Online story scheduling in web advertising

Transcription

1 Onlne tory chedulng n web advertng Anrban Dagupta Arpta Ghoh Hamd Nazerzadeh Prabhakar Raghavan Abtract We tudy an onlne job chedulng problem motvated by toryboardng n web advertng, where an adverter derve value from unnterrupted equental acce to a uer urfng the web. The uer ceae to browe wth probablty at each tep, ndependently. Store job) arrve onlne; job ha length l and per-unt value v. A value v obtaned for every unt of the job that cheduled conecutvely wthout nterrupton, dcounted for the tme at whch t cheduled. Job can be preempted, but no further value can be derved from the redual uncheduled unt of the job. We eek an onlne algorthm whoe total reward compettve agant that of the offlne cheduler that know all job n advance. We conder two model baed on the maxmum delay that can be allowed between the arrval and chedulng of a job. In the frt, a job can be cheduled anytme after t arrval; n the econd a job lot unle cheduled mmedately upon arrval, preemptng a currently runnng job f needed. The two ettng correpond to two natural model of how long an adverter retan nteret n a relevant uer. We how that there, n fact, a harp eparaton between what an onlne cheduler can acheve n thee two ettng. In the frt ettng wth no deadlne, we gve a natural determntc algorthm wth a contant compettve rato agant the offlne cheduler. In contrat, we how that n the harp deadlne ettng, no determntc or randomzed) onlne algorthm can acheve better than a polylogarthmc rato. 1 Introducton Onlne advertng a major ource of revenue for Internet compane, wth dplay advertng contrbutng a gnfcant 21% [10]) and growng fracton. In dplay advertng, the content of a webpage and ncreangly, the browng htory of a uer ued for targetng ad. One paradgm for targetng dplay ad, baed Yahoo! Reearch, Santa Clara, CA, emal: {anrban, arpta, pragh}@yahoo-nc.com Stanford Unverty, Stanford, CA, emal: hamdnz@tanford.edu on th ablty to track a uer acro the web, toryboardng, alo referred to a equence advertng or urround eon [11, 8] 1. Here, a ngle adverter get excluve acce to a uer for a equence of conecutve page vewed by the uer, wth no nterrupton from other adverter. Th equence of lot can be ued by the adverter ether to how a et of unrelated ad for renforcement of h meage, or to creatvely ue a tory lne acro everal page. Conder a ettng where multple adverter each ve for acce to a equence of contguou tep durng a uer browng eon. Each adverter appear at ome tep n the eon, when the uer vt a webpage wth content relevant to that adverter. The number of lot ought by the adverter, a well a h value from havng h requet granted, vare by the combnaton of adverter and uer. A adverter requet arrve onlne each trggered by the current tate of the uer browng htory), an ad erver mut decde whch requet to grant, nce at any tme only a ngle adverter can have acce to the uer. The objectve of the ad erver to maxmze total value on a uer. For our purpoe, t uffce to conder the decon made by the ad erver for a ngle uer, nce the overall value can be ummed over the ndvdual uer. Addtonally, a uer may ext h browng eon at any tep a clacal aumpton n web uer modelng. Thu, the ad erver face a tradeoff between adverter requet contng of hort equence wth a hgh per-unt value and longer one of hgh total value. The prototypcal decon for the ad erver then become: for each requet, decde whether and when to erve the requet, and whether to preempt an adequence currently n progre from a pror requet. 1.1 Model We abtract the followng onlne job chedulng problem from th ettng. At each tep, the uer top urfng wth probablty o the urfng tme a geometrc random varable). Job, whch correpond to adverter tore, arrve onlne. A job 1 In 2002 [11], the New York Tme offered adverter the opportunty to completely control the ad een by a partcular uer for a mall number 4-8) of conecutve page vewed by th uer the frt ntance of onlne toryboardng Copyrght by SIAM. Unauthorzed reproducton of th artcle prohbted.

2 ha an arrval tme a, a length l and a per-unt or per mpreon) value v. We ometme ue value to mean per-unt value. In general, the per-unt value need not be the ame for all ad n a tory; we dcu th n Secton 4.) An nput equence Q a et of job {v, a, l )}, and correpond to the et of adverter that arrve a the uer urf page relevant to them. Multple realtc aumpton can be made about value derved from upendng and retartng nterrupted job n th paper we aume that uch upenon not allowed. That, a job can be nterrupted, but once nterrupted, no value can be obtaned from t remanng unt. We nvetgate two model that dffer n the delay that can be allowed between the arrval and chedulng of a job, correpondng to two natural model of toryboard. In the no deadlne model, a job can be cheduled anytme after t arrval here, adverter are wllng to adverte to a uer at any tme t a, after dcoverng that the uer relevant). In the harp deadlne model, a job lot unle cheduled mmedately upon arrval at t = a here, adverter loe nteret n the uer a oon a he navgate away from the relevant page). A chedule feable f all cheduled unt of a job are conecutve, ubject to the deadlne contrant. The dcounted reward V S from a feable chedule S t expected value t=0 βt vt), where vt) the perunt value of the job n progre at tme t. We want to degn an onlne cheduler whoe dcounted reward compettve agant the dcounted reward of the optmal) offlne cheduler that know the entre nput equence n advance Reult and organzaton We how that there a harp eparaton n the power of an onlne cheduler that ha no deadlne veru one wth harp deadlne: we gve a contant compettve algorthm for the frt cae, wherea we how that no randomzed onlne algorthm can acheve better than a polylogarthmc rato n the econd. Note that whle an onlne cheduler wth no deadlne can clearly obtan hgher reward than an onlne cheduler wth harp deadlne, the offlne cheduler correpondngly advantaged a well, o t not a pror obvou how the compettve rato n the two model wll compare. 2 Note that the offlne oluton doe not know exactly when the uer top urfng, but only maxmze expected value. An onlne cheduler very lmted wth repect to the offlne cheduler whch know the toppng tme of the uer. Thee lmtaton are alo oberved n other onlne ettng [1]. In the no-deadlne model, we gve a natural determntc algorthm that 7-compettve agant the offlne cheduler Secton 2). If the dcount factor β = 1, the algorthmc problem n the no-deadlne model trval: never preempt. For any β < 1, there a preemptondelay tradeoff the cheduler ether ha to preempt the current job and loe all t remanng value, or ha to delay the newly arrved job and pay a nonzero) delay cot due to the dcount factor. Oberve that preempton cheap for β near 0 and delay expenve, wherea the convere true for β cloe to 1.) In fact, the problem ha a dcontnuty at β = 1: whle at β = 1, the trval onlne) algorthm that never preempt optmal, no determntc algorthm can have a compettve rato better than 2 3ɛ) for β = 1 ɛ, ɛ > 0, a the followng example how. The frt job arrve at tme 0 wth per-unt value 1 and nfnte length. If the determntc onlne algorthm begn th job at tme t, the adverary ntroduce a job at tme t + 1 wth value 1/) and length 1. The offlne cheduler get a total dcounted reward of β t β+β 2 ). The determntc algorthm can get no ) more than max β t + β t+1 1, β t = βt, whch gve a compettve rato of β + β 2 2 3ɛ for β = 1 ɛ. The ntuton behnd the dcontnuty that for any β < 1, there alway a job wth hgh enough value that make preempton worthwhle; th not true for β = 1. In the harp deadlne model, we adapt the proof from Cannett and Iran [3] to how that depte partal credt and the dcount factor, no onlne cheduler can acheve a reward better than a polylogarthmc factor of the offlne cheduler Secton 3). In Secton 4 we conder a natural extenon n the toryboardng applcaton, where job have ncreang rather than contant) per-unt value. For the extreme cae where value obtaned only when the job fnhed, even wth no deadlne, we how that no onlne determntc algorthm can acheve any contant compettve factor wth repect to the offlne cheduler. However, contant compettvene can be obtaned when the length of the job are bounded, and we ue th to gve a logarthmc approxmaton. 1.3 Related Work See [6] for a urvey of the vat lterature on job chedulng. The man dfference wth our work the nfnte horzon wth dcount 1276 Copyrght by SIAM. Unauthorzed reproducton of th artcle prohbted.

3 factor β. Our man algorthmc reult are n the nodeadlne model, whch trval for β = 1 and ha conequently not receved attenton n the lterature. Woegnger [12] tude the model cloet to our n term of the feablty of an allocaton th, n fact, the harp deadlne model wth no partal credt at β = 1. However, the algorthmc reult n [12] are under retrcton on the nput that are napplcable n our ettng; the hardne reult are for determntc algorthm for β = 1 and no partal credt n the harp deadlne model. A mentoned n Secton 1.2, Cannett and Iran [3] tudy lower bound n the model wth harp deadlne and no partal credt at β = 1. There ha alo been work n mechanm degn whch nvolve onlne allocaton problem [4, 9, 7, 5]. Thee model dffer from our n multple way, mot notably n not havng a tme-dcounted nfnte horzon. The dcounted reward cenaro ha been condered prevouly n ettng other than job-chedulng [2]. 2 A greedy algorthm for the no-deadlne model We now preent a greedy algorthm for onlne job chedulng n the no-deadlne model. There are two apect to conder the preempton-delay tradeoff nce job cannot be reumed after nterrupton), and the fact that job arrve onlne. In fact, the preemptondelay tradeoff alo faced by an offlne cheduler; we how n Appendx 5 that the offlne problem NP-hard, by a reducton from ubet-um. The onlne arrval of job further comfound matter n the no-deadlne cae the onlne algorthm mut decde whether to chedule a long job, or wat for mmnent hgh per-unt value job. When all job are avalable at tme 0, the problem eay: Lemma 2.1. If all job are preent at tme 0, chedulng job n decreang order of per-unt value optmal. For a et of job S, let V S) denote the reward obtaned by chedulng job n S a n Lemma 2.1: n computng V S) we pretend that all job n S have arrval tme 0. In general, of coure, there wll be new arrval whle a job n progre. At tme t, let be the currently cheduled job wth value v and l remanng uncheduled unt. Let A t be the et of all job wth value hgher than v. The greedy decon at tme t baed on comparng the reward from preemptng and mmedately chedulng job n A t, to the reward from completng and then chedulng A t. That, we preempt f 1 + β + β β l 1 )v + β l V At ) = l v + β l V At ) < V A t ). Rearrangng the above nequalty gve u the followng rule for preempton: Preempt a job f and only f v < )V A t ). Th rule mply compare the beneft from chedulng another unt of the current job, to the cot of delayng the job n A t by one tep. The correpondng greedy algorthm, G, gven n Fgure 1. Note that A t now only contan job that arrve after the current job cheduled by G. 2.1 Analy In th ecton, we prove that G 7-compettve agant the offlne cheduler Theorem 2.1). In fact, we prove that G 7-compettve agant a tronger optmal cheduler O whch allowed to reume job after nterrupton. The tronger cheduler equvalent to modfyng the nput equence Q to replace each job v, a, l ) by l job wth length 1 and value v : the optmal cheduler O greedly chedule the avalable unt wth hghet value at each tme. For mplcty of preentaton, we aume that all value v are dtnct 3. We denote by AQ) the chedule returned by a cheduler A on an nput Q, and by V AQ) the dcounted reward from th chedule. When t clear from the context, we ue A to denote both the chedule and the cheduler. There are two factor that make G chedule uboptmal: preempted unt that are never cheduled, and unt that are delayed. To analyze thee, we frt ntroduce a new nput equence Q wth utably delayed arrval, a defned below. The dea behnd the contructon of Q that, roughly peakng, G chedule the ame on Q and Q, and G uboptmal wth repect to OQ ) due only to preempted unt. In Lemma 2.6, we how V OQ ) 2V G. To account for the delay cot, we now only need to relate V OQ) to V OQ ) we do th by ntroducng another nput equence Q, wth ome unt arrvng earler than n Q, uch that V OQ ) V OQ). We then relate V OQ ) to V OQ ) through Lemma 2.2, 2.3, 2.4, and 2.5 to get the reult n Corollary 2.1. Note that wth repect to the cheduler O, we can thnk of Q a a equence of unt ntead of a equence of 3 One can perturb all value wth arbtrary mall noe Copyrght by SIAM. Unauthorzed reproducton of th artcle prohbted.

4 Algorthm G: At each tme t: If no job n progre, Schedule a job wth the hghet per-unt value. Otherwe, let be the current job. Let A t be the et of all job wth per-unt value hgher than v. If v < )V A t ) then Preempt Schedule a job wth the hghet per-unt value. Otherwe, contnue wth the current job. Fgure 1: A greedy algorthm for the no-deadlne model job. The arrval of the k th unt of job n Q defned a a + k 1. For job, let r be the tme at whch G begn, and e be the tme at whch the lat unt of cheduled,.e., completed at e or preempted at e + 1. Alo, defne D to be a et of unt uch that each unt D atfe the followng properte: 1. OQ) chedule ometme between r and e. 2. v > v. 3. If belong to job, then a > r. Now, baed on the chedule G for the nput Q, we contruct two equence wth modfed arrval, denoted by Q and Q. If unt belong to D for a job, then arrve n Q at tme r, and n Q at tme e + 1. If doe not belong to D for any, then t arrval tme the ame n Q, Q and Q. Let OQ ) and OQ ) denote the chedule of O on the nput equence Q and Q. Alo, let r rep. r ) be the tme at whch cheduled n OQ ) rep. OQ )). Lemma 2.2. If unt doe not belong to any et D, then r r. Proof. We prove the lemma by contradcton. Aume the frt unt cheduled n OQ ) uch that r < r. Let j be the unt cheduled n OQ ) at tme r. If doe not belong to any et D, then t arrve at the ame tme n Q and Q. Therefore, for nput Q and at tme r, O could have cheduled unt. Hence, we have v j v. Conder the followng cae. 1. Suppoe v j > v. Note that the arrval of every unt Q no later than t arrval n Q. Alo, O alway chedule a unt wth the hghet value. Therefore, r j < r = r j, whch contradct the aumpton that the frt unt uch that r < r. 2. If v j = v, then thee unt belong to the ame job. Snce O frt chedule the unt wth the earler arrval, we have r j < r = r j, whch agan lead to a contradcton. The lemma above gve u the followng nequalty. V OQ ) 2.1) v + V OQ ). β r Now we wll how that β r v V G + 2V OQ ). Let l = e r + 1 denote the number of unt of job cheduled by G. Note that β r v = l ) β r v + β l β r v. We prove the clam by boundng each term n the r.h. above. Lemma 2.3. l ) β r v V G. Proof. Recall that A t the et of all job that are avalable at tme t and have per-unt value hgher than the current job. Let t = e. Oberve that all unt n D belong to the job n A t. Therefore, V A t ) V D ). Becaue the algorthm dd not preempt job at tme t, we have Therefore, v )V A t ) )V D ). l v l )V D ). Summng up th nequalty over all job, we have V G l ) v. β r 1278 Copyrght by SIAM. Unauthorzed reproducton of th artcle prohbted.

5 Now, we fnd a matchng between unt n D and unt n OQ ). Defne m j to be the unque lot that a unt j D matched to. The matchng contructed a follow. Let U t be the orted lt of all unmatched unt up to tme t; U 0 ntalzed to be the empty et. At each tme t, add all unmatched unt from U t 1 plu all unt uch that D and r + l = t to U t. Then, let j be the unt wth the hghet value n U t. Te are broken n favor of unt that arrve earler n Q. Fnally, let m j = t. Lemma 2.4. For unt D, we have m r. Proof. Frt oberve that for D, r r + l. We prove the lemma by nducton on the chedulng tme n OQ ) of a unt U t. The ba of the nducton trval. Now, uppoe for all unt j, r j < r, we have m j r j. Suppoe ha not been matched yet. Hence, belong to U r. Oberve that by nducton hypothe, all unmatched unt n U r are avalable to OQ ). Snce OQ )) chedule at th tme r ), ha the earlet arrval among avalable unt wth the hghet value. Therefore, alo ha the earlet arrval among unt wth the hghet value n U r. Thu, would be matched at th tep and we have m = r r By the lemma above we have, 2.2) β m v V OQ ). Lemma 2.5. β r +l β m )v V OQ ). Proof. Conder unt j. Suppoe at tep t, j belong to U t. If no other unt added to U t, then j would be matched at tme t + π j 1, where π j the poton of j n the orted lt U t. Now uppoe unt added to U t at th pont unt are added to U t n the order of ncreang r ). Then, the poton to whch j would be matched ncreae by 1. Let n j be the lot that unt j would be matched to rght before added. Alo, let n j be the lot that unt j would be matched to rght after added. Note that for j U t, n j n j 1, where the equalty hold f when added to U t, j alo n U t and ha le value. Alo, note that for all unt j uch that n j n j = 1, all value of n j are unque, and are between r + l, and m. Therefore, t eay to ee that we have β r +l β m )v =. β nj β n j )v. j D Renamng the varable we have β r +l β m )v = β nj β n j )vj. j Conder unt D, where the job durng whch unt arrve n Q. Let be the unt that cheduled n OQ ) at tme r. Suppoe for unt j we have n j n j = 1. Note that j / D, and hence t wa avalable at tme r n Q. By Lemma 2.4, j ha not been cheduled up to tme r. Therefore, we have v v j. Hence, β nj β n j )vj j Therefore, we have β nj β n j )v. β nj β n j )v = j )β t v t r +l β r +l v β r v. But, by defnton of, β r v V OQ ). Therefore, β r +l β m )v V OQ ). Puttng all lemma above together we get a bound on the delay cot of G: Corollary 2.1. V OQ ) V G + 3V OQ ). Next we handle the preempton cot. Lemma 2.6. V OQ ) 2V G. Proof. Note that every unt n OQ ) ha the followng property: ether G chedule t no later than O, or G never chedule t at all,.e., the unt preempted. Hence, we complete the proof of the lemma by howng that the value of the preempted unt bounded by V G. A preempton chan 1,, n ) a equence of job defned by the followng properte. ) 1 doe not belong to any prevou chan. ) Story, 1 n, preempted at tme t. ) Let k be the um of length of the job avalable at tme t that have hgher perunt value than. For each 1 < n, t < t +1 t + k 1. ) G doe not preempt any job between t n and t n + k n 1. Becaue the number of job fnte, chan are of fnte length. j 1279 Copyrght by SIAM. Unauthorzed reproducton of th artcle prohbted.

6 Conder a chan 1,, n ). Let A n be the job wth hgher per-unt value than n that are avalable at tme t n. Snce n preempted, we have )V A n ) > v n. Alo, becaue there no preempton between t n and t n + k n 1, t eay to ee that the reward G obtan from the unt cheduled durng th tme at leat 2.3) β tn V A n ) β tn v n /)). Note that the per-unt value of the job ncreang along a chan. Therefore, n OQ ) all preempted unt of 1,, n are cheduled after t n + k n 1. Alo, oberve that the total reward O could obtan from thee job at mot β tn+kn v n /)). Pluggng nto 2.3) we get β tn+kn v n /)) β tn v n /)) β tn V A n ). The lemma follow from ummng th nequalty over all chan. Theorem 2.1. Algorthm G 7-compettve agant the offlne cheduler. Proof. Combnng the reult n Corollary 2.1 and Lemma 2.6, and ung the fact that V OQ) V OQ ), we get the above clam. The lower bound of 2 for β 1 Secton 1) can be exhbted very mply for th algorthm, wth one job of nfnte length and value v 1 = 1 arrvng at tme 0, 1 and a econd job of length 1 and value arrvng at tme 1. 3 A lower bound for the harp-deadlne model We now derve a lower bound on the compettvene of any randomzed onlne algorthm n the model wth harp deadlne. Our contructon the ame a that n [3]. However, our proof requre more effort due to partal credt and the dcount factor; a oppoed to [3], the onlne algorthm can now potentally utlze the partal credt to get cloer to optmal, and the offlne cheduler telf become le advantaged n the preence of a dcount factor, mply becaue the value from the future get dcounted e.g., when β 0, the greedy algorthm optmal). We how that, when β at leat 3/4, depte partal credt and the dcount factor, we can derve a poly-logarthmc lower bound on the performance of any randomzed onlne algorthm n the harp-deadlne model. The underlyng ntuton that partal credt and the dcount factor only add lower order term to the reward obtaned by the onlne algorthm, n the lower-bound contructon of [3]. Let v max and v mn denote the maxmum and mnmum per-unt value, and l max and l mn be the maxmum and mnmum length of the job. Defne µ = max lmax l mn, vmax v mn ). We contruct a famly of example for whch no randomzed onlne algorthm) can acheve compettvene better than Ω log µ log log µ on thee example. The contructon of the lower bound decrbed n Appendx 6. It cont of a equence of oblvou adverare, ADV, defned recurvely. Adverary ADV act a follow: t order new job of type ; alo f at any tme the probablty of the onlne algorthm workng a job of type hgher than a certan threhold, t call ADV 1 to obtan job of horter length but hgher per-unt value. The threhold functon and the formal trategy of ADV are defned n the Appendx, where we alo gve the proof of the followng theorem: Theorem 3.1. The compettve rato of any random- zed preemptve cheduler bounded by Ω log µ ) log log µ l for β 3/4, where µ = max max l mn, vmax v mn ). Oberve that f the per-unt value of all job dffer by at mot a contant factor λ, then the algorthm that preempt the current tory only f the new job longer, λ-compettve. Therefore, ung tandard technque, one can degn a Olog vmax v mn )-compettve algorthm, ee Secton Increang per-unt value In the toryboardng applcaton t natural for adverter to have ncreang value for the ad dplayed n ther equence, partcularly when the equence of ad form a tory. We now conder the extreme cae where adverter derve value only when ther entre tory hown wthout nterrupton. Formally, each job tory) pecfed by a length l and a fnal value f. Note that the value obtaned from the frt l 1 unt of the job zero. The total value of job f tarted at tme zero) equal to β l 1 f. We focu here on the no-deadlne model, nce the harp-deadlne model wth thee valuaton mlar to the model tuded n [12, 3], depte the dcount factor. 4.1 A lower bound We frt how that no determntc algorthm can be contant compettve n th model. For every c we contruct an nput equence contng of one long job and everal hort job, uch that preemptng the long job at any tep would lead to a rato wore than c, and fnhng the job lead to a bad compettve rato a well Copyrght by SIAM. Unauthorzed reproducton of th artcle prohbted.

7 Theorem 4.1. For any contant c > 0, no determntc algorthm can acheve a compettve rato of c wth repect to the offlne cheduler OPT. Proof. Suppoe a determntc algorthm clam a rato c. Conder an nput n whch there a job of length L pecfed later) and total value V = 1, arrvng at tme 0. Startng from tme 1 and untl the determntc algorthm preempt the long job, at each tme ntance we get a job of length 1 and value α/β 1. If the determntc algorthm never preempt the long job, the arrval never top. The value of α a functon of c and wll be choen later. Wth thee arrval, at tme, OPT obtan a reward of β 1 α β = α, o that t 1 total value grow lnearly wth tme. We chooe α to enure that preemptng at any pont caue the determntc algorthm to get value below c tme what OPT ha already got o far: at tme l, let OP T l) denote the total value that OPT get ung arrval o far ncludng the long job); let V p l) denote the value from preemptng the long job at tme l; V p l) = β l 1 1 α β l 1 + β β l βl 1 ) = α1 + β β 2l 2 ) = α2l+2 ) 2. Then, nce V = 1, 4.4) 4.5) OP T l) V p l) = αl + βl V α 2l+2 2 = l2 ) 2l+2 + βl+1 2 ) α 2l+2 ). We want to chooe α uch that th greater than c for all l. For l > c/ 2 ), 4.4) greater than c, nce the econd term potve. For maller value of l, we chooe αc) to atfy β c ) > αc. Th enure that for all value of l, preemptng the long job lead to an approxmaton factor that wore than c. We chooe L to be large enough to enure that the approxmaton factor on delayng alo wore than c,.e., uch that OP T L) V d = αl + βl > c. 1 + α 2l A compettve algorthm Now we preent an algorthm for the cae when the value of a job obtaned only after t lat unt cheduled. Our algorthm Olog fmax f mn )-compettve wth repect to the offlne cheduler, where f max = max {f } and f mn = mn {f }. We frt conder a pecal cae of the model, n whch all the value of f are equal to 1. Oberve that f β 1 2 then the algorthm that alway preempt n the favor of a job that fnhe earler acheve a compettve 1 rato of 2. For β 1 2, we propoe the followng algorthm, called C: If there no job n progre, chedule a job of the hortet length. Otherwe, let be the current job. Upon the arrval of a new job j, preempt and chedule j f l > 1 l j. Lemma 4.1. For β 1 2e 1 2, algorthm C 12 e 1 - compettve wth repect to the offlne cheduler OP T. Proof. Mot part of the proof are mlar to Theorem 2.1 and, therefore, are dcued brefly. Frt aume the length of all job are bounded by 1. Th aumpton wll be removed later. For job, defne D to be the et of unt that are cheduled by OPT at the tme C chedule. Let r denote the tme when OPT chedule unt. Smlar to Theorem 2.1, we fnd a matchng from unt n D to unt cheduled by C. Note that th optmal oluton mut chedule job wthout nterrupton; we refer to unt only for convenence. Let U be the lt of all unmatched job, ntalzed to be the empty et. Each unt, n a et D, added to U at tme r + l. At each tme t, unt are removed from U n two way: ) unt dplayed by C at tme t. ) unt ha the lowet r n U. In both of the cae above, removed from U and we have m = t. By defnton, we have m r, where r the tme that C chedule unt. Alo, for every t, there are at mot two unt and j uch that m = m j = t. Note that we can replace each job of length l wth another job of per-unt value v uch that l v = β l. Becaue nether C nor OPT preempt any job, thee replacement do not change the value of the chedule. Therefore, 4.6) β m v 2V C. Oberve that after the replacement, at tme t, the maxmum value of a unt n U at mot the value of the unt cheduled by C at tme t. The reaon that unt n U belong to the tore that are avalable to C or are currently beng dplayed); and C alway chooe a job wth the hghet value-per-unt. Therefore, mlar to Lemma 2.5, we have that 4.7) β r +l β m )v V C Copyrght by SIAM. Unauthorzed reproducton of th artcle prohbted.

8 Alo, becaue all length are bounded by 1, for β 1 2, we have β l β Therefore, 4.8) +l v 1 4 V OPT. β r Now we remove that aumpton that job length are bounded. For each length l, the total reward that could be obtaned from all job of length l at mot β l 1 1 l β l e )β l 1. Therefore, f the algorthm obtan the value of a job of length l, t can dcard all job of length > l, and tll be contant compettve wth repect to the offlne cheduler. Now by pluggng n 4.6), 4.7), and 4.8), t eay to ee that th algorthm 12 2e 1 e 1 compettve. We can buld on the above argument to gve a Olog fmax f mn )-compettve algorthm, A, for the nopartal-credt cae. The argument rather tandard. We frt partton the job n the optmal oluton nto bucket baed on f nto bucket [2, 2 +1 ). There are logf max /f mn ) uch bucket. Let W be the value that the optmal oluton get from job n bucket, OPT = W. We randomly chooe a bucket j. The prevou argument how u that retrctng the nput e 1 to job n th bucket gve u value at leat W j 122e 1), nce the optmal oluton wth th nput at leat W j. Now, the expected value we get = 1 e 1 logf max /f mn ) 122e 1) W e 1 W 122e 1) logf max /f mn ) e 1 OPT 122e 1) logf max /f mn ). Theorem 4.2. Algorthm A Ologf max /f mn ))- compettve wth repect to the offlne cheduler n the model wth no deadlne and no partal credt. Concluon. In th paper, we gve algorthm and lower bound for the onlne job chedulng problem wth dcounted reward n an equal per-unt value model: wth no deadlne, a natural greedy algorthm gve a contant factor approxmaton, wherea no randomzed cheduler can acheve better than a logarthmc factor wth harp deadlne. When per-unt value are not equal, however, the problem much harder to analyze; the mot nteretng open problem when value obtaned from a job only upon completon. We how that no determntc algorthm can be contant compettve; the queton of ether provdng a randomzed algorthm that contant compettve or howng a lower bound for all randomzed algorthm reman open. Reference [1] S. Ben-Davd, A. Borodn, R. Karp, G. Tardo, and A. Wgderon, On the Power of Randomzaton n On- Lne Algorthm. Algorthmca, 111): 2-14, [2] A. Blum, S. Chawla, D. R. Karger, T. Lane, A. Meyeron, and M. Mnkoff. Approxmaton algorthm for orenteerng and dcounted-reward tp. SIAM Journal of Computng, 372): , [3] R. Canett and S. Iran. Boundng the power of preempton n randomzed chedulng. SIAM Journal of Computng, 274): , [4] M. T. Hajaghay, R. Klenberg, M. Mahdan, and D. Parke. Onlne aucton wth re-uable good. In EC 05: Proceedng of the 6th ACM conference on Electronc commerce, page , New York, NY, USA, [5] M. T. Hajaghay, R. Klenberg, and D. C. Parke. Adaptve lmted-upply onlne aucton. In EC 04: Proceedng of the 5th ACM conference on Electronc commerce, page 71 80, New York, NY, USA, [6] A. W. Kolen, J. K. Lentra, C. H. Papadmtrou, and F. C. Spekma. Interval chedulng: A urvey. Naval Reearch Logtc, 54: , [7] R. Lav and N. Nan. Onlne acendng aucton for gradually exprng tem. In SODA 05: Proceedng of the xteenth annual ACM-SIAM ympoum on Dcrete algorthm, page , Phladelpha, PA, USA, [8] marketngterm.com. Surround eon. marketngterm.com/dctonary/urround eon. [9] R. Porter. Mechanm degn for onlne real-tme chedulng. In EC 04: Proceedng of the 5th ACM conference on Electronc commerce, page 61 70, New York, NY, USA, [10] PrcewaterhoueCooper. IAB nternet advertng revenue report. IAB PwC 2007 full year.pdf. [11] N. Y. Tme. What urround eon? [12] G. J. Woegnger. On-lne chedulng of job wth fxed tart and end tme. Theor. Comput. Sc., 1301):5 16, The offlne problem n no-deadlne model NP-Hard Theorem 5.1. The problem of computng the optmal chedule of job, even when all arrval are known n advance, weakly) NP-Hard Copyrght by SIAM. Unauthorzed reproducton of th artcle prohbted.

9 Proof. The proof by reducton from ubet-um. The ubet-ub problem defned a follow: gven a et S of nteger, and a um b, there a ubet of S whoe um b? Gven an ntance of ubet um, for each n S, defne a job of arrval tme 0, v = 1, and length n. Alo, defne a job of arrval tme b and value M, where M choen large enough o that the optmal chedule ha to dplay M at b. If there a ubet of S wth um b, then the value of the optmal chedule b 1 β t + Mβ b + t=0 t=b+1 β t. If there no uch ubet, the reward of the offlne cheduler trctly le than th value. 6 Lower bound for the harp-deadlne model The contructon of our lower bound follow that of [3]; the proof of the correpondng lemma dffer. We outlne the contructon for completene. We defne a equence of adverare recurvely: ADV, when deployed agant the randomzed algorthm, create a famly of example gvng a lower bound of γ defned n equaton 6.9). Adverary ADV ue + 1 type of job - the th job ha length l = k 4 and per-unt value v = k 2 2, o that the total value V = v l. An - perod the tme nterval between any two conecutve nteger multple of l. An onlne randomzed algorthm can be defned by a tme varyng probablty dtrbuton, {p t)}, whch the probablty that the algorthm workng on a job of type at tme t. We aume that the adverary ha acce to the probablty dtrbuton {p t)} for all t, but not to the actual con-toe of the algorthm. We demontrate the lower bound for dcount factor β 3/4. Adverary ADV k order a new job of type k every l k tep; alo f the probablty of the onlne algorthm workng a job of type k hgher than a certan threhold at any tep, t call ADV k 1 to obtan job of horter length but hgher per-unt value. The trategy of ADV preented n Fgure 2, and the threhold functon for ADV defned below: f x) = { where α = 21 γ + 2 k ) and 2 6.9) γ = γ 1 1 α e γ 1 V x, f x V 1 α e γ 1, f x > V e γ γ 1 e γ 1 ) + 1 k 2. Adverary ADV : At each tme t: If l dvde t, Schedule a tory of type. If > 0 and q t) f O t)), call ADV 1. Fgure 2: Strategy for ADV By Lemma 2 n [3], we have γ k 4/ k. Let O t) denote the offlne gan that can be accrued by the job of type j, j <, ordered n the -perod contanng tme t, untl t, approprately dcounted by the factor β. The defnton of -crtcal tme unt and -tep are the ame a n [3]. For a k-tep τ = [ τ, f τ ], let h k τ) be defned a O k+1 τ ) O k+1 τ 1). We partton the et of job from the nput equence S nto regular and nonregular a n [3], and defne EA τ S) to be randomzed) cheduler A dcounted reward, n expectaton, from the regular job requeted durng the tme nterval τ n the requet equence S. We frt bound the cheduler gan from thee regular job. Lemma 6.1. EA τ S) γ k h k τ), where γ k defned a n Equaton 6.9. Proof. The nductve proof follow that of [3], Lemma 3. The cae for 0 th and 1 th adverary are eay. For the k th adverary, we aume a contradcton. We then contruct a cheduler A that mmc the cheduler A agant the k 1) th adverary and how that th volate the nductve hypothe. We do a cae analy a n [3] accordng to the nature of the nterval τ; here we decrbe only the cae that dffer from [3]. Cae 1. Step τ not the frt k-tep. Cae 1a) trval. For cae 1b), we note that for each unt of a type job, the probablty that A get the value of p the unt t) 1 p k t), whch 1 1 p k t) of the probablty of A gettng the unt. If th unt from the job c wth endng tme e c, we have that p k t) p k e c ), and thu 1 p k t) 1 p k e c ) γ k γ k 1, and thu the bound for cae 1b) hold a n [3]. Cae 2. In th cae, τ the frt k-tep, and thu the gan h k τ) from the frt k-type job that ordered at tme 0,.e., h k τ) = V k. Replacng Equaton 3.16 n [3] by EA τ S) V k f k O k t 0 )) + l =1 EA τ S) + t0 v k t=0 βt 1 p k t), and argung mlarly, we get the 1283 Copyrght by SIAM. Unauthorzed reproducton of th artcle prohbted.

10 analog of 3.17), EA τ S) V k f k O k t 0 )) + t 0 + v k β t 1 p k t) t=0 V k f k O k t 0 )) + t 0 + v k β t 1. t=0 l 1 p k e τ ))γ k 1 h k 1 τ ) =1 l 1 p k e τ ))γ k 1 h k 1 τ ) =1 By defnton of a k-tep, f τ denote the end of k- τ tep τ, then v k 1 t=0 βt 1 = V k. Thu, nce t 0 ether fall nde a k-tep or the very next tep after t, t0 v k t=0 βt 1 V k v k /v k 1 V k /k 2. Hence, EA τ S) bounded by V k f k O k t 0 )) + l 1 p k e τ ))γ k 1 h k 1 τ ) + V k /k 2, =1 and we need to how that the above expreon at mot γ k V k. We prove th n two part; wrte the above expreon a F + G where F = V k f k O k t 0 )) + G = l 1 p k τ ))γ k 1 h k 1 τ ), =1 l p k τ ) p k e τ ))γ k 1 h k 1 τ ) + V k /k 2. =1 We wll how that F γ k 2/k 2 )V k and G 2V k /k 2. To tart wth, F V k f k O k t 0 )) + V k f k O k t 0 )) + l 1 p k τ ))γ k 1 h k 1 τ ) =1 l 1 f k O k τ )))γ k 1 h k 1 τ ) =1 V k f k O k t 0 )) + γ k 1 β ln1/β) V k 1 α k e γ 1 V k O k t 0) ) + γ k 1 β ln 1 β ) Ok t 0) 0 α k e Ok t 0) 0 γ k 1 V y k dy 1 f k y))dy ntegraton become γ k 1 γ V x F V k 1 α k e 0 V k k 1 k ) + γ k 1 β ln 1 V x β ) e 0 k 1) γ k 1 )) γ V k 1 α k β ln 1 β ) + α k 1 V x 0 ke k β ln 1 β ) 1. γ k 1 v x Now, note that n th range, e 0 k e. Alo, 1. So the above expreon at mot β ln 1 β ) )) V k 1 α k β ln 1 β ) + α ke β ln 1 β ) 1. Now, for β 3/4, Thu, β ln 1 β ) e 0.5 e 1. F V k 1 α k /2) = v k γ k 2 k 2 ) by defnton of α k, provng the bound on F. For the bound on G, ung exactly the argument n [3] how that G V k /k 2 + V k /k 2 = 2V k /k 2. Th gve u the lemma. Followng Lemma 4 n [3], we know that the cheduler reward from non-regular job alo bounded, nce the value the offlne cheduler can extract from uch job at mot 1 k of the optmal value. Thu, the reward that can be obtaned by the our randomzed cheduler certanly bounded by O k k, whch gve u the the followng theorem retated from Secton 3. Theorem 6.1. The compettve rato of any random- zed preemptve cheduler bounded by Ω log µ ) log log µ l for β 3/4, where µ = max max l mn, vmax v mn ). Proof. From Lemma 6.1 n Appendx 6, f we can employ an adverary of the k th level, then we can how that the randomzed cheduler at mot γ k 4/ k compettve agant optmal for the requet equence that generated by th adverary. Now, for th adverary, vmax v mn = k 2k. Thu, f m = k 2k, we have that γ k 4 k log µ lmax log log µ. Smlarly, l mn = k 4k alo, gvng u the factor n the clam. after convertng the um to an ntegral and ubttutng for f k n th range, nce x V k for the frt k- tep. Denote x 0 = O k t 0 ). The above expreon after 1284 Copyrght by SIAM. Unauthorzed reproducton of th artcle prohbted.