton models and desgn bddng languages and algorthms for the access allocaton problem. These can be used for the objectve of maxmzng the revenue of the auctoneer or for maxmzng the socal welfare of the bdders n conjuncton wth the VCG, second-chance or extended second-chance mechansm, as approprate. For notatonal convenence, throughout the paper, we assume that z (.) are the bds expressed by bdder and vew the access allocaton problem as the problem of maxmzng the revenue of the auctoneer. However, our framework apples wthout change to the problem of maxmzng socal welfare. 4. AUCTION WITH DEPENDENT BIDS A prmary or secondary network on a channel shares the channel wth other networks and hence ts actual utlty from the channel depends on the transmssons of those networks. A network may have some knowledge or belefs about the typcal transmsson patterns of the other bddng networks. For example, the agency ownng the network may conduct a survey on the typcal transmsson patterns of the other networks n ts regon or, f auctons are perodcally conducted to allocate spectrum n the regon, the agency may gan ths knowledge about the networks wth whom t shared channels prevously. Thus, the conjectured utltes and hence the bds of a network would depend on whch networks t wll share dfferent channels wth. 4.1 Basc Model In the basc model wth dependent bds, we consder the model descrbed n Secton 2 wth the followng addtonal assumptons. Assumpton 1. There s only one secondary network on each channel. Assumpton 2. Each network can be ether the prmary or the secondary network on only one channel. We explore the effect of relaxng ether of these assumptons n Secton 4.2. We assume that N 2M, sothatall M channels can be allocated. A secondary network on a channel can use the channel whenever the prmary network s not usng t. So the throughput and delay of the secondary network on the channel depends on the channel usage behavor of the prmary on the channel,.e. on the rate of ts transmssons on the channel and how these transmssons are spread over tme. On the other hand, the prmary network on a channel has prortzed access to the channel. That s, when the secondary network wants to transmt on the channel, t senses the channel and can transmt only f t fnds that the prmary network s not transmttng. However, due to the mperfect nature of sensng, the secondary network wll sometmes transmt whle the prmary network s transmttng, resultng n a collson. Hence the prmary network s utlty depends on the channel usage behavor of the secondary network on the channel. Thus, the actual utlty of a prmary or secondary network depends on whch network t shares a channel wth. As explaned above, a network may n general have certan belefs about the channel usage behavor of other networks and hence may wsh to express bds dependent on the network wth whom t shares the channel. To model ths, let z p (j), j {1,...,N}\{} be the bd of network for the case when t s the prmary network on a channel and network j s the secondary network on the channel. Smlarly, let z s (j), j {1,...,N}\{} be the bd of network for the case when t s the secondary network on a channel and network s the prmary network. Let k = {( 1,j 1),...,( M,j M )} be an allocaton of the M channels to a set of networks. k s a set of M orderered pars ( t,j t) such that network t s the prmary network on channel t and network j t s the secondary network on channel t. Note that the revenue of the allocaton k s: M (z p t (j t)+zj s t ( t)) t=1 We descrbe an algorthm for determnng k, the allocaton that maxmzes the revenue, by reducton to a maxmum weght matchng problem n a graph. Let G be a weghted undrected graph wth N nodes, one node correspondng to each network. G s a complete graph,.e. between every par of nodes, there s an edge. Let the weght of the edge jonng nodes and j be w j =max(z p (j)+zs j (),z p j ()+zs (j)) (3) Note that the weghts are nonnegatve real numbers. The nterpretaton of the weghts w j s as follows. If network (respectvely network j) s the prmary network on a channel and network j (respectvely network ) s the secondary network, then the sum of the amounts pad by networks and j s z p (j) +zs j () (respectvely,z s (j) +z p j ()). So wj, the greater of these two quanttes, s the maxmum sum of payments of networks and j f they are the two networks on the same channel.

5 A matchng M n a graph s defned to be a subset of the edges such that no two edges n the subset share a common node. The weght of a matchng s the sum of the weghts of ts edges. The followng algorthm fnds the channel allocaton k that maxmzes the revenue: STEP1: In graph G, fnd a matchng M M of maxmum weght among matchngs wth exactly M edges (we say how later). STEP2: Let e 1,...,e M be the M edges n the matchng M M. Let e 1 t and e 2 t be the two endponts of edge e t. The allocaton k s chosen such that for t =1,...,M,networks e 1 t and e 2 t become the two networks (prmary and secondary) on channel t. If z p (e 2 e 1 t )+z s e 2(e1 t ) z p (e 1 t t e 2 t )+z s e 1(e2 t ) t t then network e 1 t becomes the prmary network on channel t and network e 2 t becomes the secondary network, otherwse network e 2 t becomes the prmary network on channel t and network e 1 t becomes the secondary network. Theorem 1. The allocaton k found from the matchng M M n the above algorthm s the one that maxmzes the revenue. Proof. (Outlne) There s a many-to-one correspondence between the set of channel allocatons and the set of matchngs wth exactly M edges. (It s many-to-one snce the allocatons obtaned from any allocaton by swappng the roles of the prmary and secondary networks on one or more channels correspond to the same matchng). From the nterpretaton of the weght of an edge gven above, t follows that the weght of a matchng M M has the maxmum revenue among the revenues of the channel allocatons that correspond to t. Moreover, Step 2 of the above algorthm selects the prmary network on each channel from the two networks on the channel so as to maxmze the sum of payments of the two networks. Fnally, snce M M s the maxmum weght matchng among all matchngs wth exactly M edges, we get the desred result that the channel allocaton k obtaned from the matchng M M s the one that maxmzes the revenue. Now, t remans to show how to fnd the matchng M M. Edmonds [13] gave a polynomal-tme algorthm for fndng the maxmum weght matchng (wth any number of edges) n a graph. However, we are nterested n a maxmum weght matchng among matchngs wth M edges, whch cannot be drectly obtaned by Edmonds algorthm. It can be obtaned n O(M 4 + M 2 N 2 ) tme usng Whte s modfcaton [14], [15] to the algorthm. 4.2 Intractablty of Extensons We now explore the effect of relaxng ether one of Assumptons 1 and 2. Suppose Assumpton 1 s relaxed and Assumpton 2 s retaned. That s, we assume that each network can be the prmary or a secondary network on only one channel. However, there can be multple secondary networks on a channel. We show that even f there are two secondary networks on a channel, the problem of fndng a channel allocaton that maxmzes the revenue s NP-Complete. Suppose there s one prmary network and r 1 secondary networks on each channel. Let z p (j1,...,jr 1) be the bd of network for the case n whch t s prmary on a channel and networks j 1,...,j r 1 are secondary. Let z s j 1 (, j 2,...,j r 1) be the bd of network j 1 for the case n whch network s the prmary and networks j 1,...,j r 1 are the secondary networks. We now defne the r-network Dependent Bd Access Allocaton Problem (r-ndbaa). Defnton 1 (The r-ndbaa Problem). Suppose M channels are to be allocated to N bdders such that on each channel, one network s prmary and r 1 networks are secondary, where r s a fxed postve nteger. Each bdder can be a prmary or secondary network on at most one channel and the bds of networks are as gven above. Fnd the allocaton that maxmzes the revenue. The decson verson of r-ndbaa s as follows: gven a bound D, s there a channel allocaton such that the revenue under the allocaton s at least D? We next show that (the decson verson of) 3-NDBAA s NP-Complete. Theorem 2. 3-NDBAA s NP-Complete. Proof. (Outlne) Gven an allocaton of the M channels, we can verfy n polynomal tme whether the revenue under the allocaton s at least D. Ths shows that 3-NDBAA s n the class NP. Next, we show that the 3-Dmensonal Matchng problem (3DM), whch s known to be NP-complete [18], s polynomaltme reducble to 3-NDBAA,.e. 3DM p 3-NDBAA. An nstance of 3DM s as follows [18]: Gven dsjont sets A, B, C of m elements each and a set T of ordered trples of the form (a, b, c), where a A, b B and c C, dothereexst asetofm trples n T so that each element of A B C s contaned n exactly one of these trples? From ths nstance of 3DM, we construct an nstance of 3-NDBAA as follows. Let there be M = m channels and 3m networks one network correspondng to each element of A B C. For every set {, j, l} of three networks such that (, j, l) (or one of ts permutatons (j, l, ), (l, j, ) etc.) s a trple n T, defne all of the followng bds to be equal to 1 : 3 zp (j, l), zp j (, l), zp l (, j), zs (j, l), z s (l, j), zj s (, l), zj s (l, ), zl s (, j), zl s (j, ). For every set {, j, l} of three networks such that no permutaton of (, j, l) s a trple n T,letallofthe above bds be equal to 1. In ths 3-NDBAA problem, we 6 ask: s there a channel allocaton of the m channels wth revenue of at least D = m? Itcanbeshownthattheanswer s yes f and only f the answer n the orgnal 3DM problem s yes. Ths shows that 3DM p 3-NDBAA and hence that 3-NDBAA s NP-Complete. By an analogous reducton from r-dmensonal Matchng, t can be shown that r-ndbaa s NP-Complete for fxed r > 3. Note that for r > 3, r-dmensonal Matchng s NP-Complete, whch follows from a trval reducton from 3-Dmensonal Matchng. Moreover, f r s unbounded, then each bdder would have to submt an exponental number of bds z p (j1,...,jr 1) andzs (j 1,...,j r 1). Also, we assumed that exactly r 1 networks are the secondary networks on a channel. If dfferent numbers of networks can be the secondary networks, then each network would have to submt an even greater number of bds. Now, suppose we relax Assumpton 2 and retan Assumpton 1. Then each network can become a prmary or secondary network on up to M channels. As explaned above, the utlty of a network from the prmary or secondary rghts

6 on a gven channel depends upon the channel usage behavor of the network t shares the channel wth. However, the channel usage behavor of ths network on the channel may n turn depend upon the number of channels on whch t has prmary and secondary rghts and the channel usage behavor of the networks t shares those channels wth and so on. Thus, n general, the utlty of a network may depend upon whch networks are the prmary and secondary networks on each channel. The number of possble ways of choosng the prmary and secondary networks on the M channels s clearly exponental. Thus, relaxng Assumpton 2 n the aucton wth dependent bds would requre a network to express an exponental number of bds, whch s nfeasble. 5. AUCTION WITH INDEPENDENT BIDS In Secton 4, we noted that when networks have some knowledge of the channel usage behavor of other networks, they would lke to express bds dependent on whch networks they wll share channels wth. However, t s qute possble n some scenaros that networks have no knowledge of the channel usage behavor of the other bddng networks. In ths case, ther conjectures about the utlty that they wll actually get from a channel allocaton would be based only on the number of channels on whch they wll get prmary and secondary rghts and the number of other networks they wll share these channels wth n the allocaton and would be ndependent of whch other networks they wll share channels wth. Thus, they would submt bds, based on these conjectured utltes, that are ndependent of whch networks share dfferent channels wth them. Moreover, n Secton 4.2, we showed that bds of exponental sze are needed n the aucton wth dependent bds when Assumptons 1 and 2 are relaxed. Ths motvates the dea that even when networks have some knowledge of the channel usage behavor of the other networks, we can obtan a compact bddng language, that s, a means for networks to approxmately convey ther bds, by mposng the restrcton that the bds of a network be ndependent of whch other networks t shares dfferent channels wth. We study the aucton resultng from mposng ths restrcton n ths secton. We descrbe the model n Secton 5.1 and provde an optmal dynamc-programmng algorthm for the access allocaton problem n Secton Model Consder the model n Secton 2 wth the followng addtons. On each channel, one network can be the prmary network and m 1, m 2,..., m (n 1) or m n networks can be the secondary networks, where 1 m 1 <m 2 <... < m n. Note that n s the number of possble cardnaltes of the set of secondary networks on a channel. When the results of the aucton are declared, let n,0 be the number of channels on whch bdder s the prmary network. Let n,j,j =1,...,n be the number of channels on whch bdder s a secondary network along wth m j 1 other secondary networks. Suppose there are m j secondary networks on a channel. Recall from Secton 2 that each of these m j secondary networks have equal rghts on the channel. The share of each of these networks n the secondary rghts on the channel s called a secondary part of type j. Also, the channel s sad to be dvded nto m j secondary parts of type j. Smlarly, snce exactly one network becomes a prmary network on a channel, f a network s the prmary network on l channels, we say that t s allocated l prmary parts. Also, were- fer to the throughput receved by a network as a secondary network as ts secondary throughput. In general, network s utlty may depend not only on the total expected secondary throughput that t gets, but also on the dstrbuton of ths secondary throughput over the M channels. For example, t may get the same expected secondary throughput f (a) t s the secondary network on two channels wth one other secondary network on each and (b) f t s the sole secondary network on one channel. But t may prefer one of these scenaros over the other. Ths s because a network has to sense dfferent channels on whch t has secondary rghts for ongong transmssons and also communcate on them. There may be costs due to delays for swtchng the antennas of the network s nodes between dfferent channels. To take nto account ths possblty, n ths secton, we assume that the utlty of network depends not just on the expected secondary throughput (and the number of prmary parts) t receves, but on the vector (n,0,n,1,...,n,n). We allow bdder to submt bds as a functon of ths vector. Each bdder submts the followng bd vector to the auctoneer: {z (n,0,n,1,...n,n) :0 n,0,n,1,...n,n M, n,1 + n, n,n M; n,j nteger,j =0, 1,...n} where z (n,0,n,1,...n,n) snetwork s bd for becomng the prmary network on n,0 channels and becomng a secondary network on n,j channels along wth m j 1other secondary networks, for j =1, 2,...n. The followng result can be easly proved. Lemma 1. The sze of the bd vector submtted by each network s O(M n+1 ). We say that an allocaton {n,j : =1,...,N; j =0,...,n} s feasble f t s possble to assgn to networks, the rghts to be prmary and secondary on each of the M channels such that network, =1,...,N s allocated n,0 prmary parts and n,j secondary parts of type j for j =1,...,n.Thefollowng lemma descrbes necessary and suffcent condtons for an allocaton to be feasble. Lemma 2. An allocaton {n,j : =1,...,N; j =0,...,n} s feasble f and only f n,0,n,1...n,n for =1,...,N are ntegers such that for some nonnegatve ntegers M j,j = 1,...n satsfyng M M n = M: 0 n,0 M, =1,...,N (4) N n,0 = M (5) =1 0 n,j M j,=1,...,n; j =1,...,n (6) N n,j = m jm j,j =1,...,n (7) =1 Note that the nteger M j n the above lemma corresponds to the number of channels that are dvded nto m j secondary parts of type j. We assume that the number of bdders s at least m 1 so that a feasble allocaton exsts.

7 From a feasble allocaton {n,j : =1,...,N; j =0,...,n}, t s easy to construct a consstent specfcaton of the prmary and secondary networks on each channel. Hence, the access allocaton problem reduces to fndng the feasble allocaton {n,j : =1,...,N; j =0,...,n} that maxmzes the auctoneer s revenue gven the submtted bd vectors z (.). 5.2 Algorthm to fnd the Optmal Feasble Allocaton In ths secton, we present an exact algorthm for fndng the feasble allocaton that maxmzes the auctoneer s revenue. The algorthm s polynomal-tme when n, thenumber of possble cardnaltes of the set of secondary networks on a channel, s fxed (and m n s allowed to grow wth the problem sze). Ths specal case can be useful n practce because even wth small n, flexblty n channel allocaton can be acheved by choosng m 1,..., m n judcously. For example, wth n =3,wecanchoosem 1 =1,m 2 =4andm 3 =8. In ths case, large chunks of secondary throughput can be allocated to a network by havng t the sole secondary network on several channels and small chunks can be allocated to networks by havng 4 or 8 networks share a channel. A dynamc programmng algorthm s gven n [12] and [11] for the wnner determnaton problem n a combnatoral aucton wth multple unts of a fxed number of dfferent types of objects. We generalze the algorthm n [12], [11] n two drectons: (a) the objects n a combnatoral aucton are ndvsble, whereas we need to decde nto how many secondary parts to dvde each channel and (b) n our aucton, the allocaton has to be feasble accordng to the condtons n Lemma 2. Due to space constrants, we only outlne our algorthm wthout gvng detals. Gven the bds z (.), our goal s to fnd the feasble allocaton {n,j : =1,...N; j =0,...,n} whch maxmzes revenue. Fx M 1,...,M n satsfyng M M n = M such that M j channels are dvded nto m j secondary parts of type j, forj =1,...,n. For these fxed values, let T (j 0,j 1,...j n,) denote the maxmum possble revenue from all partcpatng networks when j 0 prmary parts and j t secondary parts of type t, t = 1,...,n, are to be allocated and networks 1,..., are partcpatng n the aucton. Thus, T (M,m 1M 1,...m nm n,n)sthemaxmum revenue from all N networks when M j channels are dvded nto m j secondary parts of type j, forj =1,...,n and can be found usng the followng dynamc programmng algorthm. Intalzaton T (j 0,j 1,...j n, 1) = z 1(j 0,j 1,...,j n, 1) f j 0 M,j t M t,t=1,...,n = otherwse (8) Recurrence T (j 0,j 1,...,j n,)=max( T (j 0 l 0,j 1 l 1,...,j n l n, 1) + z (l 0,l 1,...,l n): l 0 {0, 1,...,mn(j 0,M)},l v {0, 1,...,mn(j v,m v)}, v =1,...,n) (9) It can be shown that T (M,m 1M 1,...m nm n,n) found from the above recurrence s the revenue of the feasble allocaton {n,j : =1,...N; j =0,...,n} that acheves the maxmum revenue for the fxed values M 1,...,M n assumed. Also, the revenue maxmzng feasble allocaton tself can be found from the array T (j 0,j 1,...j n,). For all sets M 1,...,M n such that M M n = M, T (M,m 1M 1,...m nm n,n) and the revenue maxmzng feasble allocaton are found as explaned above. Then the optmal set (M 1,...,M n) s found as follows: (M1,...,Mn) = argmax M M n=m T (M,m 1M 1,...m nm n,n) (10) The revenue maxmzng feasble allocaton wth M 1 = M1,..., M n = Mn s the one that maxmzes revenue among all feasble allocatons. Lemma 3. The runnng tme of the above algorthm s O(M 3n+2 m n nn). Lemma 4. The maxmum amount of storage requred at any gven tme durng the executon of the algorthm s O(M n+1 m n nn). Note that the runnng tme and space complexty of the algorthm are polynomal for fxed n. 6. A GREEDY 2-APPROXIMATION ALGORITHM The scheme descrbed n Secton 5 s feasble for fxed n, the number of possble cardnaltes of the set of secondary networks on a channel. However, f n s allowed to grow, the set of bds of a network s exponental n sze as Lemma 1 shows and hence the scheme s nfeasble. In ths secton, we frst provde a compact bddng language for the case wth large n,.e. a means for networks to approxmately convey ther bds. We conjecture that under ths bddng language, the access allocaton problem s NP-hard. We gve a bass for ths conjecture n Secton 7. We provde a polynomal-tme algorthm that approxmates the maxmum revenue of the auctoneer wthn a factor of 2. Note that ths algorthm can also be used to approxmate the maxmum socal welfare of the bddng networks wthn a factor of 2 usng the extended second-chance mechansm descrbed n Secton 3. We descrbe the bddng language n Secton 6.1. In Secton 6.2, we ntroduce resdual bd functons, a concept used n the approxmaton algorthm. We descrbe the algorthm n Secton 6.3 and prove that t acheves an approxmaton rato of 2 n Secton Bddng Language Consder the model n Secton 5 wth the followng changes. Let the bandwdth of each of the M channels be W bps. We assume that the prmary network on a channel uses the channel for an expected fracton of tme α, where 0 <α<1. When auctons are repeated perodcally to assgn spectrum, α can be estmated based on long-term measurements of the prmary networks channel usage. Alternatvely, t can be estmated va smulatons. Snce secondary networks can use the channel whenever the prmary s not usng t, an expected bandwdth of W (1 α) s avalable on a channel for the secondary networks. So when m j secondary networks

8 share a channel, each one of them can get an expected secondary throughput of W (1 α) m j on the channel. In ths secton, we allow a network to express bds as a functon of the number of channels n,0 on whch t s prmary and ts total expected secondary throughput T s on all M channels. Note that: n T s n,jw (1 α) = (11) m j j=1 In the sequel, for brevty, we smply say secondary throughput nstead of expected secondary throughput. Moreover, we assume that the utlty, and hence the bd z (n,0,t s ), of each network when t s prmary on n,0 channels and has T s unts of secondary throughput, s separable,.e. of the form: z (n,0,t s )=w (n,0)+y (T s ) (12) where w (n,0) s ts bd for beng prmary on n,0 channels and y (T s ) s ts bd for T s unts of throughput as a secondary network. Ths assumpton s a good approxmaton snce networks transmt dfferent knds of traffc (delay-senstve and elastc respectvely) as a prmary and secondary network. Under ths assumpton, the access allocaton problem separates out nto two ndependent problems allocatng the prmary parts and allocatng the secondary parts. The problem of allocatng the prmary parts can be optmally solved n O(M 2 N) tme usng the dynamc programmng algorthm n Secton 5.2 wth n = 0. In ths secton, we focus on gvng a 2-approxmaton algorthm for the problem of allocatng the secondary parts so as to maxmze the auctoneer s revenue. In the rest of the secton, revenue refers to the auctoneer s revenue from sellng the secondary rghts on the M channels. Assume that y (.) s a concave ncreasng functon for each network. We use pecewse lnear concave functons to compactly represent the bd functons of the networks. They can be used to closely approxmate arbtrary concave functons [16] and have been prevously used n the context of spectrum auctons n [9]. Each network specfes ts bd for at most P dfferent levels of secondary throughput, for a postve nteger P.Moreprecsely,letP P be a postve nteger and let: 0=q,1 <q,2 <...<q,p (13) For v =1,...,P,network specfes y (q,v), whch s ts bd for q,v unts of secondary throughput. Network s bd for q unts of secondary throughput, where q,v <q<q,v+1 s found by lnear nterpolaton: ( ) y(q,v+1) y (q,v) y (q) =y (q,v)+ (q q,v) (14) q,v+1 q,v We assume that for each network, q,1 =0,thaty (q,1) = y (0) = 0 and that q,p MW(1 α). (15) Snce MW(1 α) s the total secondary throughput avalable on the M channels, the second assumpton means that network s bd for any amount of secondary throughput on the M channels can be found by lnear nterpolaton. 6.2 Resdual Bd Functons Our algorthm uses the followng concept. Defnton 2. Let q 0. The q-resdual bd functon of network s the functon ỹ (.) gven by: ỹ (q) =y ( q + q) y ( q) (16) We wll sometmes say, the resdual bd functon after accountng for q nstead of the q-resdual bd functon. Informally, once network has been allocated q unts of secondary throughput, ỹ (.) acts as ts bd functon for allocatons of addtonal secondary throughput. It can be shown that the resdual bd functon can be effcently computed from the bd functon. We omt the proof due to space constrants. The followng lemma gves some smple propertes about the q-resdual bd functon. Lemma 5. Let ỹ (q) be the q-resdual bd functon of network for some q 0. Then 1. ỹ (q) y (q) q ỹ (q) s a concave ncreasng functon. The sgnfcance of the q-resdual bd functon s gven by the followng lemma. Lemma 6. Suppose the bd functon of network s y (.) and t s successvely allocated secondary throughputs of q 1,q 2,...,q f.lety v (.) denote the (q q v)-resdual bd functon of network, forv =1,...,f.Then y (q q f )=y (q 1)+y 1 (q 2)+...+ y f 1 (q f ) (17) Thus, the sgnfcance of the resdual bd functon s that f anetwork s successvely allocated chunks q 1,..., q f of secondary throughput (e.g. by successve steps of an algorthm), then we can keep track of ts resdual bd functon after every allocaton so that the extra money that network s wllng to pay for the v th allocaton q v s smply y v 1 (q v). Moreover, ths trackng can be done usng the update rule n part 1 of the followng lemma to calculate y v+1 (.) from y v (.). Lemma 7. Let ỹ (.) and y + (.) be the q-resdual bd functon and ( q +ˆq)-resdual bd functon of network respectvely. Then 1. y + (q) =ỹ(q +ˆq) ỹ(ˆq) q 0 2. y + (q) ỹ(q) q 0. Note that y + (.) stheˆq-resdual bd functon correspondng to the bd functon ỹ (.). 6.3 Algorthm Descrpton We now descrbe the 2-approxmaton algorthm. For each network, the resdual bd functon ỹ (.) s ntalzed to y (.). The algorthm successvely determnes nto how many secondary parts channel l s to be dvded and whch networks are to be the secondary networks on that channel, for l = 1,...,M, one channel at a tme. Suppose we have taken these decsons for channels 1, 2,...,l 1andforeachnetwork, havesetỹ (.) to be equal to ts resdual bd functon after accountng for the secondary throughput allocated to t n the frst l 1 channels. Assgn channel l usng the followng steps: STEP1: For j =1,...,n, fnd the maxmum ncrease n revenue Rj l obtanable from channel l by dvdng the channel

9 nto m j secondary ( parts usng ) the followng rule. Sort the set of numbers ỹ W (1 α) m j, =1,...,N nto decreasng ( ) order. Let ỹ W (1 α) (v) m j denote the v th largest element. Then Rj l s gven by: m j ( ) W (1 α) Rj l = ỹ (v) m j v=1 STEP2: Fnd the maxmum among R1,...,R l n. l Suppose Rj l s the maxmum. Then dvde the l th channel nto m j secondary parts. On the l th channel, the m j networks wth m j ) ( W (1 α) m j ) be- ( the m j largest values ỹ W (1 α) (1),...,ỹ (mj ) come secondary networks. STEP3: Update the functon ỹ (.) of each one of the m j networks that become the secondary networks on channel l to ts resdual bd functon after accountng for the secondary throughput allocated to t n the frst l channels Comments on Algorthm Once channels 1,...,l 1 have been allocated, steps 1 and 2 allocate channel l so as to obtan the maxmum possble ncrease n revenue over the revenue from channels 1,...,l 1. In Step 3, the rule n part 1 of Lemma 7 can be used to update the resdual bd functons of the networks who become secondary networks on channel l. 6.4 Approxmaton Rato Theorem 3. Let R be the maxmum possble revenue under any allocaton of the rghts to be secondary networks on the M channels and let R G be that acheved by the above greedy algorthm. Then R G R. 2 Proof. Let R l be the ncrease n revenue obtaned by the greedy algorthm from allocatng the l th channel. Denote by q,l, G the amount of secondary throughput allocated by the greedy algorthm to network n the l th channel. Let y(.) l bethe(q,1 G +...+q,l)-resdual G bd functon of network, that s, ts resdual bd functon after accountng for the secondary throughput allocated to t n channels 1 to l. By part 2 of Lemma 7: y(q) l y l 1 (q) q 0 (18) From the dscusson after Lemma 6, t follows that after channels 1,...,l were allocated, the extra money network s wllng to pay for ts share n channel (l +1) s y(q l,l+1). G Moreover, f the greedy algorthm were to allocate the l th channel to the same set of networks to whom t actually allocated the (l+1) st channel, then (a) after channels 1,...,l 1 were allocated, the extra money network would be wllng to pay for ts share n channel l would have been y l 1 (q,l+1) G and hence by (18), (b) the ncrease n revenue from the l th channel would have been at least R l+1. But the actual ncrease n revenue from the l th channel, R l, s by defnton of the greedy rule, the maxmum possble from allocatng the l th channel. Hence R l R l+1.thus,weget: R 1 R 2... R M Snce R G = R R M,weget: R M RG M (19) Now, let q be the total secondary throughput allocated by the optmal algorthm to network and q G be that allocated by the greedy algorthm. Also, let Sl be the set of secondary networks on the l th channel, l =1,...,M,nthe optmal allocaton. Next, we wll upper bound R R G,the excess revenue of the optmal allocaton over the greedy allocaton. To ths end, for each network, we account for ts payment for max(q q G, 0), the excess secondary throughput f any, of the optmal allocaton over the greedy algorthm s allocaton, by accountng for ts payments for the chunks q,l,l e =1,...,M. Here, q,l e s the contrbuton of channel l to the excess max(q q G, 0), once the contrbutons of channels 1,...,l 1 have been accounted for and s gven by: ( W (1 α) q,l e =mn Sl, ) max(q q G q,1 e... q,l 1, e 0), Sl (20) q e,l =0, / S l (21) We get the expresson n (20) as follows. Frst, snce channel l s shared by Sl networks, q,l e W (1 α) S l. The second term n the mn s equal to the as yet unaccounted for excess, f any, obtaned by subtractng the contrbutons q,1,...,q e,l 1 e of channels 1,...,l 1 from the total excess throughput max(q q G, 0). Let y,l(.) e bethe(q G +q,1 e +...+q,l)-resdual e bd functon of network. That s, y,l(.) e s the resdual bd functon after accountng for the amount of secondary throughput allocated to network by the greedy algorthm (q G )andthe contrbutons q,1,...,q e,l e of the frst l channels to the excess max(q q G, 0). Now, R R G M y,l 1(q e,l) e (22) Sl l=1 M y M 1 Sl l=1 (q e,l) (23) M ( ) y M 1 W (1 α) S Sl l (24) l=1 M R M (25) l=1 = MR M (26) ( ) R G M (27) M = R G (28) from whch t follows that R 2R G. We get (22) as follows. The excess revenue R R G arses from the extra payment that each network makesfortheexcesssecondary throughput max(q q G, 0), f any, that t receves under the optmal allocaton over the greedy allocaton. Expresson (22) accounts for these extra payments by addng, for channels l =1,...,M, the total payments (see the dscusson after Lemma 6) of all networks S l for the contr-

10 butons q,l e of channel l to the excess secondary throughput max(q q G, 0). There s an nequalty nstead of equalty n (22) because q G may be greater than q for some networks. Inequalty (23) follows by part 2 of Lemma 7, snce (.) s the resdual bd functon of network after accountng for the amount of secondary throughput allocated to t by the greedy algorthm n the frst M 1 channels, whereas y,l 1(.) e s the resdual bd functon after accountng for the secondary throughput allocated to t n all M channels as well as the contrbutons q,1,...,q e,l 1 e of the frst l 1 channels to the excess max(q q G, 0). Inequalty (24) follows from the fact that q,l e W (1 α) and snce S l y M 1 (.) s ncreasng by part 2 of Lemma 5. In nequalty (25), we use the fact that ( ) W (1 α) Sl R M (29) y M 1 y M 1 S l whch s true because when the greedy algorthm was about to allocate channel M, the ncrease n revenue t would have got from the channel f t allocated the channel to the Sl networks n the set Sl s equal to the expresson on the left hand sde of (29). Ths expresson s at most R M, snce the greedy algorthm allocates the M th channel so as to maxmze the ncrease n revenue from t. Fnally, nequalty (27) follows from (19). Lemma 8. The runnng tme of the above greedy algorthm s O(nMN log(pn)+mpm n). 7. CONCLUSIONS AND FUTURE WORK We developed a comprehensve framework for access allocaton n cogntve rado networks. We proposed three bddng languages, each less expressve than the prevous one, but sutable for larger auctons. Also, we developed a number of algorthms for the access allocaton problem for these bddng languages. In Secton 6, we gave a polynomal-tme 2-approxmaton algorthm for the problem, descrbed n Secton 6.1, of allocatng secondary rghts so as to maxmze the auctoneer s revenue. We conjecture that t s NP-hard to solve t optmally. Our conjecture s motvated by the facts that (a) the bd functon of each network can be an arbtrary realvalued functon satsfyng the condtons n Secton 6.1, (b) the number of secondary networks on each channel can be selected from a possbly large set {m 1,...,m n} and (c) the set of secondary networks on each channel can be an arbtrary subset of the set of all N networks. We leave the queston of NP-hardness as an open problem for future research. Also, we consdered the case when the M channels are dentcal. The extenson to non-dentcal channels can be consdered as part of future work. 8. ACKNOWLEDGMENTS The contrbutons of both authors have been supported by NSF grants NCR , CNS , ECS [2] FCC Spectrum Polcy Task Force Report of the Spectrum Effcency Workng Group. Nov Avalable at: [3] FCC Auctons [4] I. Akyldz, W.-Y. Lee, M. Vuran, S. Mohanty NeXt generaton/dynamc spectrum access/cogntve rado wreless networks: a survey. In Computer Networks, Vol. 50, Issue 13, pp , [5] D. Bertsekas, R. Gallager, Data Networks. Prentce Hall, Englewood Clffs, New Jersey, second edton, [6] Z. J, K.J. Ray Lu, Belef-Asssted Prcng for Dynamc Spectrum Allocaton n Wreless Networks wth Selfsh Users. InProc.ofIEEESECON, [7] J. Huang, R. Berry, M. Hong Aucton Mechansms for Dstrbuted Spectrum Sharng. In Proc. of 42nd Allerton Conference, [8] S. Sengupta, M. Chatterjee Sequental and Concurrent Aucton Mechansms for Dynamc Spectrum Access. In Proc. of CROWNCOM, [9] S. Gandh, C. Buragohan, L. Cao, H. Zheng, S. Sur, Towards Real-Tme Dynamc Spectrum Auctons. In Computer Networks, Vol. 52, Issue 4, March 2008, Pp [10] J. Peha, Emergng Technology and Spectrum Polcy Reform In Proc. of ITU Workshop on Market Mechansms for Spectrum Management, Jan [11] P. Cramton, Y. Shoham, R. Stenberg, Combnatoral Auctons. MIT Press, 2006 [12] M. Tennenholtz, Some Tractable Combnatoral Auctons. Proc. of Natonal Conference on Artfcal Intellgence (AAAI), 2000 [13] J. Edmonds Maxmum Matchng and a Polyhedron wth 0,1-Vertces. In Journal of Research of the Natonal Bureau of Standards, 69B, , [14] L. Whte A Parametrc Study of Matchngs and Coverngs n Weghted Graphs Ph.D. Dssertaton, Unversty of Mchgan, Ann Arbor, [15] L. Whte An Effcent Algorthm for Maxmum k-matchng n Weghted Graphs. In Proc.of12th Allerton Conference on Crcuts and Systems Theory, [16] D. Bertsmas and J. Tstskls Introducton to Lnear Optmzaton. Athena-Scentfc, [17] A. Mas-Colell, M. Whnston and J. Green Mcroeconomc Theory. Oxford Unversty Press, [18] J. Klenberg and E. Tardos Algorthm Desgn. Addson Wesley, [19] N. Nsan and A. Ronen Computatonally Feasble VCG Mechansms. In Proc. of ACM Conference on Electronc Commerce, [20] A. Ronen Mechansm Desgn wth Incomplete Languages. In Proc. of ACM Conference on Electronc Commerce, REFERENCES [1] Q. Zhao and B. Sadler A Survey of Dynamc Spectrum Access. In IEEE Sgnal Processng Magazne, Vol. 24, Issue 3, pp , 2007.