Efficient and Strategyproof Spectrum Allocations in Multi-Channel Wireless Networks

Transcription

1 Efficient and Strategyproof Spectru Allocations in Multi-Channel Wireless Networks Ping Xu, Xiang-Yang Li, Senior Meber, IEEE, ShaoJie Tang, and JiZhong Zhao Abstract In this paper, we study the spectru assignent proble for wireless access networks. We assue that each secondary user will bid a certain value for exclusive usage of soe spectru channels for a certain tie period or for a certain tie duration. A secondary user ay also require the exclusive usage of a subset of channels, or require the exclusive usage of a certain nuber of channels. Thus, several versions of probles are forulated under various different assuptions. For the ajority of probles, we design PTAS or efficient constant-approxiation algoriths such that overall profit is axiized. Here the profit is defined as the total bids of all satisfied secondary users. As a sideproduct of our algoriths, we are able to show that a previously studied Scheduling Split Interval Proble (SSIP) [], in which each job is coposed of t intervals, cannot be approxiated within O(t ɛ ) for any sall ɛ > 0 unless NP=ZPP. Opportunistic spectru usage, although a proising technology, could suffer fro the selfish behavior of secondary users. In order to iprove opportunistic spectru usage, we then propose to cobine the gae theory with wireless odeling. We show how to design a truthful echanis based on all of these algoriths such that the best strategy of each secondary user to axiize its own profit is to truthfully report its actual bid. Index Ters Wireless networks, spectru, disk graph, interval graph, PTAS, approxiation, strategyproof. I. INTRODUCTION Wireless technology is expected to play a bigger and ore fundaental role in the new Internet than it has today. The radio frequency spectru has been chronically regulated with static spectru allocation policies since the early 0th century. With the recent fast growing spectru-based services and devices, the reaining spectru available for future wireless services is being exhausted, known as the spectru scarcity proble. The current fixed spectru allocation schee leads to significant spectru white spaces where any allocated spectru blocks are used only in certain geographical areas and/or in brief periods of tie. A huge aount of precious spectru (below 5GHz), perfect for wireless counications P. Xu and S.-J. Tang are with Departent of Coputer Science, Illinois Institute of Technology, Chicago, IL, USA (e-ail: {pxu3,stang7}@iit.edu). X.-Y. Li is with Institute of Coputer Application Technology, Hangzhou Dianzi University, Hangzhou, 3008, PRC. He is also with Departent of Coputer Science, Illinois Institute of Technology, Chicago, IL, USA (e-ail: xli@cs.iit.edu). He is a visiting/adjunct professor of Xi An JiaoTong University. J. Zhao are with Departent of Coputer Science and Tech., Xi an Jiaotong Univ., China, Eail: zjz@ail.xjtu.edu.cn. The research of authors are partially supported by NSF CNS-0830, National Natural Science Foundation of China under Grant No , the Natural Science Foundation of Zhejiang Province under Grant No.Z080979, National Basic Research Progra of China (973 Progra) under grant No. 00CB3800, the National High Technology Research and Developent Progra of China (863 Progra) under grant No. 007AA0Z80. that is worth billions of dollars, sit there silently. Recognizing that the traditional spectru anageent process can stifle innovation, and it is difficult to provide a certain quality of service (QoS) for systes operated in unlicensed spectru, the FCC has proposed new spectru anageent odels and the use of a easure of interference teperature. Current spectru anageent ethods include coand and control (e.g., for public safety), exclusive usage based on license (e.g., for cellular counication), coons (e.g., ISM bands), interference teperature (also called opportunistic usage), and fast coand and control [4]. One proising technology is the opportunistic spectru usage. In opportunistic spectru usage, the secondary users observe the channel availability dynaically and explore it opportunistically. Secondary users are cognitive devices that can sense the environent and adapt to appropriate frequency, power, and transission schees. They can opportunistically access unused spectru vacated by idle priary users, who have strict priority on spectru access and will share spectru with others under certain protections. While opportunistic spectru has several advantages, it suffers fro the selfish behavior of the secondary users. For secondary users, usually they are selfish. Obtaining ore spectru could ean ore benefit econoically. Therefore, it is difficult, if not possible, to require each secondary user faithfully confor to a distributed channel assignent ethod that ay reduce its own chance to use the spectru, for the overall syste benefit. It is then reasonable and appropriate to odel the as rational agents. Thus, we propose to cobine the gae theory [0], specifically, echanis design theory with wireless counication odeling. More specifically, we study how to share the spectru and how to charge the secondary users such that the overall social benefit or profit by spectru owner is axiized even in the presence of selfish behavior. In this paper, we assue that each secondary user v i will bid a value b i for usage of soe channels vacated by the priary users. A secondary user ay also iply (by its locations) or specify soe additional constraints for the usage of the spectrus, such as the region where it will use the channels. The first constraint is where the spectrus will be used. We assue that each user v i will specify a two-diensional space requireent, which is typically a disk D(v i, r i ) centered at node v i with a radius r i. In other words, secondary user v i wants an exclusive usage of F i inside the disk D(v i, r i ). Notice that the requireent that the space required by each secondary user is a disk is iaterial to results presented in this paper. Our results will hold as along as regions required by all secondary users have siilar size and constant-bounded

2 aspect ratio. Here the aspect ratio of a region (without hole) is defined as its area divided by the square of its perieter s length. For siplicity, all our proofs assue that the space requireent by a secondary user is a disk with unit radius. The second constraint is the tie constraint which specifies a tie interval [s i, e i ] or a tie duration d i that node v i wants an exclusive usage of F i for that tie interval or duration. In other words, generally, we assue that for a bundle F i F of channels, a space requireent specified by a disk D(v i, r i ), and a tie constraint (tie interval [s i, e i ] or a duration d i ) user v i places a bid b i when it is single-inded. Otherwise, it could bid b i,j for each channel f j separately with possible different tie constraints for each channel. The central authority will design a echanis which, upon receiving bidding requests fro different secondary users, coputes how to share the channels aong the secondary users, and coputes a schedule of channels for the secondary users subject to the tie-space restriction: no two secondary users that are interfered with each other are given the sae channel at any tie. Last but not the least, the echanis should copute how uch we should charge the secondary users. The difficulty of this proble arises fro two aspects. Firstly, the correlation of the tie and space aong requests fro secondary users introduces high coplexity copared with traditional auctions [3], [6], [8], [6], [], [5]. The proble is NP-coplete by the reduction fro the proble the axiu weighted independent set of disk graphs even the tie requireents of all agents over all channels are the sae or fro the set packing proble even when all nodes are co-located. Interference along already akes nuerous related probles untractable, not to ention the scheduling of the channels. The ain contributions of this paper are as follows. First we design efficient algoriths to allocate channels in different cases such that the social efficiency (defined as the total valuations of all satisfied bidders) are approxiately axiized. Specifically, we design a polynoial tie approxiation schee (PTAS) for axiizing the social efficiency when all secondary users are co-located, not single-inded, but could have ixed tie requireents. Here a secondary user is singleinded if it requests a subset of channels and the assignent is valid only if the exact subset of channels is allocated to this user at the exact requested tie. All users are co-located if the space requireents of all users overlap at soe point. We design PTAS for axiizing the social efficiency when () all secondary users are not single-inded (they can be allocated any nuber of channels fro their required channels), and () the tie-requireent by each secondary is specified by a tie-interval. Constant approxiation algoriths are designed when users are not single-inded but with possible ixed tie-requireents. When users are single-inded and each user will bid for a subset of channels, the proble does not have an approxiation ratio within / ε for any ε > 0, unless NP=ZPP, ZPP (Zero-error Probabilistic Polynoial tie) is the coplexity class of probles for which a probabilistic Turing achine exists with these properties: () It always returns the correct YES or NO answer; and () The running tie could be unbounded, but is polynoial on average for any input. where is the total nuber of channels in F [9], [6]. In this case, we then design approxiation algoriths with ratio Θ( ) when the space requireents by all secondary users are unit disks. As a side-product of our algoriths, we are able to show that a previously studied Scheduling Split Interval Proble (SSIP) [] cannot be approxiated within O(t ɛ ) for any sall ɛ > 0 unless N P = ZP P. There is a t approxiation ethod for SSIP []. Based on these approxiation algoriths, we then design strategyproof echaniss to charge the secondary users such that, under our echaniss, each secondary user will axiize its own profit if it truthfully reported its own bid, no atter what others will do. We essentially show that our approxiation algoriths satisfy a onotone property: if a secondary user is satisfied with bid b i then it will still be satisfied with a bid b i > b i while the bids of all other secondary users reain the sae. Then a strategyproof echanis can always be designed, in couple with our algoriths with onotone property, based on results fro [3]. Notice that here we focus on strategyproof echanis solution concept to address the noncooperative gae aong secondary users. A nuber of other interesting solution concepts could be used to study these gaes, such as Nash equilibria. The rest of the paper is organized as follows. In Section II, we define in detail the probles to be studied in this paper. Then we review related results on those spectru assignent probles in Section III. Fro Section IV to Section VII, we discuss algoriths for several versions of probles described in Section II. In section VIII, we show how to design strategyproof echaniss based the algoriths we designed. And we do siulations in section IX to study the perforance of our algoriths in experient. At last we conclude the paper in Section X with the discussion of soe possible future works. A. Network Model II. PRELIMINARIES Consider a wireless network syste fored by soe priary users U = {u, u,, u p } who hold the right of soe spectru channels, secondary users V = {v, v,, v n } who want to lease the right to use soe channels in soe region for soe tie period. In certain applications, each secondary user v i ay provide service to κ i clients Z i = {z i,, z i,,, z i,κi } within a geoetry region. Here u j, v i and z i,k also represent the fixed two-diensional geoetry location of these users. For siplicity, we treat all priary users as one unified central authority. Let F = {f, f,, f } be the set of frequencies that can be used by soe secondary users for a given tie interval [0, T ]. For soe wireless network systes, it is possible that the priary users will only lease a spectru frequency for a certain tie interval in a certain geographical region. If this is the case, we assue that for each f i F, we associate it with a region Ω i and a set of tie intervals T i that it is available. We assue that every channel will be available everywhere and at all tie slots. Our results can easily deal with a general Ω i and T i. We divide the tie into ultiple tie intervals and ake allocation on each interval at the beginning of that interval.

3 3 New secondary users ay join the syste. But they cannot lease channels until next allocation begins. Existing secondary users ay leave the syste at anytie. Observe that the total tie duration required by a user ay last for several tie intervals. Thus, at the beginning of each tie-interval, soe cobination of channels and locations could be hold by users fro previous allocations. Our channel allocation schees will not use these channels at soe specific regions. We assue that a secondary user v i ay wish to lease a set of channels F i F. For a bidding, the secondary user will also specify two additional constraints: space requireent and tie condition. We assue that each user v i will specify a twodiensional region which is typically a disk D(v i, r i ) centered at node v i with a radius r i. In other words, secondary user v i wants an exclusive usage of F i inside the disk D(v i, r i ). Additionally, user v i also specifies a tie interval [s i, e i ] or a tie duration d i that node v i wants an exclusive usage of F i for that tie interval or duration. Here it is assued that 0 s i < e i T and 0 < d i T. A tie interval requireent iplies that the secondary user wants the exclusive usage of the channels F i fro tie s i to tie e i. A tie duration requireent d i iplies that the secondary user wants the exclusive user of channels for a tie that lasted continuously for d i tie units. It can be started fro any tie. Generally, we use T i to denote the tie constraint of user v i, where T i is either [s i, e i ] or a scalar d i > 0. Two different odels of secondary users will be studied in this paper. The first odel assues that every secondary user is single-inded: when a secondary user v i bids for F i, the valuation of v i over an assignent is 0 if the assignent does not satisfy all requireents and not all frequencies in F i are allocated. The secondary user v i will be called flexible if it will pay the central authority based on the frequencies allocated. For a flexible user v i, we assue that for each channel f j F i, user v i will bid b i,j for the right to use the channel f j for a certain tie requireent and geographical requireent. In this case, we use b i = {b i,, b i,,, b i, } to denote the bid vector of user i, where b i,j = 0 if v i did not bid for f j. Thus, a bidding by a user v i will be written as follows B i = [b i, F i, D(v i, r i ), T i ] Upon receiving the bids fro secondary users, the central authority decides an allocation ethod ) An allocation ethod X = {x, x,, x n } where x i {0, } denotes whether user v i s bid will be satisfied, and also a tie-interval [s i, e i ] with e i s i = d i when user v i required a tie-duration d i in the bid B i. ) A payent schee P = {p, p,, p n } where p i denotes how uch user v i should pay. The allocation ust be conflict free aong satisfied bids. Here two bids B i and B j conflict if F i F j, D(v i, r i ) D(v j, r j ), and [s i, e i ] [s j, e j ]. The objective of an allocation is to axiize i= x ib i. For siplicity, given a set of bids Y, we use ω(y ) to denote the total weight of bids in Y, i.e., B b i Y i. B. Introduction to Gae Theory and Mechanis Design A gae includes a finite set of players (decision akers), a set of actions and a set of utility functions that players wish to axiize. Other gaes ay include additional coponents, such as the inforation available to each player and counication echaniss. For exaple, in a sequential gae, one player akes his decision before others do so and the others ay have soe inforation of the first decision which will affects their strategies. And in a repeated gae, players are allowed to observe the actions of the other players, reeber past actions, and attepts to predict future actions of players. Traditional applications of gae theory attept to find equilibria in these gaes. In an equilibriu each player has adopted a strategy that they do not want to change. And any equilibriu concepts, for exaple, Nash equilibriu, have been developed. Equilibriu analysis ainly attepts to atheatically capture behavior in strategic situations, in which an individual s success in aking decision depends on the decision of others. On the other hand, echanis design, another branch of gae theory, is the study of designing rules of a gae to achieve a specific outcoe [7], [8]. Unlike equilibriu analysis, echanis design ainly focus on how to design rules to achieve a axiized social benefit even though each user is selfish and how to design a echanis such that selfish users have incentive to follow the echanis truthfully. In a word, echanis design uses the techniques of gaes theory to develop rules for a gae. A standard odel for echanis design is as following. There are n agents,,, n. Each agents i has soe private inforation t i, which is called type, and a set of strategy A i fro which it can choose. For each input vector a = (a, a,, a n ) where a i A i is the strategy played by agent i, the echanis M = (O, P ) coputers and output o = O(a) and a payent vector p(a) = (p (a), p (a),, p n (a)). Here the payent p i (a) is the oney we charge agent i depending on the strategies vector a. In our case, each secondary user v i is an agent i. The valuation of request by v i is private type t i. Moreover, bid b i, frequencies F i and region D(v i, r i ) are the strategy played by secondary user v i. We need to design a strategyproof echanis M = (O, P ), i.e., for each secondary user, reporting its valuation truthfully will axiize its profit. Strategyproof echaniss should satisfy both incentive copatible (IC) and incentive rational (IR) properties. An incentive copatible echanis is a echanis in which an agent will axiize its utility by reporting its private type truthfully. An individual rational echanis is a echanis in which the utilities of the agents participating in the output is not less than that of those no participating in the output. In Section VIII, we will show how to design strategyproof echaniss based on the algoriths we designed. C. Probles Forulation In this paper, we study several versions of spectru assignent proble by separately assuing whether the secondary users are single-inded or not, whether their required regions

4 4 D(v i, r i ) overlap or not, whether all secondary users ask for a fixed tie-interval, or all users ask for soe duration, or the tie-requireents are ixed aong users. Then we design strategyproof echaniss based on those approxiation algoriths. Based on those algoriths and perforance analysis, the network syste could decide an assuption cobination which is feasible in practice and results in ore benefits theoretically. For notational convenience, we use CRT to denote a proble, where C denotes choice of spectrus by secondary users. Here C will be either S (denoting that soe secondary users require a subset of channels and they are single-inded), or F (denoting that every secondary user will bid separately for each channel and is flexible), or Y (denoting that there is only one channel available in the syste, i.e., = ). R denotes region requireent by secondary users. Here R will be either O (denoting that the required regions overlap at soe single point) or U (denoting that the required regions are denoted by unit disks in D). T denotes tie requireent by secondary users. Here T will be either I (denoting that the required tie by every user v i will be an interval [s i, e i ]) or D (denoting that the required tie by every user v i will be a duration d i ) or M (denoting that soe users required a tie interval and soe users required a tie duration). For exaple, proble SUI represents the case that each user v i will bid for a subset of channels F i and is single-inded, will require a unit disk region D(v i, ), and a fixed tieinterval [s i, e i ]. For each proble where every secondary user will bid separately for each channel (called proble with C = Y), we have a corresponding C = Y proble when considering each user requires k channels as k duplicated users that each of the requires channel. Therefore, we don t discuss these C = F probles as they are special cases of C = Y probles. Notice that each version of proble includes two parts, the allocation ethods and payent schee as entioned in subsection II-A. In rest of this paper, we ainly focus the allocation ethods. Therefore, when we ention a version of proble, we indicate its allocation proble without special explanation. We would like to point out soe relations between the coplexity of different probles. For any proble in which users have a ixed tie requireent, i.e., probles in the forat T=M, it can be solved if the corresponding probles in the forat T=D and T=I are solved. For exaple, proble SUM can be solved if proble SUD and proble SUI are solved. More details and perforance analysis will be given in section IV. Soe versions of the probles turn out to be soe wellstudied probles in the literature and soe well-studied probles turns out to be a special case of the above probles. Proble YOI is essentially to copute the axiu weighted independent set in interval graphs, which has a well-known polynoial tie algorith by dynaic prograing. Proble YOD is essentially a knapsack proble, which has a wellknown polynoial tie approxiation schee (PTAS) [5]. In this paper, we will ainly focus on the probles YOM, YUI, YUD, and SUI. Probles YUM can be solved by algorith based on our ethods for proble YUI and YUD. Also observe that SOI, SOD, SOM are special cases of SUI, SUD, SUM respectively. And SUM can be solved based our ethod for SUI and a ethod for SUD. Thus, the only challenging question that is left unsolved is SUD. III. LITERATURE REVIEWS In this section, we briefly review literature related to the probles studied in following sections. The allocation of spectrus to users is essentially the cobinatorial auctions, which have been well-studied in the literature and a nuber of algoriths [6], [] have been proposed. And gae theory is also applied widely in resource anageent probles [], [4]. Specifically, the probles we will discuss are at the intersection of a lot of faous probles, such as knapsack, rectangle packing, set packing, cobinatorial auctions, and so on. Here we review results for soe of these probles. Knapsack proble, which is sae as siple proble YOD in our definition, has a classic fully polynoial tie approxiation schee (FPTAS) by the eans of rounding [5], []. However, we cannot design a strategyproof echanis using this FPTAS since it is not onotone. An alternative rounding schee was proposed by Patrick Briest in [4], which gave a new rounding schee leading to a onotone FPTAS for knapsack proble. Rectangle packing proble is that, there is a set of sall rectangles with specific profits and a big rectangle, we try to packing sall rectangles into the big rectangle with axial total profit and no overlap aong sall rectangles. In [], Jansen and Zhang presented a (+ɛ)- approxiation algorith for rectangle packing proble which is siilar with our proble YUI. Specifically, unit-height rectangle packing proble is a special case of YUI if we don t consider intersections in space. Kovaleva described a PTAS for unit-height rectangle packing proble in [5]. We extend this PTAS to a PTAS for proble YUI as described in following section. For the proble SUI such that secondary users require a subset of channels and they are single-inded, it is a special case of set packing proble. In [9], Hastad proved that set packing proble cannot be approxiated within ɛ, unless NP = ZP P. Another special case of set packing proble is Scheduling Split Intervals Proble (SSIP), which considers how to schedule weighted jobs, where each job is given as a group of non-intersecting t segents on the real line. Bar- Yehuda et al. [] gave an algorith with t-approxiation ratio. We show how to convert the proble SUI to SSIP and our ethod achieves an approxiation ratio, which is asyptotically tight, in following sections. In this paper, we also discuss how to design truthful echanis based on our algoriths to copute payent schee. The field of echanis design [7], [8] ais to study how privately known preferences of any people can be aggregated towards a social choice. Soe tools of echanis design [7], [9] were applied to optiization probles that involve selfish participants, which is siilar to our proble.

5 5 When users arrive in an online fashion and the syste need ake a decision iediately or within a tie-liit, recently Xu et al. [6], [7] developed soe echaniss that can ensure certain truthfulness and perforance guarantee under soe assuptions. Using soe concepts of Myerson auction, Jia et al. [] designed truthful spectru auction for spectru access to (approxiately) axiize the revenue. IV. ALGORITHM FOR PROBLEM YOM In this section, we design an approxiation algorith for proble YOM, which is equivalent to the following proble. Assue that we are given n secondary users whose tie requireents are tie intervals [s i, e i ] for 0 s i < e i T, i n, with a weight b i > 0 for each interval; and n secondary users whose tie requireents are tie durations d i for 0 < d i T, where n < i n + n, with a weight b i > 0 for each duration. Here n = n + n. Our objective is to select soe secondary users such that their requests can be satisfied in tie segent [0, T ] without intersections while the total weight is axiized. Algorith provides our ethod that essentially uses algoriths for proble YOD, and YOI. Algorith Constant Approxiation for YOM Input: n secondary users requiring tie intervals and n secondary users requiring tie duration. Output: Soe secondary users that can be satisfied in tie segent [0, T ] without intersections. : Find the optial solution with the first n secondary users by using dynaic prograing. The detail of the dynaic prograing is oitted here due to its siplicity and space liit. Let S denote the solution. : Find the approxiated optial solution with all users that request tie duration by using the PTAS for knapsack proble, e.g., [5], []. Let S denote the solution. 3: Return one of the solution with larger total weight fro S and S. Theore : Algorith is ( ɛ)/-approxiation for any > ɛ > 0. Proof: For the n secondary users who request tie intervals, the optial solution can be found in polynoial tie by dynaic prograing. For the n secondary user who request tie duration, the optial solution can be approxiated within /( + ɛ ) for any > ɛ > 0 by the PTAS for knapsack proble. Here we also use S, S and OP T to denote the total weight of these solutions. Then S + +ɛ S > OP T. According to pigeonhole principle, ax(s, S ) ɛ ɛ OP T For each > ɛ > 0, let ɛ = ɛ ɛ, we finish the proof. Moreover, here we show a general ethod for those probles in which users have a ixed tie requireent, i.e., probles in the forat T=M, when the corresponding probles in the forat T=D and T=I are solved. The algorith is as following. Theore : Algorith is at least α α α +α -approxiation. Algorith Approxiation Method for Probles with Mixed Tie Requireent : Find a channel allocation to all users who require for a tie-interval, i.e., T=I, using a ethod with an approxiation ratio α ; : Find a channel allocation to all users who require for a tie-duration, i.e., T=D, using a ethod with an approxiation ratio α ; 3: Return the better solution of these allocations as the solution of the proble. Proof: In the optial solution, suppose that β(0 β ) of the total profit coes fro the users who require for a tie duration; and β of the total profit coes fro the users who require for a tie interval. Algorith will give us a ax(βα, ( β)α ) approxiation by pigeonhole principle. The iniu value of ax(βα, ( β)α ) is α α α +α when β = α α +α. Therefore we finish the proof. V. PTAS FOR PROBLEM YUI In this section, we present a polynoial tie approxiation schee for the proble YUI, where there is only a single channel available, the required region by every secondary user v i is a unit disk, and each user v i asks for a tieinterval [s i, e i ]. The PTAS runs in O(n ɛ ) tie and provides an approxiation factor of ( ɛ) where n is the nuber of secondary users. Fig.. An illustration of cylinder graph. Notice that finding the set of bids with the axiu value is equivalent to solve the axiu weighted independent set in the following intersection graph of cylinders. Each bid B i = [b i, F i, D(v i, r i ), T i ] defines a three diensional cylinder B i = (D(v i, r i ) [s i, e i ]) with weight b i. See Figure for illustration. For siplicity, we assue that the three axes are X, Y and Z and the axis Z denotes the tie diension. The disk D(v i, r i ) is called the base of the cylinder B i. The intersection graph has all the cylinders as its vertices, and two vertices for an edge if the corresponding two cylinders overlap. The weight of a vertex is the bid value of the corresponding secondary user. Our PTAS is based on the shifting strategy developed in [0]. We will partition the space using hyperplanes perpendicular to X-axis and hyperplanes perpendicular to Y -axis.

6 6 For each i, j belong to {0,,, k }, we copose a sub-instance I i,j containing all cylinders except those being hit by a hyperplane fro {x = p p i od k} or hit by a hyperplane fro {y = p p j od k}. There are k different sub-instances. For each sub-instance, we calculate its optial solution using the dynaic prograing in every of the k k grids, which will be described in detail in the next subsection. We first establish a technical lea for the perforance guarantee, then prove our algorith achieve ( ɛ)-approxiation for any > ɛ > 0. Lea 3: For at least one sub-instance I i,j, 0 i, j < k, Fig.. An illustration of partition of space using hyperplanes (k = 6). This is a view fro the top (Z-axis). See Figure for illustration, which shows a slice produced by cut the space using the hyperplane Z = 0. In such a partition, we will throw away soe cylinders that intersect soe special hyperplanes X a od k and Y b od k for soe special values of a and b and a given integer k, and then solve the sub-instances of cylinders contained in each cell individually. Here a cell is a 3-diensional cuboid defined by {(x, y, z) a+ik x a+(i+)k, a+jk y a+(j+)k} for soe integers i and j. Let instance I be the set of all n cylinders. For any given integer k >, we derive k polynoially solvable sub-instances fro the given instance I in polynoial tie. The best value of those sub-instance solutions is at least ( k )ω(op T (I)), where OP T (I) is the optial solution on I and ω(op T (I)) is the value of the solution. Then, solve these sub-instances using a dynaic prograing procedure and return the solution with best value. Algorith 3 A PTAS for Proble YUI Input: A given instance I and a given integer k > 0. Output: A group of secondary users that can be satisfied without intersections. : Derive k sub-instances fro the given instance I in polynoial tie. The procedure of deriving will be shown in subsection V-A. : Solve each sub-instances in polynoial tie. The procedure of solving will be shown in subsection V-B. 3: Return the solution with the largest total weight in those solutions to sub-instances. A. Deriving sub-instances For siplicity, we assue that the diaeter of all disks D(v i, r i ) is and we also assue that the disk is open disk. Draw a grid consisting of hyperplanes x = i (for i Z) perpendicular to X-axis and hyperplanes Y = j (for j Z) perpendicular to Y -axis. The distance between every two parallel neighbor hyperplanes is the diaeter of unit disk. So each cylinder will be hit by at ost one hyperplane perpendicular to X-axis and at ost one hyperplane line perpendicular to Y -axis. ω(op T (I i,j )) ( k + )ω(op T (I)). k Proof: Let us project all cylinders to the plane Z = 0 and we only focus on this plane Z = 0. If a disk is hit by a horizontal line i, it will be dropped k ties in the sub-instances I i,0, I i,,, I i,k. Sae result also applies to the case that it is hit by a vertical line j. Here we double count the nuber of dropping once when the disk is hit by a horizontal line and a vertical line at sae tie. Therefore each disk is included in at least k k + sub-instances. Therefore the disks in the optial solution of I are also included in at least k k + sub-instances, which eans ω(i i,j OP T (I)) (k k+)ω(op T (I)) i,j {0,,,k } The best value of these sub-instance solutions ax 0 i,j<k ω(i i,j OP T (I)) k 0 i,j<k ω(i i,j OP T (I)) ( k + k )ω(op T (I)). The lea then follows directly fro the fact that ω(op T (I i,j )) ω(i i,j OP T (I)). Theore 4: Algorith 3 is ( ɛ)-approxiation for any > ɛ > 0. Proof: According to previous lea, by setting k = /ɛ, our algorith iplies that we have a PTAS (i.e., finding a solution whose total value is at least ɛ ties of the optiu) for proble SUI in tie n ɛ. For each sub-instance, each disk is in a k k grid. The disks in different grid won t intersect each other. So the solution in each k k grid is independent. In the following subsection, we show that the proble in a k k grid is polynoially solvable. B. Dynaic prograing Here we describe a dynaic prograing approach to find a axiu weighted independent set for an instance I i,j. We only consider cylinders contained in one k k cell of I i,j. It is easy to show that the following siple greedy ethod does not work. A siple greedy ethod works as follows: it sorts the secondary users in the decreasing order of their bids, say {v, v,, v n } is the sorted list, and for i = to n, a user v i is granted the channel only if it will not cause any conflict with previously assigned users. Since the tie requireent of that secondary user could intersect with a group of secondary users whose total bids are arbitrarily larger, the approxiation ratio could be arbitrarily bad.

7 7 For cylinders in each k k cell, our dynaic prograing ethod first sort the in non-decreasing order of their ending tie e i. For siplicity, let B, B, B i,, B n be the n cylinders contained in one cell in the sorted order. In the rest of subsection, we will use i to denote the cylinder B i. Definition : Pile: A pile is an ordered collection j, j,, j q of pairwise non-intersecting cylinders that intersect a coon hyperplane z = b (for soe value b) perpendicular to Z-axis. Here j i is the ending tie of a cylinder B ji and j t < j t+. Given a hyperplane z = b for soe fixed value b, in a pile that was hit by z = b, there are at ost k cylinders in the pile. Notice that all cylinders in this pile intersect the hyperplane z = b and are disjoint fro each other. Then an area arguent iplies that the nuber of cylinders in a pile is at ost k / π 4 < k. Lea 5: The total nuber of all possible piles in each k k grid is polynoial of n. Proof: There are k hyperplanes perpendicular to X-axis and k hyperplanes perpendicular to Y -axis in a k k grid. So there are k intersections defined by these hyperplanes in each cell. Also there are k center points in the k sall squares in each grid. Each unit disk in the k k grid will be hit by either an intersection or a center point. The disks hit by the sae intersection or center points ust intersect with each other. For each intersection or center point, we enuerate nuber of disks hit by the intersection or center point. Then we can get O(n k ) cobinations which include all possible piles in the grid. Since there are O(k ) cylinders in a pile, we can check whether a cobination of at ost k cylinders is a pile in tie O(k 4 ). Our dynaic prograing approach will first sort the piles based on an order defined below; then find the optiu solution of all cylinders that are ordered in front of a pile. Definition : Define an order on the set of piles as following: j, j,, j l i, j,, i if () j l < i, or () j l = i and j, j,, j l i, j,, i, or (3) = 0. Definition 3: Let OP T (X j, j,, j l ) be the axiu total weight of pairwise non-overlapping cylinders fro set X given that cylinders j, j,, j l already occupy their places. For a set of non-overlapping cylinders {j, j,, j l } and a cylinder t with t < j, we define the arginal contribution, denoted as R j,j,,j l (t), of cylinder t to OP T (i : i t j, j,, j l ) as (OP T (i t j, j,, j l ) OP T (i < t j, j,, j l )) +. Here (x) + = ax{0, x}. Based on the above definition, a cylinder t has positive arginal contribution R j,j,,j l (t) will clearly be used in an optiu solution OP T (i t j, j,, j l ). If its arginal contribution is 0, then it eans that there is one optiu solution OP T (i t j, j,, j l ) that will not use the cylinder t. As proved in [5], it is easy to show that t OP T (i t j, j,, j l ) = R j,j,,j l (i) i= Here, R j,j,,j l (t) = b t + OP T (i < t j,, j l, t) OP T (i < t j,, j l ). We then present our dynaic prograing to find an optial solution in each k k cell as following Algorith 4. Algorith 4 Find Maxiu Weighted Independent Set of Cylinders in Each Cell Input: A set S of weighted cylinders contained in a cell. Output: An optiu axiu weighted independent set in S. : Find all the piles P of cylinders fro S and sort the in the order of ; : Take the piles one by one in the order starting fro the least one. For each pile j, j,, j l calculate and store in the eory the value R j,,j l (j ) according to the forula: ( R j,,j l (j ) = b j + R j,j,,j l (i) R j,,j l (i) i:i<j i:i<j 3: After all the piles corresponding values of R j,j 3,,j l (j ) is calculated for every possible pile p = j, j,, j l P, we schedule the cylinders in the following way. Consider cylinders fro S in the order of decreasing nuber. Take a cylinder j next in that order. Suppose that cylinders {j, j,, j l } are already scheduled, then schedule j iff R j,j,,j l (j) is positive. Theore 6: The running tie of our dynaic prograing is at ost O(n k + ) in the worst case. Proof: Fro lea 5, we know that the total nuber of piles is at ost n k. Thus, it takes at ost O(n k lg n) to sort the. To find each cylinder s arginal contribution R p (j), it costs n n k. Thus, the total tie is at ost O(n k + ). Notice that O(n k lg n) is the worst case tie coplexity in which we assue that, in the coputation of the dynaic prograing, there is a certain hyperplane such that there are Θ(n) cylinders, defined by the bids with bases contained in a k k region, which intersect this hyperplane. The actual tie coplexity actually depends on the nuber of independent set of cylinders intersecting a hyperplane, which could be uch saller than n in practice. For exaple, if the arrivals of requests fro users are randoly distributed, the network syste could set a larger interval [0, T ] coparing with the users tie requireent. The nuber of independent set of cylinders intersecting a hyperplane would decrease significantly and total tie coplexity would be uch lower than the theoretical bound. Our siulation studies verified our clai and showed that the algorith actually runs efficiently. VI. ALGORITHM FOR PROBLEM YUD In this section, we design an approxiation algorith with a constant approxiation ratio for proble YUD, where there is only a single channel available, the required region by every secondary user v i is a unit disk and each user v i asked for a tie duration d i. We can use the sae graph of cylinders which is entioned in the previous section. See Figure for ) +

8 8 illustration. Notice that the cylinders can ove along the axis Z in this proble since each user asks for a tie duration. The ain idea here is to partition cylinders into g groups such that we can solve the axiu weighted independent set in each group and then take the group with the best solution. By pigeonhole principle, we know that it will give us /g approxiation. We ainly will focus on designing a partition with iniu constant value g. For siplicity, we assue that the radius of every disk D(v i, r i ) is and we also assue that the disk is open disk. Draw a grid consisting of hyperplanes x = + 3 i (for i Z) perpendicular to X-axis and hyperplanes y = + 3 j (for j Z) perpendicular to Y -axis. The distance between every two parallel neighbor hyperplanes is + 3 ties the radius of unit disk. Algorith 5 Constant Approxiation Method for SUI Input: An instance for proble SUI. Output: A group of secondary users who could be satisfied without intersections. : For each i, j {0,, }, find a group that contains all cylinders except those being hit by a hyperplane fro {x = + 3 p p i od 3} or hit by a hyperplane fro {y = + 3 p p j od 3}. Denote the group as G i,j. : Solve the subproble for each group. 3: Return the solution with largest total weight in those solutions to all groups. Lea 7: All cylinders in instance are included in at least one of these groups. Proof: Consider the X-Y plane. All cylinders are unit disks on the plane. Since the radius of these disks is and the distance between every two parallel neighbor hyperplanes is + 3, a disk is hit by at ost parallel neighbor hyperplanes perpendicular to one axis. Therefore all disks ust be in a 3 3 cell. Group G 0,0, G 0,,,G, represents all different groups with 3 3 cell on the plane. So all cylinders are included in at least one of these groups. Theore 8: Algorith 5 is /9-approxiation and polynoial tie solvable. Proof: For each group, the disks in different cells cannot intersect with each other. So the solution in each cell is independent. We can solve these subprobles in each cell one by one and achieve the final result. Considering the subproble in each 3 3 cell, the disks ust intersect with each other, which eans that we can select only one disk at any tie. Therefore the subproble becoes a knapsack proble. We can solve it with a FPTAS. So each group can be solved with a FPTAS. We can then solve the 9 groups in polynoial tie and have a /9 approxiation by pigeonhole principle. VII. ALGORITHM FOR PROBLEM SUI In this section, we design an approxiation algorith with approxiation ratio Θ( ) for proble SUI where is the total nuber of available channels. Here we say an algorith for SUI has approxiation ratio α, if for all instances it will return a solution at least /α of the optiu. Notice that the set packing proble is a special case of the proble SUI due to the following observation. Recall that in a set packing proble, we are given a universal set E (with size ) and a set of subsets S = {S, S,, S n }, where each subset S i E is associated with a weight w i. We need to find a subset of S such that the selected subsets are disjoint and the suation of their weights is axiized. Given a set-packing proble, we can easily define a proble SUI as follows: F E, F i S i, b i w i, and D(v i, r i ) centered at the point (0, 0). These two probles will have the sae optiu solution. Recall that set packing proble cannot be approxiated within / ε for any ε > 0, unless NP=ZPP, i.e., soe NP proble can be solved in probabilistic polynoial tie. Thus, we have Lea 9: Proble SUI cannot be approxiated within / ε for any ε > 0, unless NP=ZPP. Using the sae partition ethod in Section 5, we can get an asyptotical optiu approxiation based on the approxiation optial solution in each group. Next, we will focus on designing an algorith for each group with approxiation ratio Θ( ). In the rest of the section, we assue that the disks D(v i, r i ) of all bidders conflict with each other, e.g., having a coon point. For set packing proble, a greedy algorith was proposed in [6] that will find a solution whose total weight is at least fraction of the optiu solution. Their ethod, by interpreting using the ters in this paper, will first sort b the bidders in decreasing order of i, where F Fi i denotes the nuber of frequencies in F i. The ethod processes the bidders in this order and a bidder i is selected only if it will not cause conflict with any of the previously selected bidders. In addition, since the set packing proble is essentially the axiu weighted independent set (MWIS) proble, we could also use heuristics for MWIS to find a good solution for set packing. In [3], Sakai et al. studied the perforances of three siple greedy approaches: GWMIN, GWMAX, and in-weight-ratio. The heuristic GWMIN always recursively W (v) d Gi (v)+ adds a vertex v iniizing aong all vertices in graph G i ; and then reove v and all adjacent vertices fro G i to get graph G i+. It starts with G = G for the first step. The heuristic GWMAX always recursively adds a vertex W (v) v iniizing d Gi (v)(d Gi (v)+) aong all vertices in graph G i ; and reove v and all its adjacent vertices. The heuristic in-weight-ratio will always adds a vertex v iniizing u N Gi (v) W (u) W (v), where N Gi (v) is the set of adjacent vertices of v in graph G i. It was proved in [3] that heuristics GWMIN and GWMAX both have approxiation ratio Θ( ) and the heuristic in-weight-ratio has approxiation ratio W (v) v V u N G (v) W (u). Here is the axiu node degree and w(u) is the weight of the vertex u. We begin our investigation of algorith by first showing that a siple straightforward extension of the above algoriths Notice that here the sorting is not based on naive criterion b i F i.

9 9 could result in an arbitrarily large approxiation ratio for our SUI proble. First, let us consider the ethod in [6]. Sorting the bidders b by descending i is a very reasonable idea. But this Fi d i approach ay perfor arbitrarily bad. Consider the following exaple. Assue that there are only two bidders, i and j: b b i b j and d i is so large that i b < j, if we Fi d i Fj d j choose bidder j but not bidder i, the total weight we get is b j, that is arbitrarily saller than the optiu solution b i. Second naive ethod is to sorting the bidders by nondecreasing order of b(ni) b i. Here we use b(n i ) to denote the total weight bided by the bidders who intersect bidder i, and we call N i the neighbors of bidder i. Unfortunately, a siple exaple shows that this approach ay also perfor quite poorly; see Figure 3 for illustration. Assue that bidder i bids n+ε and each i s neighbor bids exactly. According to nondecreasing order of b(n i) b i neighbors since, we will choose bidder i instead of all its < n+ε. The total weight we get is n n+ε n + ε, but the optiu solution is to select all nodes bidder, which give us n. This exaple shows that this naive ethod will have approxiation ratio at least Ω( n), i.e., the solution returned is at ost O( n ) of the optiu for soe instances. Fig = tie interval bidded by i = tie intervals bidded by Ni This is a view fro the lateral face(xy -axis). In the rest of section, we will design a polynoial tie algorith for proble SUI with approxiation ratio Θ( ). A. Θ( )-Approxiation Algorith For SUI with Single Tie Interval Consider each frequency set bided by different bidders. Assue that every bidder bids k frequencies and single tie interval. When k =, obviously the optiu solution is the union of the optiu solutions OP T i, where OP T i is the optiu solution for the set of users who do bid for frequency f i. Notice that we can copute the optial solution OP T i using dynaic prograing. Next, we will focus on the scenario in which k >, i.e., each bidder bids for at least k different frequencies. In the following part, we will use OP T to denote the global optial solution. For each frequency f i, let OP T i be the set of users in OP T that bid for frequency f i. Obviously, we have OP T i k OP T, i= since each user in OP T will appear in at least k different OP T i. Thus, ax{op T, OP T,, OP T } k OP T. Algorith 6 Asyptotically Optiu Method for SUI with Single Tie Interval Input: An instance for proble SUI Output: A group of secondary users who could be satisfied without intersections. : Partition the bidders into two groups: ) Z contains all the bidders that bid at least frequencies; and ) Z contains all the other bidders, i.e., bid less than frequencies. : Approxiate the optial solution for Z and Z using best available polynoial-tie ethod. 3: Return the larger solution. We will show algoriths with approxiation ratio Θ( ) for both groups respectively. Then obviously, the axiu of these two solutions will give us Θ( ) approxiation for SUI. Lea 0: For SUI with group Z, there is a polynoialtie ethod with approxiation ratio Θ( ). Proof: First for users in group Z, we just use ax{op T, OP T,, OP T } as our solution. Since each bidder bids at least k frequencies, ax i= OP T (Z ) i OP T (Z ) = OP T (Z ). Here OP T (Z j ) is the optial solution for users in group Z j for j =,. For bidders in Z, we use I i to denote the set of bidders in Z that bid for frequency f i. Then we find the axiu weighted independent set OP T i of I i using standard dynaic prograing. Obviously, OP T i OP T (Z ) i. Thus, we have ax OP T i ax OP T (Z ) i OP T (Z ). i= i= This finishes the proof. Lea : For users in group Z, there is a polynoialtie ethod with approxiation ratio Θ( ). Proof: We will convert this proble into the Scheduling Split Intervals Proble (SSIP) []. The ordinary SSIP considers the proble of scheduling jobs that are given as groups of non-intersecting segents on the real line. Each job J i is associated with an interval, I j, which consists of up to t segents, for soe t, and a positive weight, w j. Two jobs are in conflict if any of their segents intersect. The objective is to schedule a subset of non-conflicting jobs with axiu total weight. In [], they gave an algorith with t-approxiation ratio for proble SSIP. Here we create a special instance for SSIP proble as follows. Let [0, T ] be the original tie period where bidders can place their tie-interval [s i, e i ], i.e., 0 s i e i T for every bidder i. We then create a bigger tie period [0, T ] where is the total nuber of

10 0 frequencies. Then a user i will be associated with following t i = F i < segents in [0, T ]: {[(j ) T + s i, (j ) T + e i ] f j F i, j } In other words, we duplicate the period [0, T ] ties for each of the frequencies in F, and a user i will have a segent in the jth duplication if it bids for frequency f j. Then there are at ost segents for every bidder. Then based on algoriths in [], we get -approxiation solution, i.e., find a solution with value at least OP T (Z ). Theore : Algorith 6 has approxiation ratio Θ( ) for proble SUI with single tie interval. Proof: Let OP T (Z ) and OP T (Z ) be the optiu solution for group Z and group Z respectively. Then the axiu of these two solutions is at least ( ) OP T (Z ) ax, OP T (Z ) 3 OP T, since either OP T (Z ) 3 OP T, or OP T (Z ) 3 OP T fro OP T (Z ) + OP T (Z ) OP T. We finish the proof. Notice that by using a different group partition, where group Z contains the bidders that bid for at least, we get an algorith with approxiation ration 4. As a byproduct of our result, we show that for proble SSIP, there is no approxiation algorith with ratio O(t ɛ ) for any ɛ > 0 unless NP = ZP P. To the best of our knowledge, we are the first one to prove this. Theore 3: For proble SSIP, there is no polynoialtie approxiation algorith with ratio O(t ɛ ) for any ɛ > 0 unless NP = ZP P. Proof: We prove this by contradiction. Assue that there is a polynoial-tie algorith A with ratio O(t ɛ ) for soe ɛ > 0. We will use this algorith to design an algorith for solving the proble SUI with approxiation ratio O( ɛ ) for soe ɛ > 0. We partition the bidders into two groups: group Z contains all bidders requiring at least ɛ frequencies; group Z contains all other bidders. We then solve the SUI proble as our previous discussion: finding a axiu independent set in Z using ax i OP T (I i ); finding a axiu independent set in Z using a SSIP transforation. Obviously, we have ɛ OP T (Z ) = ax OP T (I i) i and for soe constant c > 0, OP T (Z ) c ɛ OP T. ( ɛ) OP T, It is easy to show that ax{op T (Z ), OP T (Z )} will give us a solution that is at least O( ) ties of the 4 ɛ ɛ optiu. Notice that here ɛ is a certain fixed constant. This iplies that we have an algorith for SUI (and thus setpacking proble) with approxiation ratio ɛ, where constant ɛ = ɛ 4 ɛ > ɛ 4. This is clearly ipossible if NP is not equal to ZPP. This finishes the proof. B. Θ(t )-Approxiation Algorith For SUI With t Tie Intervals Different fro the above case, assue that every secondary user could requires t tie intervals (or at ost t tie intervals) and bids for a set of frequencies. We need to find a subset of non-conflicting users with the axiu total bid value. Obviously, when all users bid for ore than F / frequencies, every pair of users will have a coon requested frequency. Thus, for this case, coputing the approxiation solution for each frequency is exactly the traditional SSIP. And we can get t-approxiation solution for this special case. Here we show that Algorith 6 also works for this case. Lea 4: For users in group Z, there is a polynoialtie algorith for SUI with approxiation ratio Θ(t ). Proof: For group Z, for each frequency f i, let OP T (Z ) i be the set of users in OP T (Z ) that bid for frequency f i and let OP T i be the optiu solution for users in Z whose bid contains frequency f i. Clearly, ω(op T i ) ω(op T (Z ) i ). We just use ax i= OP T i as our solution. Assue that each user bids at least k frequencies. Then we have i= OP T (Z ) i k OP T (Z ), which iplies that ax i= OP T (Z ) i k OP T (Z ). Since each bidder bids at least k frequencies, ax OP T i i= OP T (Z ) = OP T (Z ). The SSID proble has a polynoial-tie algorith with approxiation ratio t, which iplies an algorith with approxiation ratio t for proble SUI with users in Z. Lea 5: For users in group Z, there is a polynoialtie algorith with approxiation ratio Θ(t ). Proof: We can also convert this proble into SSIP []. Siilarly, we duplicate the period [0, T ] ties for each of the frequencies in F, and a user i will have t segents in the jth duplication if it bids for frequency f j. Then there are at ost t segents for every bidder. Then based on algoriths in [], we get t -approxiation solution for OP T (Z ). Theore 6: Algorith 6 is Θ(t ) approxiation for proble SUI with t tie intervals. Proof: Let OP T (Z ) and OP T (Z ) be the optiu solution for group Z and group Z respectively. Then the axiu of the above two solutions is at least ax ( OP T (Z ) t, OP T (Z ) t ) 3t OP T, since either OP T (Z ) 3 OP T, or OP T (Z ) 3 OP T fro OP T (Z ) + OP T (Z ) OP T. Notice that by using a different group partition, where group Z contains the bidders that bid for at least, we can also get an algorith that finds a set of secondary users with profit that is at least fraction of the optiu. 4t

11 VIII. STRATEGYPROOF MECHANISM DESIGN Fro section IV to section VII, we ainly focus on the allocation ethods, i.e. who will get the channels, for different versions of probles. Recall that we also need to copute payent schees, i.e., how uch each secondary user should pay. In this section, we show how to design a truthful echanis based on the algoriths discussed in previous sections to copute payent schees. The echanis should be strategyproof, i.e., for each secondary user, reporting its valuation truthfully is its doinant strategy. A strategyproof echanis should satisfy both IC and IR properties which was introduced in Section II-B. To satisfy these properties, the underlying allocation ethod ust be a onotone output algorith O. Then we can design the payent based on a critical payent schee P. It was proved that all strategyproof echaniss should have a onotone allocation ethod and their payent schees are based on the cut value [3]. Here an allocation algorith is onotone if, for an agent that is allocated, the agent will still participate in the output when it increases its bid. We define a critical value θ i, i.e., the iniu valuation v i that akes agent i participate in the output. A critical value payent schee P O for an algorith O is that p i = θ i if agent i is in the output, otherwise, p i = 0. If O is a onotone algorith and P O is a critical value payent schee for O, M = (O, P O ) is a strategyproof echanis [3]. Now let us consider whether algoriths described in previous sections are onotone. If they are onotone, we also have relevant strategyproof echanis. In these algoriths, we used technology of dynaic prograing, grouping and the classic FPTAS for knapsack proble. Dynaic prograing and grouping clearly do not affect the onotone property. However, the classic FPTAS [5], [] for knapsack proble is not onotone. We know that classic FPTAS for knapsack uses npi ɛp to round all values down to a saller value, where n is the nuber of ites, p i is the profit of ite i, P is the axial possible profit. Then solve the knapsack by using dynaic prograing. For ore details, see [5], []. Here is a counterexaple why these ethods are not always onotone. Suppose a knapsack with size 300, ite, and 3 s profits and sizes are all 00, ite 4, 5, 6 and 7 s profits and sizes are all 75. We choose ɛ = 7 5. After rounding, ite, and 3 s profits are both 5, ite 4, 5, 6 and 7 s profits are all. So the solution of knapsack is ite, and 3. However, if ite increases its profit to 0, after rounding, ite s profit is 5, ite and 3 s profits are both 4, ite 4, 5, 6 and 7 s profits are all still. Therefore solution will be ite 4, 5, 6 and 7. Ite will not participate in the solution when increasing its bid, which eans that this FPTAS is not onotone. Now our question becoes whether there is a onotone FPTAS for knapsack proble. If we have such FPTAS, all algoriths in previous sections can be onotone, because cobinations of onotone ethods is still onotone. In [4], Briest proposed an alternative rounding schee that transfors a pseudopolynoial algorith into a onotone FPTAS for knapsack proble. Using this FPTAS for knapsack proble, all algoriths presented in previous sections will be onotone. Therefore we can design strategyproof echaniss M = (O, P O ) for all probles discussed in previous sections. Observe that the payent of any agent i in a strategyproof echanis is always at ost its bid b i. Due to space liit and its siplicity, we oit all echaniss here. Dynaic User Join and Leaving: So far we assued that when we ake spectru allocation and decide the payent fro secondary users, we know all the requests by all secondary users. This is also the assuption ade by the ajority results in the literature. In practice, it is possible that secondary users ay arrive in an online fashion and we are required to ake a decision on whether to grant the request of a secondary user iediately or within a certain tie liit. To address such challenge, we propose a siple solution here. We divide the tie into ultiple tie intervals (the length of a tie interval depend on applications) and ake allocation on each interval at the beginning of that interval. New secondary users ay join the syste anytie. However, they cannot lease channels until next allocation begins, i.e., they will be considered at the beginning of next tie interval. Existing secondary users ay leave the syste at anytie since we already charged the existing secondary users. After a secondary user leaves, it releases the right to the channels allocated at its required region. If a secondary user leaves earlier than its reserved tie, here we assue that no copensation fro the syste will be given. At the beginning of any tie interval, we run our echaniss for only newly arrived secondary users without using the channels at regions already occupied by existing users. When secondary users will not lie about their arrival tie, we can show that our echaniss are still truthful and efficient. IX. SIMULATIONS We conduct extensive siulations to study the perforances of our algoriths, ainly algoriths for proble YUI. We also study the perforance of the echaniss by coparing the payent charged to all users and the total valuations of all allocated users. In our siulations, we generate rando requests with rando bid, rando tie requireent and rando space requireent. The bid of each request is uniforly drawn fro all integers in [, 00]. The length of each tie interval is uniforly drawn fro all integers in [, 0] and the start tie of each tie interval is uniforly distributed in [0, 00]. The space requireent of each user is a unit disk. Each unit disk is uniforly distributed in a square area. One aspect ay affect the results is the degree of copetition. We use the average nuber of requests generated to represent the degree of copetition. For echanis design, ore copetition often eans that the collected payent by our strategyproof echanis is closer to the total valuations of all allocated users. We first study the approxiation ratio of our ethod for proble YUI in siulations. Fro Fig 4, obviously, the approxiation ratio increases when k increases, where k is the size of a cell used in our ethod. This is because when k increases, less cylinders defined by users requests are thrown

12 Approxiation Ratio Fig. 4. Efficiency Ratio Fig Theoritical Bound ,000 requests in average 00,000 requests in average 00,000 requests in average Paraeter k The approxiation ratio of algorith for proble YUI ,000 requestsin average ,000 requests in average 00,000 requests in average Paraeter k The efficiency ratio of echanis for proble YUI away during the step of dynaic prograing. Therefore, the approxiation ratio increases significantly. On the other hand, we found that the degree of copetition does not affect approxiation ratio uch. Three curves which represent different degree of copetition are very close to each other. It eans the approxiation ratio ainly depends on paraeter k. We can observe that the approxiation ratio is a litter bit worse when the copetition is intense. This is because when there are ore cylinders defined by users requests, they tend to distribute in different area evenly. Of course, the perforances of our ethod in siulations are always better than the theoretical bound in perforance analysis. We can see that even with a sall paraeter k, i.e. k = 0, our ethod can achieve ore than 80% of the optiu. And the tie coplexity in siulations is not as bad as our theoretical analysis in Section V-B. Most results can be coputed within a few inutes (< 0) when there are totally 00, 000 requests. Then we study the efficiency ratio of the strategyproof echanis that were described in Section VIII. Here the efficiency ratio is defined as the total payent charged fro users who are granted the usage of channels over their total valuations. Clearly, for a strategyproof echanis, its efficiency ratio is always at ost. Fro Fig 5, we can see that the efficiency ratio increases significantly when the copetition is harder. This is because it is easier for our ethod to find a good replaceent of users (without reducing the payent fro users) when a user ay reduce its bid when the copetition is hard. On the other hand, the paraeter k also affect the efficiency ratio. The efficiency ratio increases slightly when paraeter k increases. This is because less cylinders, defined by the bids of users, have been thrown away before the step of dynaic prograing. This is a positive factor for finding a good replaceent. X. CONCLUSIONS In this paper, we cobine the gae theory with counication odeling to solve soe channel allocation probles. We study how to assign the spectru and how to charge the secondary users such that the overall social benefits are axiized. More specifically, we forulate several versions of spectru assignent probles by separately assuing whether the secondary users are single-inded or not, whether their required regions overlap or not, whether they ask for fixed tie-intervals or tie durations. When secondary users require only one channel, we show a siple ɛ approxiation algorith for proble YOM. For proble YUI, we present a polynoial tie approxiation schee by deriving k polynoially solvable sub-instances fro the given instance, where the best value of those subinstances solutions is at least a k + k approxiation of optiu. For proble YUD, we design a polynoial-tie algorith with a constant approxiation ratio. For proble YUM, we also give a polynoial-tie algorith with a constant ratio by cobining the results of probles YUD and YUI. On the other hand, for those probles such that soe secondary users require a subset of channels and they are single-inded, we show an Θ( )-approxiation algorith for proble SUI. Furtherore, if at ost t tie intervals are required by a user, there is an approxiation algorith with ratio Θ(t ). We also show how to design strategyproof echaniss based on those described algoriths that have a onotone property. There are still a nuber of interesting probles to be studied. We leave it as a future work whether we can design efficient approxiation algoriths for proble SUD, where single-inded secondary users request unit disk region and a tie duration. The second interesting proble is to study the non-cooperative gaes aong second users using soe other solution concepts such as Nash equilibriu, or study the repeated gaes by these users. Another interesting gae odel is when the priary users are also non-cooperative and a secondary user requireent can be satisfied by several priary users. Then how to design schees to achieve arket clearance between the priary users and secondary users. The last, but not the least, question is that we should design spectru allocation ethods for the case when the channels are available to secondary users for only soe tie intervals.

13 3 REFERENCES [] AHMAD, ISHFAQ AND LUO, JIANCONG On using Gae Theory for Perceptually Tuned Rate Control Algorith for Video Coding. IEEE Transactions on Circuits and Systes for Video Technology (vol.3, no., Feb. 006), pp [] BAR-YEHUDA, R., HALLDÓRSSON, M. M., NAOR, J. S., SHACHNAI, H., AND SHAPIRA, I. Scheduling split intervals. In SODA 0: Proceedings of the thirteenth annual ACM-SIAM syposiu on Discrete algoriths (00), Society for Industrial and Applied Matheatics, pp [3] BARTAL, Y., GONEN, R., AND NISAN, N. Incentive copatible ulti unit cobinatorial auctions. In TARK 03: Proceedings of the 9th conference on Theoretical aspects of rationality and knowledge (003), ACM Press, pp [4] BRIEST, P., KRYSTA, P., AND VÖCKING, B. Approxiation techniques for utilitarian echanis design. In STOC 05: Proceedings of the thirty-seventh annual ACM syposiu on Theory of coputing (005), ACM Press, pp [5] CHEKURI, C., AND KHANNA, S. A ptas for the ultiple knapsack proble. 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Journal of Finance (96), [6] XU, PING AND LI, XIANG-YANG Online Market Driven Spectru Scheduling and Auction, CoRoNet workshop of ACM MobiCo, 009. [7] XU, PING AND LI, XIANG-YANG SOFA: Strategyproof Online Frequency Allocation for Multihop Wireless Networks. International Syposiu on Algoriths and Coputation (ISAAC) 009. Ping Xu has been a PhD student of Coputer Science Departent at the Illinois Institute of Technology since 006. He received BS and MS degrees in Electronic Engineering fro Shanghai Jiaotong University, China, in 999 and 003 respectively. His current research interests include algorith design and analysis for wireless ad hoc networks, wireless sensor networks, online algoriths, and algorithic gae theory. Xiang-Yang Li (M 99, SM 08) has been an Associate Professor (since 006) and Assistant Professor (fro 000 to 006) of Coputer Science at the Illinois Institute of Technology. He received MS (000) and PhD (00) degree at Departent of Coputer Science fro University of Illinois at Urbana-Chapaign. He received the Bachelor degree at Departent of Coputer Science and Bachelor degree at Departent of Business Manageent fro Tsinghua University, China, both in 995. His research interests span wireless ad hoc and sensor networks, gae theory, coputational geoetry, and cryptography and network security. He served as a co-chair of ACM FOWANC 008 workshop, a co-chair of AAIM 007 conference, a TPC co-chair of WTASA 007, and TPC ebers of ACM MobiCo, ACM MobiHoc, IEEE INFOCOM, IEEE ICDCS, and so on. He is an editor of Ad Hoc & Sensor Wireless Networks: An International Journal, and Journal of Networks. He has served as a guest editor of IEEE JSAC and ACM MONET. ShaoJie Tang has been a PhD student of Coputer Science Departent at the Illinois Institute of Technology since 006. He received BS degree in Radio Engineering fro Southeast University, China, in 006. His current research interests include algorith design and analysis for wireless ad hoc networks, wireless sensor networks, and online social networks. Dr. JiZhong zhao is a Professor of Coputer Science and Technology Departent, Xi an Jiaotong University, China. He received his BS degree and MS degree in Matheatic Departent fro Xi an Jiaotong University, China. He received a Ph.D. degree in Coputer Science, focus on Distributed Syste, fro Xi an Jiaotong University in 00. His research interests include coputer software, pervasive coputing, distributed systes, network security. He is a eber of IEEE coputer society, and a eber of ACM.