Approximately-Strategyproof and Tractable Multi-Unit Auctions

Transcription

1 Approxmatey-Strategyproof and Tractabe Mut-Unt Auctons Anshu Kothar Davd C. Parkes Subhash Sur ABSTRACT We present an approxmatey-effcent and approxmateystrategyproof aucton mechansm for a snge-good mut-unt aocaton probem. The bddng anguage n our auctons aows margna-decreasng pecewse constant curves. Frst, we deveop a fuy poynoma-tme approxmaton scheme for the mut-unt aocaton probem, whch computes a (1 + ǫ)- approxmaton n worst-case tme T = O(n 3 /ǫ), gven n bds each wth a constant number of peces. Second, we embed ths approxmaton scheme wthn a Vckrey-Carke-Groves (VCG) mechansm and compute payments to n agents for an asymptotc cost of O(T og n). The maxma possbe gan from manpuaton to a bdder n the combned scheme s bounded by ǫ/(1+ǫ)v, where V s the tota surpus n the effcent outcome. Categores and Subject Descrptors F.2 [Theory of Computaton]: Anayss of Agorthms and Probem Compexty; J.4 [Computer Appcatons]: Soca and Behavora Scences Economcs. Genera Terms Agorthms, Economcs. Keywords Approxmaton Agorthm, Mut-unt Auctons, Strategyproof. Department of Computer Scence, Unversty of Caforna at Santa Barbara, CA Ema: {kothar, sur}@cs.ucsb.edu. Supported n part by NSF grant IIS Dvson of Engneerng and Apped Scences, 33 Oxford Street, Harvard Unversty, Cambrdge, MA Ema: parkes@eecs.harvard.edu. Supported n part by NSF grant IIS Permsson to make dgta or hard copes of a or part of ths work for persona or cassroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commerca advantage and that copes bear ths notce and the fu ctaton on the frst page. To copy otherwse, to repubsh, to post on servers or to redstrbute to sts, requres pror specfc permsson and/or a fee. EC 03, June 9 12, 2003, San Dego, Caforna, USA. Copyrght 2003 ACM X/03/ $ INTRODUCTION In ths paper we present a fuy poynoma-tme approxmaton scheme for the snge-good mut-unt aucton probem. Our scheme s both approxmatey effcent and approxmatey strategyproof. The aucton settngs consdered n our paper are motvated by recent trends n eectronc commerce; for nstance, corporatons are ncreasngy usng auctons for ther strategc sourcng. We consder both a reverse aucton varaton and a forward aucton varaton, and propose a compact and expressve bddng anguage that aows margna-decreasng pecewse constant curves. In the reverse aucton, we consder a snge buyer wth a demand for M unts of a good and n suppers, each wth a margna-decreasng pecewse-constant cost functon. In addton, each supper can aso express an upper bound, or capacty constrant on the number of unts she can suppy. The reverse varaton modes, for exampe, a procurement aucton to obtan raw materas or other servces (e.g. crcut boards, power suppers, toner cartrdges), wth fexbe-szed ots. In the forward aucton, we consder a snge seer wth M unts of a good and n buyers, each wth a margna-decreasng pecewse-constant vauaton functon. A buyer can aso express a ower bound, or mnmum ot sze, on the number of unts she demands. The forward varaton modes, for exampe, an aucton to se excess nventory n fexbe-szed ots. We consder the computatona compexty of mpementng the Vckrey-Carke-Groves [22, 5, 11] mechansm for the mutunt aucton probem. The Vckrey-Carke-Groves (VCG) mechansm has a number of nterestng economc propertes n ths settng, ncudng strategyproofness, such that truthfu bddng s a domnant strategy for buyers n the forward aucton and seers n the reverse aucton, and aocatve effcency, such that the outcome maxmzes the tota surpus n the system. However, as we dscuss n Secton 2, the appcaton of the VCG-based approach s mted n the reverse drecton to nstances n whch the tota payments to the seers are ess than the vaue of the outcome to the buyer. Otherwse, ether the aucton must run at a oss n these nstances, or the buyer cannot be expected to vountary choose to partcpate. Ths s an exampe of the budget-defct probem that often occurs n effcent mechansm desgn [17]. The computatona probem s nterestng, because even wth margna-decreasng bd curves, the underyng aocaton probem turns out to (weaky) ntractabe. For nstance, the cassc 0/1 knapsack s a speca case of ths probem. 1 We mode the 1 However, the probem can be soved easy by a greedy scheme f we remove a capacty constrants from the seer and a

2 aocaton probem as a nove and nterestng generazaton of the cassc knapsack probem, and deveop a fuy poynomatme approxmaton scheme, computng a (1+ǫ)-approxmaton n worst-case tme T = O(n 3 /ε), where each bd has a fxed number of pecewse constant peces. Gven ths scheme, a straghtforward computaton of the VCG payments to a n agents requres tme O(nT). We compute approxmate VCG payments n worst-case tme O(αT og(αn/ε)), where α s a constant that quantfes a reasonabe no-monopoy assumpton. Specfcay, n the reverse aucton, suppose that C(I) s the mnma cost for procurng M unts wth a seers I, and C(I \ ) s the mnma cost wthout seer. Then, the constant α s defned as an upper bound for the rato C(I \)/C(I), over a seers. Ths upper-bound tends to 1 as the number of seers ncreases. The approxmate VCG mechansm s ( ε )-strategyproof for 1+ε an approxmaton to wthn (1 + ǫ) of the optma aocaton. Ths means that a bdder can gan at most ( ε )V from a nontruthfu bd, where V s the tota surpus from the effcent aoca- 1+ε ton. As such, ths s an exampe of a computatonay-tractabe ε-domnance resut. 2 In practce, we can have good confdence that bdders wthout good nformaton about the bddng strateges of other partcpants w have tte to gan from attempts at manpuaton. Secton 2 formay defnes the forward and reverse auctons, and defnes the VCG mechansms. We aso prove our cams about ε-strategyproofness. Secton 3 provdes the generazed knapsack formuaton for the mut-unt aocaton probems and ntroduces the fuy poynoma tme approxmaton scheme. Secton 4 defnes the approxmaton scheme for the payments n the VCG mechansm. Secton 5 concudes. 1.1 Reated Work There has been consderabe nterest n recent years n characterzng poynoma-tme or approxmabe speca cases of the genera combnatora aocaton probem, n whch there are mutpe dfferent tems. The combnatora aocaton probem (CAP) s both NP-compete and napproxmabe (e.g. [6]). Athough some poynoma-tme cases have been dentfed for the CAP [6, 20], ntroducng an expressve excusve-or bddng anguage qucky breaks these speca cases. We dentfy a non-trva but approxmabe aocaton probem wth an expressve excusveor bddng anguage the bd taker n our settng s aowed to accept at most one pont on the bd curve. The dea of usng approxmatons wthn mechansms, whe retanng ether fu-strategyproofness or ε-domnance has receved some prevous attenton. For nstance, Lehmann et a. [15] propose a greedy and strategyproof approxmaton to a snge-mnded combnatora aucton probem. Nsan & Ronen [18] dscussed approxmate VCG-based mechansms, but ether appeaed to partcuar maxma-n-range approxmatons to retan fu strategyproofness, or to resource-bounded agents wth nformaton or computatona mtatons on the abty to compute strateges. Fegenmnmum-ot sze constrants from the buyers. 2 However, ths may not be an exampe of what Fegenbaum & Shenker refer to as a toeraby-manpuabe mechansm [8] because we have not tred to bound the effect of such a manpuaton on the effcency of the outcome. VCG mechansm do have a natura sef-correctng property, though, because a usefu manpuaton to an agent s a reported vaue that mproves the tota vaue of the aocaton based on the reports of other agents and the agent s own vaue. baum & Shenker [8] have defned the concept of strategcay fathfu approxmatons, and proposed the study of approxmatons as an mportant drecton for agorthmc mechansm desgn. Schummer [21] and Parkes et a [19] have prevousy consdered ε-domnance, n the context of economc mpossbty resuts, for exampe n combnatora exchanges. Eso et a. [7] have studed a smar procurement probem, but for a dfferent voume dscount mode. Ths earer work formuates the probem as a genera mxed nteger near program, and gves some emprca resuts on smuated data. Kaagnanam et a. [12] address doube auctons, where mutpe buyers and seers trade a dvsbe good. The focus of ths paper s aso dfferent: t nvestgates the equbrum prces usng the demand and suppy curves, whereas our focus s on effcent mechansm desgn. Ausube [1] has proposed an ascendng-prce mut-unt aucton for buyers wth margna-decreasng vaues [1], wth an nterpretaton as a prma-dua agorthm [2]. 2. APPROXIMATELY-STRATEGYPROOF VCG AUCTIONS In ths secton, we frst descrbe the margna-decreasng pecewse bddng anguage that s used n our forward and reverse auctons. Contnung, we ntroduce the VCG mechansm for the probem and the ε-domnance resuts for approxmatons to VCG outcomes. We aso dscuss the economc propertes of VCG mechansms n these forward and reverse aucton mut-unt settngs. 2.1 Margna-Decreasng Pecewse Bds We provde a pecewse-constant and margna-decreasng bddng anguage. Ths bddng anguage s expressve for a natura cass of vauaton and cost functons: fxed unt prces over ntervas of quanttes. See Fgure 1 for an exampe. In addton, we sghty reax the margna-decreasng requrement to aow: a bdder n the forward aucton to state a mnma purchase amount, such that she has zero vaue for quanttes smaer than that amount; a seer n the reverse aucton to state a capacty constrant, such that she has an effectvey nfnte cost to suppy quanttes n excess of a partcuar amount. Prce Forward Aucton Bd Quantty Prce Reverse Aucton Bd Quantty Fgure 1: Margna-decreasng, pecewse constant bds. In the forward aucton bd, the bdder offers $10 per unt for quantty n the range [5,10), $8 per unt n the range [10, 20), and $7 n the range [20, 25]. Her vauaton s zero for quanttes outsde the range [10,25]. In the reverse aucton bd, the cost of the seer s outsde the range [10, 25]. In deta, n a forward aucton, a bd from buyer can be wrtten as a st of (quantty-range, unt-prce) tupes, ((u 1, p 1 ), (u 2, p 2 ),..., (u m 1, p m 1 )), wth an upper bound u m on the quantty. The nterpretaton s that the bdder s vauaton n the

3 (sem-open) quantty range [u j, uj+1 ) s p j for each unt. Addtonay, t s assumed that the vauaton s 0 for quanttes ess than u 1 as we as for quanttes more than u m. Ths s mpemented by addng two dummy bd tupes, wth zero prces n the range [0, u 1 ) and (u m, ). We nterpret the bd st as defnng a prce functon, p bd, (q) = qp j, f uj q < uj+1, where j = 1, 2,..., m 1. In order to resove the boundary condton, we assume that the bd prce for the upper bound quantty u m s p bd, (u m ) = u m p m 1. A seer s bd s smary defned n the reverse aucton. The nterpretaton s that the bdder s cost n the (sem-open) quantty range [u j, uj+1 ) s p j for each unt. Addtonay, t s assumed that the cost s for quanttes ess than u 1 as we as for quanttes more than u m. Equvaenty, the unt prces n the ranges [0, u 1 ) and (u m, ) are nfnty. We nterpret the bd st as defnng a prce functon, p ask, (q) = qp j, f uj q < uj VCG-Based Mut-Unt Auctons We construct the tractabe and approxmatey-strategyproof mutunt auctons around a VCG mechansm. We assume that a agents have quasnear utty functons; that s, u (q, p) = v (q) p, for a buyer wth vauaton v (q) for q unts at prce p, and u (q, p) = p c (q) for a seer wth cost c (q) at prce p. Ths s a standard assumpton n the aucton terature, equvaent to assumng rsk-neutra agents [13]. We w use the term payoff nterchangeaby for utty. In the forward aucton, there s a seer wth M unts to se. We assume that ths seer has no ntrnsc vaue for the tems. Gven a set of bds from I agents, et V (I) denote the maxma revenue to the seer, gven that at most one pont on the bd curve can be seected from each agent and no more than M unts of the tem can be sod. Let x = (x 1,..., x N) denote the souton to ths wnner- determnaton probem, where x s the number of unts sod to agent. Smary, et V (I \ ) denote the maxma revenue to the seer wthout bds from agent. The VCG mechansm s defned as foows: 1. Receve pecewse-constant bd curves and capacty constrants from a the buyers. 2. Impement the outcome x that soves the wnner-determnaton probem wth a buyers. 3. Coect payment p vcg, = p bd, (x ) [V (I) V (I \ )] from each buyer, and pass the payments to the seer. In ths forward aucton, the VCG mechansm s strategyproof for buyers, whch means that truthfu bddng s a domnant strategy,.e. utty maxmzng whatever the bds of other buyers. In addton, the VCG mechansm s aocatvey-effcent, and the payments from each buyer are aways postve. 3 Moreover, each buyer pays ess than ts vaue, and receves payoff V (I) V (I \ ) n equbrum; ths s precsey the margna-vaue that buyer contrbutes to the economc effcency of the system. In the reverse aucton, there s a buyer wth M unts to buy, and n suppers. We assume that the buyer has vaue V > 0 to purchase a M unts, but zero vaue otherwse. To smpfy the mechansm desgn probem we assume that the buyer w truthfuy announce ths vaue to the mechansm. 4 The wnner- 3 In fact, the VCG mechansm maxmzes the expected payoff to the seer across a effcent mechansms, even aowng for Bayesan-Nash mpementatons [14]. 4 Wthout ths assumpton, the Myerson-Satterthwate [17] mpossbty resut woud aready mpy that we shoud not expect an effcent tradng mechansm n ths settng. determnaton probem n the reverse aucton s to determne the aocaton, x, that mnmzes the cost to the buyer, or forfets trade f the mnma cost s greater than vaue, V. Let C(I) denote the mnma cost gven bds from a seers, and et C(I \ ) denote the mnma cost wthout bds from seer. We can assume, wthout oss of generaty, that there s an effcent trade and V C(I). Otherwse, then the effcent outcome s no trade, and the outcome of the VCG mechansm s no trade and no payments. The VCG mechansm mpements the outcome x that mnmzes cost based on bds from a seers, and then provdes payment p vcg, = p ask, (x )+[V C(I) max(0, V C(I\))] to each seer. The tota payment s coected from the buyer. Agan, n equbrum each seer s payoff s exacty the margna-vaue that the seer contrbutes to the economc effcency of the system; n the smpe case that V C(I \ ) for a seers, ths s precsey C(I \ ) C(I). Athough the VCG mechansm remans strategyproof for seers n the reverse drecton, ts appcabty s mted to cases n whch the tota payments to the seers are ess than the buyer s vaue. Otherwse, there w be nstances n whch the buyer w not choose to vountary partcpate n the mechansm, based on ts own vaue and ts beefs about the costs of seers. Ths eads to a oss n effcency when the buyer chooses not to partcpate, because effcent trades are mssed. Ths probem wth the sze of the payments, does not occur n smpe snge-tem reverse auctons, or even n mut-unt reverse auctons wth a buyer that has a constant margna-vauaton for each addtona tem that she procures. 5 Intutvey, the probem occurs n the reverse mut-unt settng because the buyer demands a fxed number of tems, and has zero vaue wthout them. Ths eads to the possbty of the trade beng contngent on the presence of partcuar, so-caed pvota seers. Defne a seer as pvota, f C(I) V but C(I\) > V. In words, there woud be no effcent trade wthout the seer. Any tme there s a pvota seer, the VCG payments to that seer aow her to extract a of the surpus, and the payments are too arge to sustan wth the buyer s vaue uness ths s the ony wnnng seer. Concretey, we have ths partcpaton probem n the reverse aucton when the tota payoff to the seers, n equbrum, exceeds the tota payoff from the effcent aocaton: V C(I) [V C(I) max(0, V C(I \ ))] As stated above, frst notce that we requre V > C(I \ ) for a seers. In other words, there must be no pvota seers. Gven ths, t s then necessary and suffcent that: V C(I) (C(I \ ) C(I)) (1) 5 To make the reverse aucton symmetrc wth the forward drecton, we woud need a buyer wth a constant margna-vaue to buy the frst M unts, and zero vaue for addtona unts. The payments to the seers woud never exceed the buyer s vaue n ths case. Conversey, to make the forward aucton symmetrc wth the reverse aucton, we woud need a seer wth a constant (and hgh) margna-cost to se anythng ess than the frst M unts, and then a ow (or zero) margna cost. The tota payments receved by the seer can be ess than the seer s cost for the outcome n ths case.

4 In words, the surpus of the effcent aocaton must be greater than the tota margna-surpus provded by each seer. 6 Consder an exampe wth 3 agents {1, 2, 3}, and V = 150 and C(123) = 50. Condton (1) hods when C(12) = C(23) = 70 and C(13) = 100, but not when C(12) = C(23) = 80 and C(13) = 100. In the frst case, the agent payoffs π = (π 0, π 1, π 2, π 3), where 0 s the seer, s (10,20, 50, 20). In the second case, the payoffs are π = ( 10, 30, 50, 30). One thng we do know, because the VCG mechansm w maxmze the payoff to the buyer across a effcent mechansms [14], s that whenever Eq. 1 s not satsfed there can be no effcent aucton mechansm ε-strategyproofness We now consder the same VCG mechansm, but wth an approxmaton scheme for the underyng aocaton probem. We derve an ε-strategyproofness resut, that bounds the maxma gan n payoff that an agent can expect to acheve through a unatera devaton from foowng a smpe truth-reveang strategy. We descrbe the resut for the forward aucton drecton, but t s qute a genera observaton. As before, et V (I) denote the vaue of the optma souton to the aocaton probem wth truthfu bds from a agents, and V (I \) denote the vaue of the optma souton computed wthout bds from agent. Let ˆV (I) and ˆV (I \ ) denote the vaue of the aocaton computed wth an approxmaton scheme, and assume that the approxmaton satsfes: (1 + ǫ)ˆv (I) V (I) for some ǫ > 0. We provde such an approxmaton scheme for our settng ater n the paper. Let ˆx denote the aocaton mpemented by the approxmaton scheme. The payoff to agent, for announcng vauaton ˆv, s: v (ˆx ) + j ˆv j(ˆx j) ˆV (I \ ) The fna term s ndependent of the agent s announced vaue, and can be gnored n an ncentve-anayss. However, agent can try to mprove ts payoff through the effect of ts announced vaue on the aocaton ˆx mpemented by the mechansm. In partcuar, agent wants the mechansm to seect ˆx to maxmze the sum of ts È true vaue, v (ˆx ), and the reported vaue of the other agents, j ˆvj(ˆxj). If the mechansm s aocaton agorthm s optma, then a the agent needs to do s truthfuy state ts vaue and the mechansm w do the rest. However, faced wth an approxmate aocaton agorthm, the agent can try to mprove ts payoff by announcng a vaue that corrects for the approxmaton, and causes the approxmaton agorthm to mpement the aocaton that exacty maxmzes the tota reported vaue of the other agents together wth ts own actua vaue [18]. 6 Ths condton s mped by the agents are substtutes requrement [3], that has receved some attenton n the combnatora aucton terature because t characterzes the case n whch VCG payments can be supported n a compettve equbrum. Usefu characterzatons of condtons that satsfy agents are substtutes, n terms of the underyng vauatons of agents have proved qute eusve. 7 Moreover, athough there s a sma terature on maxmayeffcent mechansms subject to requrements of vountarypartcpaton and budget-baance (.e. wth the mechansm nether ntroducng or removng money), anaytc resuts are ony known for smpe probems (e.g. [16, 4]). We can now anayze the best possbe gan from manpuaton to an agent n our settng. We frst assume that the other agents are truthfu, and then reax ths. In both cases, the maxma beneft to agent occurs when the nta approxmaton s worst-case. Wth truthfu reports from other agents, ths occurs when the vaue of choce ˆx s V (I)/(1 + ε). Then, an agent coud hope to receve an mproved payoff of: V (I) V (I) 1 + ε = ε 1 + ε V (I) Ths s possbe f the agent s abe to seect a reported type to correct the approxmaton agorthm, and make the agorthm mpement the aocaton wth vaue V (I). Thus, f other agents are truthfu, and wth a (1 + ε)-approxmaton scheme to the aocaton probem, then no agent can mprove ts payoff by more than a factor ε/(1 + ε) of the vaue of the optma souton. The anayss s very smar when the other agents are not truthfu. In ths case, an ndvdua agent can mprove ts payoff by no more than a factor ǫ/(1 + ǫ) of the vaue of the optma souton gven the vaues reported by the other agents. Let V n the foowng theorem defne the tota vaue of the effcent aocaton, gven the reported vaues of agents j, and the true vaue of agent. THEOREM 1. A VCG-based mechansm wth a (1 + ε)- aocaton agorthm s ( ǫ V ) strategyproof for agent, and 1+ǫ agent can gan at most ths payoff through some non-truthfu strategy. Notce that we dd not need to bound the error on the aocaton probems wthout each agent, because the ǫ-strategyproofness resut foows from the accuracy of the frst-term n the VCG payment and s ndependent of the accuracy of the second-term. However, the accuracy of the souton to the probem wthout each agent s mportant to mpement a good approxmaton to the revenue propertes of the VCG mechansm. 3. THE GENERALIZED KNAPSACK PROB- LEM In ths secton, we desgn a fuy poynoma approxmaton scheme for the generazed knapsack, whch modes the wnnerdetermnaton probem for the VCG-based mut-unt auctons. We descrbe our resuts for the reverse aucton varaton, but the formuaton s competey symmetrc for the forward-aucton. In descrbng our approxmaton scheme, we begn wth a smpe property (the Anchor property) of an optma knapsack souton. We use ths property to deveop an O(n 2 ) tme 2-approxmaton for the generazed knapsack. In turn, we use ths basc approxmaton to deveop our fuy poynoma-tme approxmaton scheme (FPTAS). One of the major appeas of our pecewse bddng anguage s ts compact representaton of the bdder s vauaton functons. We strve to preserve ths, and present an approxmaton scheme that w depend ony on the number of bdders, and not the maxmum quantty, M, whch can be very arge n reastc procurement settngs. The FPTAS mpements an (1 + ε) approxmaton to the optma souton x, n worst-case tme T = O(n 3 /ε), where n s the number of bdders, and where we assume that the pecewse bd for each bdder has O(1) peces. The dependence on the number of peces s aso poynoma: f each bd has a maxmum

5 of c peces, then the runnng tme can be derved by substtutng nc for each occurrence of n. 3.1 Premnares Before we begn, et us reca the cassc 0/1 knapsack probem: we are gven a set of n tems, where the tem has vaue v and sze s, and a knapsack of capacty M; a szes are ntegers. The goa s to determne a subset of tems of maxmum vaue wth tota sze at most M. Snce we want to focus on a reverse aucton, the equvaent knapsack probem w be to choose a set of tems wth mnmum vaue (.e. cost) whose sze exceeds M. The generazed knapsack probem of nterest to us can be defned as foows: Generazed Knapsack: Instance: A target M, and a set of n sts, where the th st has the form B = (u 1, p 1 ),..., (u m 1, p m 1 ),(u m (), ), where u j are ncreasng wth j and pj are decreasng wth j, and u j, pj, M are postve ntegers. Probem: Determne a set of ntegers x j such that 1. (One per st) At most one x j s non-zero for any, 2. (Membershp) x j 0 mpes xj [uj, uj+1 ), È È 3. (Target) È j xj M, and È 4. (Objectve) j pj xj s mnmzed. Ths generazed knapsack formuaton s a cear generazaton of the cassc 0/1 knapsack. In the atter, each st conssts of a snge pont (s, v ). 8 The connecton between the generazed knapsack and our aucton probem s transparent. Each st encodes a bd, representng mutpe mutuay excusve quantty ntervas, and one can choose any quantty n an nterva, but at most one nterva can be seected. Choosng nterva [u j, uj+1 ) has cost p j per unt. The goa s to procure at east M unts of the good at mnmum possbe cost. The probem has some favor of the contnuous knapsack probem. However, there are two major dfferences that make our probem sgnfcanty more dffcut: (1) ntervas have boundares, and so to choose nterva [u j, uj+1 ) requres that at east u j and at most uj+1 unts must be taken; (2) unke the cassc knapsack, we cannot sort the tems (bds) by vaue/sze, snce dfferent ntervas n one st have dfferent unt costs. 3.2 A 2-Approxmaton Scheme We begn wth a defnton. Gven an nstance of the generazed knapsack, we ca each tupe t j = (u j, pj ) an anchor. Reca that these tupes represent the breakponts n the pecewse constant curve bds. We say that the sze of an anchor t j s uj, 8 In fact, because of the one per st constrant, the generazed probem s coser n sprt to the mutpe choce knapsack probem [9], where the underng set of tems s parttoned nto dsjont subsets U 1, U 2,..., U k, and one can choose at most one tem from each subset. PTAS do exst for ths probem [10], and ndeed, one can convert our probem nto a huge nstance of the mutpe choce knapsack probem, by creatng one group for each st; put a (quantty, prce) pont tupe (x, p) for each possbe quantty for a bdder nto hs group (subset). However, ths converson expodes the probem sze, makng t nfeasbe for a but the most trva nstances. the mnmum number of unts avaabe at ths anchor s prce p j. The cost of the anchor t j s defned to be the mnmum tota prce assocated wth ths tupe, namey, cost(t j ) = pj uj f j < m, and cost(t m ) = p m 1 u m. In a feasbe souton {x 1, x 2,..., x n} of the generazed knapsack, we say that an eement x 0 s an anchor f x = u j, for some anchor u j. Otherwse, we say that x s mdrange. We observe that an optma knapsack souton can aways be constructed so that at most one souton eement s mdrange. If there are two mdrange eements x and x, for bds from two dfferent agents, wth x x, then we can ncrement x and decrement x, unt one of them becomes an anchor. See Fgure 2 for an exampe. LEMMA 1. [Anchor Property] There exsts an optma souton of the generazed knapsack probem wth at most one mdrange eement. A other eements are anchors. Prce Quantty () Optma souton wth 2 mdrange bds Prce Quantty () Optma sotuton wth 1 mdrange bd Fgure 2: () An optma souton wth more than one bd not anchored (2,3); () an optma souton wth ony one bd (3) not anchored. We use the anchor property to frst obtan a poynoma-tme 2-approxmaton scheme. We do ths by sovng severa nstances of a restrcted generazed-knapsack probem, whch we ca Knapsack, where one eement s forced to be mdrange for a partcuar nterva. Specfcay, suppose eement x for agent s forced to e n ts jth range, [u j, uj+1 ), whe a other eements, x 1,..., x 1, x +1, x n, are requred to be anchors, or zero. Ths corresponds to the restrcted probem Knapsack(, j), n whch the goa s to obtan at east M u j unts wth mnmum cost. Eement x s assumed to have aready contrbuted u j unts. The vaue of a souton to Knapsack(, j) represents the mnma addtona cost to purchase the rest of the unts. We create n 1 groups of potenta anchors, where th group contans a the anchors of the st n the generazed knapsack. The group for agent contans a snge eement that represents the nterva [0, u j+1 u j ), and the assocated unt-prce pj. Ths nterva represents the excess number of unts that can be taken from agent n Knapsack(, j), n addton to u j, whch has aready been commtted. In any other group, we can choose at most one anchor. The foowng pseudo-code descrbes our agorthm for ths restrcton of the generazed knapsack probem. U s the unon of a the tupes n n groups, ncudng a tupe t for agent. The sze of ths speca tupe s defned as u j+1 u j, and the cost s defned as p j (uj+1 u j ). R s the number of unts that reman to be acqured. S s the set of tupes accepted n the current tentatve

6 souton. Best s the best souton found so far. Varabe Skp s ony used n the proof of correctness. Agorthm Greedy(, j) 1. Sort a tupes of U n the ascendng order of unt prce; n case of tes, sort n ascendng order of unt quanttes. 2. Set mark() = 0, for a sts = 1, 2,..., n. Intaze R = M u j, S = Best = Skp =. 3. Scan the tupes n U n the sorted order. Suppose the next tupe s t k,.e. the kth anchor from agent. If mark() = 1, gnore ths tupe; otherwse do the foowng steps: f sze(t k ) > R and = return mn {cost(s) + Rp j, cost(best)}; f sze(t k ) > R and cost(t k ) cost(s) return mn {cost(s) + cost(t k ), cost(best)}; f sze(t k ) > R and cost(t k ) > cost(s) Add t k to Skp; Set Best to S {t k } f cost mproves; f sze(t k ) R then add t k to S; mark() = 1; subtract sze(t k ) from R. The approxmaton agorthm s very smar to the approxmaton agorthm for knapsack. Snce we wsh to mnmze the tota cost, we consder the tupes n order of ncreasng per unt cost. If the sze of tupe t k s smaer than R, then we add t to S, update R, and deete from U a the tupes that beong to the same group as t k. If sze(t k ) s greater than R, then S aong wth t k forms a feasbe souton. However, ths souton can be far from optma f the sze of t k s much arger than R. If tota cost of S and t k s smaer than the current best souton, we update Best. One excepton to ths rue s the tupe t. Snce ths tupe can be taken fractonay, we update Best f the sum of S s cost and fractona cost of t s an mprovement. The agorthm termnates n ether of the frst two cases, or when a tupes are scanned. In partcuar, t termnates whenever we fnd a t k such that sze(t k ) s greater than R but cost(t k ) s ess than cost(s), or when we reach the tupe representng agent and t gves a feasbe souton. LEMMA 2. Suppose A s an optma souton of the generazed knapsack, and suppose that eement (, j) s mdrange n the optma souton. Then, the cost V (, j), returned by Greedy(, j), satsfes: V (,j) + cost(t j ) 2cost(A ) PROOF. Let V (, j) be the vaue returned by Greedy(, j) and et V (, j) be an optma souton for Knapsack(, j). Consder the set Skp at the termnaton of Greedy(, j). There are two cases to consder: ether some tupe t Skp s aso n V (,j), or no tupe n Skp s n V (, j). In the frst case, et S t be the tentatve souton S at the tme t was added to Skp. Because t Skp then sze(t) > R, and S t together wth t forms a feasbe souton, and we have: V (,j) cost(best) cost(s t) + cost(t). Agan, because t Skp then cost(t) > cost(s t), and we have V (, j) < 2cost(t). On the other hand, snce t s ncuded n V (,j), we have V (,j) cost(t). These two nequates mpy the desred bound: V (, j) V (, j) < 2V (, j). In the second case, magne a modfed nstance of Knapsack(, j), whch excudes a the tupes of the set Skp. Snce none of these tupes were ncuded n V (,j), the optma souton for the modfed probem shoud be the same as the one for the orgna. Suppose our approxmaton agorthm returns the vaue V (, j) for ths modfed nstance. Let t be the ast tupe consdered by the approxmaton agorthm before termnaton on the modfed nstance, and et S t be the correspondng tentatve souton set n that step. Snce we consder tupes n order of ncreasng per unt prce, and none of the tupes are gong to be paced n the set Skp, we must have cost(s t ) < V (, j) because S t s the optma way to obtan sze(s t ). We aso have cost(t ) cost(s t ), and the foowng nequates: V (,j) V (, j) cost(s t ) + cost(t ) < 2V (, j) The nequaty V (, j) V (,j) foows from the fact that a tupe n the Skp st can ony affect the Best but not the tentatve soutons. Therefore, droppng the tupes n the set Skp can ony make the souton worse. The above argument has shown that the vaue returned by Greedy(, j) s wthn a factor 2 of the optma souton for Knapsack(, j). We now show that the vaue V (,j) pus cost(t j ) s a 2-approxmaton of the orgna generazed knapsack probem. Let A be an optma souton of the generazed knapsack, and suppose that eement x j s mdrange. Let x to be set of the remanng eements, ether zero or anchors, n ths souton. Furthermore, defne x = x j uj. Thus, cost(a ) = cost(x ) + cost(t j ) + cost(x ) It s easy to see that (x, x ) s an optma souton for Knapsack(, j). Snce V (, j) s a 2-approxmaton for ths optma souton, we have the foowng nequates: V (, j) + cost(t j ) cost(tj ) + 2(cost(x ) + cost(x )) 2(cost(x ) + cost(t j ) + cost(x )) 2cost(A ) Ths competes the proof of Lemma 2. It s easy to see that, after an nta sortng of the tupes n U, the agorthm Greedy(, j) takes O(n) tme. We have our frst poynoma approxmaton agorthm. THEOREM 2. A 2-approxmaton of the generazed knapsack probem can be found n tme O(n 2 ), where n s number of tem sts (each of constant ength). PROOF. We run the agorthm Greedy(, j) once for each tupe (, j) as a canddate for mdrange. There are O(n) tupes, and t suffces to sort them once, the tota cost of the agorthm s O(n 2 ). By Lemma 1, there s an optma souton wth at most one mdrange eement, so our agorthm w fnd a 2-approxmaton, as camed. The dependence on the number of peces s aso poynoma: f each bd has a maxmum of c peces, then the runnng tme s O((nc) 2 ).

7 3.3 An Approxmaton Scheme We now use the 2-approxmaton agorthm presented n the precedng secton to deveop a fuy poynoma approxmaton (FPTAS) for the generazed knapsack probem. The hgh eve dea s fary standard, but the detas requre technca care. We use a dynamc programmng agorthm to sove Knapsack(, j) for each possbe mdrange eement, wth the 2-approxmaton agorthm provdng an upper bound on the vaue of the souton and enabng the use of scang on the cost dmenson of the dynamc programmng (DP) tabe. Consder, for exampe, the case that the mdrange eement s x, whch fas n the range [u j, uj+1 ). In our FPTAS, rather than usng a greedy approxmaton agorthm to sove Knapsack(, j), we construct a dynamc programmng tabe to compute the mnmum cost at whch at east M u j+1 unts can be obtaned usng the remanng n 1 sts n the generazed knapsack. Suppose G[, r] denotes the maxmum number of unts that can be obtaned at cost at most r usng ony the frst sts n the generazed knapsack. Then, the foowng recurrence reaton descrbes how to construct the dynamc programmng tabe: G[0, r] = 0 G[ 1, r] µ G[, r] = max {G[ 1, r cost(t j )] + uj } max j β(,r) where β(, r) = {j : 1 j m, cost(t j ) r}, s the set of anchors for agent. As conventon, agent w ndex the row, and cost r w ndex the coumn. Ths dynamc programmng agorthm s ony pseudo-poynoma, snce the number of coumn n the dynamc programmng tabe depends upon the tota cost. However, we can convert t nto a FPTAS by scang the cost dmenson. Let A denote the 2-approxmaton to the generazed knapsack probem, wth tota cost, cost(a). Let ε denote the desred approxmaton factor. We compute the scaed cost of a tupe t j, denoted scost(t j ), as scost(t j ) = n cost(tj ) εcost(a) (2) Ths scang mproves the runnng tme of the agorthm because the number of coumns n the modfed tabe s at most n, and ndependent of the tota cost. However, the computed ε souton mght not be an optma souton for the orgna probem. We show that the error ntroduced s wthn a factor of ε of the optma souton. As a preude to our approxmaton guarantee, we frst show that f two dfferent soutons to the Knapsack probem have equa scaed cost, then ther orgna (unscaed) costs cannot dffer by more than εcost(a). LEMMA 3. Let x and y be two dstnct feasbe soutons of Knapsack(, j), excudng ther mdrange eements. If x and y have equa scaed costs, then ther unscaed costs cannot dffer by more than εcost(a). PROOF. Let I x and I y, respectvey, denote the ndcator functons assocated wth the anchor vectors x and y there s 1 n poston I x[, k] f the x k > 0. Snce x and y has equa scaed cost, k scost(t k )I x[, k] = k scost(t k )I y[, k] (3) However, by (2), the scaed costs satsfy the foowng nequates: (scost(t k ) 1)εcost(A) n cost(t k ) scost(tk )εcost(a) n (4) Substtutng the upper-bound on scaed cost from (4) for cost(x), the ower-bound on scaed cost from (4) for cost(y), and usng equaty (3) to smpfy, we have: cost(x) cost(y) εcost(a) n k I y[, k] εcost(a), The ast nequaty uses the fact that at most n components of an ndcator vector are non-zero; that s, any feasbe souton contans at most n tupes. Fnay, gven the dynamc programmng tabe for Knapsack(, j), we consder a the entres n the ast row of ths tabe, G[n 1, r]. These entres correspond to optma soutons wth a agents except, for dfferent eves of cost. In partcuar, we consder the entres that provde at east M u j+1 unts. Together wth a contrbuton from agent, we choose the entry n ths set that mnmzes the tota cost, defned as foows: cost(g[n 1, r]) + max {u j, M G[n 1, r]}pj, where cost() s the orgna, unscaed cost assocated wth entry G[n 1, r]. It s worth notng, that unke the 2-approxmaton scheme for Knapsack(, j), the vaue computed wth ths FPTAS ncudes the cost to acqure u j unts from. The foowng emma shows that we acheve a (1+ε)-approxmaton. LEMMA 4. Suppose A s an optma souton of the generazed knapsack probem, and suppose that eement (, j) s mdrange n the optma souton. Then, the souton A(, j) from runnng the scaed dynamc-programmng agorthm on Knapsack(, j) satsfes cost(a(, j)) (1 + 2ε)cost(A ) PROOF. Let x denote the vector of the eements n souton A wthout eement. Then, by defnton, cost(a ) = cost(x ) + p j xj. Let r = scost(x ) be the scaed cost assocated wth the vector x. Now consder the dynamc programmng tabe constructed for Knapsack(, j), and consder ts entry G[n 1, r]. Let A denote the 2-approxmaton to the generazed knapsack probem, and A(, j) denote the souton from the dynamc-programmng agorthm. Suppose y s the souton assocated wth ths entry n our dynamc program; the components of the vector y are the quanttes from dfferent sts. Snce both x and y have equa scaed costs, by Lemma 3, ther unscaed costs are wthn εcost(a) of each other; that s, cost(y ) cost(x ) È εcost(a). È Now, defne y j = max{uj, M j yj }; ths s the contrbuton needed from to make (y, y j ) a feasbe souton. Among a the equa cost soutons, our dynamc programmng tabes chooses the one wth maxmum unts. Therefore, j y j j x j

8 Therefore, t must be the case that y j xj. Because (yj, y ) s aso a feasbe souton, f our agorthm returns a souton wth cost cost(a(,j)), then we must have cost(a(, j)) cost(y ) + p j yj cost(x ) + εcost(a) + p j xj (1 + 2ε)cost(A ), where we use the fact that cost(a) 2cost(A ). Puttng ths together, our approxmaton scheme for the generazed knapsack probem w terate the scheme descrbed above for each choce of the mdrange eement (, j), and choose the best souton from among these O(n) soutons. For a gven mdrange, the most expensve step n the agorthm s the constructon of dynamc programmng tabe, whch can be done n O(n 2 /ε) tme assumng constant ntervas per st. Thus, we have the foowng resut. THEOREM 3. We can compute an (1 + ε) approxmaton to the souton of a generazed knapsack probem n worst-case tme O(n 3 /ε). The dependence on the number of peces s aso poynoma: f each bd has a maxmum of c peces, then the runnng tme can be derved by substtutng cn for each occurrence of n. 4. COMPUTING VCG PAYMENTS We now consder the reated probem of computng the VCG payments for a the agents. A nave approach requres sovng the aocaton probem n tmes, removng each agent n turn. In ths secton, we show that our approxmaton scheme for the generazed knapsack can be extended to determne a n payments n tota tme O(αT og(αn/ε)), where 1 C(I\)/C(I) α, for a constant upper bound, α, and T s the compexty of sovng the aocaton probem once. Ths α-bound can be justfed as a no monopoy condton, because t bounds the margna vaue that a snge buyer brngs to the aucton. Smary, n the reverse varaton we can compute the VCG payments to each seer n tme O(αT og(αn/ε)), where α bounds the rato C(I\ )/C(I) for a. Our overa strategy w be to bud two dynamc programmng tabes, forward and backward, for each mdrange eement (, j) once. The forward tabe s but by consderng the agents n the order of ther ndces, where as the backward tabe s but by consderng them n the reverse order. The optma souton correspondng to C(I \ ) can be broken nto two parts: one correspondng to frst ( 1) agents and the other correspondng to ast (n ) agents. As the ( 1)th row of the forward tabe corresponds to the seers wth frst ( 1) ndces, an approxmaton to the frst part w be contaned n ( 1)th row of the forward tabe. Smary, (n )th row of the backward tabe w contan an approxmaton for the second part. We frst present a smpe but an neffcent way of computng the approxmate vaue of C(I \ ), whch ustrates the man dea of our agorthm. Then we present an mproved scheme, whch uses the fact that the eements n the rows are sorted, to compute the approxmate vaue more effcenty. In the foowng, we concentrate on computng an aocaton wth x j beng mdrange, and some agent removed. Ths w be a component n computng an approxmaton to C(I \ ), the vaue of the souton to the generazed knapsack wthout bds from agent. We begn wth the smpe scheme. 4.1 A Smpe Approxmaton Scheme We mpement the scaed dynamc programmng agorthm for Knapsack(, j) wth two aternate orderngs over the other seers, k, one wth seers ordered 1,2,..., n, and one wth seers ordered n, n 1,..., 1. We ca the frst tabe the forward tabe, and denote t F, and the second tabe the backward tabe, and denote t B. The subscrpt remnds us that the agent s mdrange. 9 In budng these tabes, we use the same scang factor as before; namey, the cost of a tupe t j s scaed as foows: scost(t j ) = ncost(tj ) εcost(a) where cost(a) s the upper bound on C(I), gven by our 2- approxmaton scheme. In ths case, because C(I \ ) can be α tmes C(I), the scaed vaue of C(I \ ) can be at most nα/ε. Therefore, the cost dmenson of our dynamc program s tabe w be nα/ε n m 1 m g F ( 1) Tabe F m 1 m h B (n ) Tabe Fgure 3: Computng VCG payments. m = nα ε B n 1 n 2 Now, suppose we want to compute a (1 + ǫ)-approxmaton to the generazed knapsack probem restrcted to eement (, j) mdrange, and further restrcted to remove bds from some seer. Ca ths probem Knapsack (, j). Reca that the th row of our DP tabe stores the best souton possbe usng ony the frst agents excudng agent, a of them ether ceared at zero, or on anchors. These frst agents are a dfferent subset of agents n the forward and the backward tabes. By carefuy combnng one row of F wth one row of B we can compute an approxmaton to Knapsack (,j). We consder the row of F that corresponds to soutons constructed from agents {1, 2,..., 1}, skppng agent. We consder the row of B that corresponds to soutons constructed from agents {+1, +2,..., n}, agan skppng agent. The rows are abeed F ( 1) and B (n ) respectvey. 10 The scaed costs for acqurng these unts are the coumn ndces for these entres. To sove Knapsack (, j) we choose one entry from row F ( 1) and one from row B (n ) such that ther tota quantty exceeds M u j+1 and ther combned cost s mnmum over a such combnatons. Formay, et g F ( 1), and h B (n 1) denote entres n each row, wth sze(g), sze(h), denotng the number of unts and cost(g) and cost(h) denotng the unscaed cost assocated wth the entry. We compute the foowng, subject 9 We coud abe the tabes wth both and j, to ndcate the jth tupe s forced to be mdrange, but omt j to avod cutter. 10 To be precse, the ndex of the rows are ( 2) and (n ) for F and B when <, and ( 1) and (n 1), respectvey, when >. n 1

9 to the condton that g and h satsfy sze(g) + sze(h) > M u j+1 : mn g F ( 1),h B (n ) Ò cost(g) + cost(h) + Ó p j max{uj, M sze(g) sze(h)} (5) LEMMA 5. Suppose A s an optma souton of the generazed knapsack probem wthout bds from agent, and suppose that eement (, j) s the mdrange eement n the optma souton. Then, the expresson n Eq. 5, for the restrcted probem Knapsack (,j), computes a (1 + ε)-approxmaton to A. PROOF. From earer, we defne cost(a ) = C(I \ ). We can spt the optma souton, A, nto three dsjont parts: x corresponds to the mdrange seer, x corresponds to frst 1 seers (skppng agent f < ), and x corresponds to ast n seers (skppng agent f > ). We have: cost(a ) = cost(x ) + cost(x ) + p j xj Let r = scost(x ) and r = scost(x ). Let y and y be the souton vectors correspondng to scaed cost r and r n F ( 1) and B (n ), respectvey. From Lemma 3 we concude that, cost(y ) + cost(y ) cost(x ) cost(x ) εcost(a) where cost(a) s the upper-bound on C(I) computed wth the 2-approxmaton. Among a equa scaed cost soutons, our dynamc program chooses the one wth maxmum unts. Therefore we aso have, (sze(y ) sze(x )) and (sze(y ) sze(x )) where we use shorthand sze(x) to denote tota number of unts n a tupes n x. Now, defne y j = max(u j, M sze(y) sze(y )). From the precedng nequates, we have y j xj. Snce (yj, y, y ) s aso a feasbe souton to the generazed knapsack probem wthout agent, the vaue returned by Eq. 5 s at most cost(y ) + cost(y ) + p j yj C(I \ ) + εcost(a) Ths competes the proof. C(I \ ) + 2cost(A )ε C(I \ ) + 2C(I \ )ε A nave mpementaton of ths scheme w be neffcent because t mght check (nα/ε) 2 pars of eements, for any partcuar choce of (, j) and choce of dropped agent. In the next secton, we present an effcent way to compute Eq. 5, and eventuay to compute the VCG payments. 4.2 Improved Approxmaton Scheme Our mproved approxmaton scheme for the wnner-determnaton probem wthout agent uses the fact that eements n F ( 1) and B (n ) are sorted; specfcay, both, unscaed cost and quantty (.e. sze), ncreases from eft to rght. As before, et g and h denote generc entres n F ( 1) and B (n ) respectvey. To compute Eq. 5, we consder a the tupe pars, and frst dvde the tupes that satsfy condton sze(g) + sze(h) > M u j+1 nto two dsjont sets. For each set we compute the best souton, and then take the best between the two sets. [case I: sze(g) + sze(h) M u j ] The probem reduces to mn g F ( 1), h B (n ) Ò Ó cost(g) + cost(h) + p j uj We defne a par (g,h) to be feasbe f sze(g) + sze(h) M u j. Now to compute Eq. 6, we do a forward and backward wak on F ( 1) and B (n ) respectvey. We start from the smaest ndex of F ( 1) and move rght, and from the hghest ndex of B (n ) and move eft. Let (g,h) be the current par. If (g, h) s feasbe, we decrement B s ponter (that s, move backward) otherwse we ncrement F s ponter. The feasbe pars found durng the wak are used to compute Eq. 6. The compexty of ths step s near n sze of F ( 1), whch s O(nα/ε). [case II: M u j+1 sze(g) + sze(h) M u j ] The probem reduces to mn g F ( 1), h B (n ) Ò cost(g) + cost(h) + p j (M sze(g) sze(h)) Ó To compute the above equaton, we transform the above probem to another probem usng modfed cost, whch s defned as: mcost(g) = cost(g) p j sze(g) mcost(h) = cost(h) p j sze(h) The new probem s to compute Ó Òmcost(g) + mcost(h) + p j M mn g F ( 1), h B (n ) The modfed cost smpfes the probem, but unfortunatey the eements n F ( 1) and B (n ) are no onger sorted wth respect to mcost. However, the eements are st sorted n quantty and we use ths property to compute Eq. 7. Ca a par (g,h) feasbe f M u j+1 sze(g) + sze(h) M u j. Defne the feasbe set of g as the eements h B (n ) that are feasbe gven g. As the eements are sorted by quantty, the feasbe set of g s a contguous subset of B (n ) and shfts eft as g ncreases Begn 4 End 5 6 F ( 1) B (n ) Fgure 4: The feasbe set of g = 3, defned on B (n ), s {2, 3,4} when M u j+1 = 50 and M u j = 60. Begn and End represent the start and end ponters to the feasbe set. Therefore, we can compute Eq. 7 by dong a forward and backward wak on F ( 1) and B (n ) respectvey. We wak on B (n ), startng from the hghest ndex, usng two ponters, Begn and End, to ndcate the start and end of the current feasbe set. We mantan the feasbe set as a mn heap, where the key s modfed cost. To update the feasbe set, when we ncrement F s ponter(move forward), we wak eft on B, frst usng End to remove eements from feasbe set whch are no onger (6) (7)

10 feasbe and then usng Begn to add new feasbe eements. For a gven g, the ony eement whch we need to consder n g s feasbe set s the one wth mnmum modfed cost whch can be computed n constant tme wth the mn heap. So, the man compexty of the computaton es n heap updates. Snce, any eement s added or deeted at most once, there are O( nα ε ) heap updates and the tme compexty of ths step s O( nα ε og nα ε ). 4.3 Coectng the Peces The agorthm works as foows. Frst, usng the 2 approxmaton agorthm, we compute an upper bound on C(I). We use ths bound to scae down the tupe costs. Usng the scaed costs, we bud the forward and backward tabes correspondng to each tupe (, j). The forward tabes are used to compute C(I). To compute C(I \ ), we terate over a the possbe mdrange tupes and use the correspondng forward and backward tabes to compute the ocay optma souton usng the above scheme. Among a the ocay optma soutons we choose one wth the mnmum tota cost. The most expensve step n the agorthm s computaton of C(I \ ). The tme compexty of ths step s O( n2 α og nα ) ε ε as we have to terate over a O(n) choces of t j, for a, and each tme use the above scheme to compute Eq. 5. In the worst case, we mght need to compute C(I \ ) for a n seers, n whch case the fna compexty of the agorthm w be O( n3 α og nα ). ε ε THEOREM 4. We can compute an ǫ/(1+ǫ)-strategyproof approxmaton to the VCG mechansm n the forward and reverse mut-unt auctons n worst-case tme O( n3 α ε og nα ε ). It s nterestng to reca that T = O( n3 ) s the tme compexty of the FPTAS to the generazed knapsack probem wth a ε agents. Our combned scheme computes an approxmaton to the compete VCG mechansm, ncudng payments to O(n) agents, n tme compexty O(T og(n/ε)), takng the no-monopoy parameter, α, as a constant. Thus, our agorthm performs much better than the nave scheme, whch computes the VCG payment for each agent by sovng a new nstance of generazed knapsack probem. The speed up comes from the way we sove Knapsack (, j). Tme compexty of computng Knapsack (, j) by creatng a new dynamc programmng tabe w be O( n2 ) ε but by usng the forward and backward tabes, the compexty s reduced to O( n og n ). We can further mprove the tme compexty of our agorthm by computng Eq. 5 more effcenty. ε ε Currenty, the agorthm uses heap, whch has ograthmc update tme. In worst case, we can have two heap update operatons for each eement, whch makes the tme compexty super near. If we can compute Eq. 5 n near tme then the compexty of computng the VCG payment w be same as the compexty of sovng a snge generazed knapsack probem. 5. CONCLUSIONS We presented a fuy poynoma-tme approxmaton scheme for the snge-good mut-unt aucton probem, usng margna decreasng pecewse constant bddng anguage. Our scheme s both approxmatey effcent and approxmatey strategyproof wthn any specfed factor ε > 0. 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