Near Optal Onlne Algorths and Fast Approxaton Algorths for Resource Allocaton Probles ABSTRACT Nhl R Devanur Mcrosoft Research Redond WA USA ndev@crosoftco Balasubraanan Svan Coputer Scences Dept Unv of Wsconsn-Madson Madson WI USA balu2901@cswscedu We present algorths for a class of resource allocaton probles both n the onlne settng wth stochastc nput and n the offlne settng Ths class of probles contans any nterestng specal cases such as the Adwords proble In the onlne settng we ntroduce a new dstrbutonal odel called the adversaral stochastc nput odel whch s a generalzaton of the d odel wth unnown dstrbutons where the dstrbutons can change over te In ths odel we gve a 1 O() approxaton algorth for the resource allocaton proble wth alost the weaest possble assupton: the rato of the axu aount of resource consued by any sngle request to the total capacty of the resource and the rato of the proft contrbuted by any sngle request to the optal proft s at ost 2 /log(1/) 2 where n s the nuber of resources log n+log(1/) avalable There are nstances where ths rato s 2 /log n such that no randozed algorth can have a copettve rato of 1 o() even n the d odel The upper bound on rato that we requre proves on the prevous upper-bound for the d case by a factor of n Our proof technque also gves a very sple proof that the greedy algorth has a copettve rato of 1 1/e for the Adwords proble n the d odel wth unnown dstrbutons and ore generally n the adversaral stochastc nput odel when there s no bound on the bd to budget rato All the prevous proofs assue A full verson of ths paper wth all the proofs s avalable at http://arxvorg Part of ths wor was done whle the author was at Mcrosoft Research Redond Part of ths wor was done whle the author was at Mcrosoft Research Redond Persson to ae dgtal or hard copes of all or part of ths wor for personal or classroo use s granted wthout fee provded that copes are not ade or dstrbuted for proft or coercal advantage and that copes bear ths notce and the full ctaton on the frst page To copy otherwse to republsh to post on servers or to redstrbute to lsts requres pror specfc persson and/or a fee EC 11 June 5 9 2011 San Jose Calforna USA Copyrght 2011 ACM 978-1-4503-0261-6/11/06 $1000 Kaal Jan Mcrosoft Research Redond WA USA aalj@crosoftco Chrstopher A Wlens Coputer Scence Dvson Unv of Calforna at Bereley Bereley CA USA cwlens@csbereleyedu that ether bds are very sall copared to budgets or soethng very slar to ths In the offlne settng we gve a fast algorth to solve very large LPs wth both pacng and coverng constrants We gve algorths to approxately solve (wthn a factor of 1 + ) the xed log n 2 pacng-coverng proble wth O( ) oracle calls where the constrant atrx of ths LP has denson n and γ s a paraeter whch s very slar to the rato descrbed for the onlne settng We dscuss several applcatons and how our algorths prove exstng results n soe of these applcatons Categores and Subject Descrptors F20 [Analyss of Algorths and Proble Coplexty]: General; J4 [Socal and Behavoral Scences]: Econocs General Ters Algorths Econocs Theory Keywords Onlne algorths Stochastc nput Pacng-Coverng 1 INTRODUCTION The results n ths paper fall nto dstnct categores of copettve algorths for onlne probles and fast approxaton algorths for offlne probles We have two an results n the onlne fraewor and one result n the offlne settng However they all share coon technques There has been an ncreasng nterest n onlne algorths otvated by applcatons to onlne advertsng The ost well nown s the Adwords proble ntroduced by Mehta et al [MSVV05] where the algorth needs to assgn eywords arrvng onlne to bdders to axze proft subject to budget constrants for the bdders The proble has been analyzed n the tradtonal fraewor for onlne algorths: worst-case copettve analyss As wth any onlne probles the worst-case copettve analyss s not entrely satsfactory and there has been a drve n the last few years to go beyond the worst-case analyss The predonant approach has been to assue that the nput satsfes soe stochastc property For nstance the rando perutaton odel (ntroduced 29
by Goel and Mehta [GM08]) assues that the adversary pcs the set of eywords but the order n whch the eywords arrve s chosen unforly at rando A closely related odel s the d odel: assue that the eywords are d saples fro a fxed dstrbuton whch s unnown to the algorth Stronger assuptons such as d saples fro a nown dstrbuton have also been consdered Frst Result A ey paraeter on whch any of the algorths for Adwords depend s the bd to budget rato For nstance n Mehta et al [MSVV05] and Buchbnder Jan and Naor [BJN07] the algorth acheves a worst case copettve rato that tends to 1 1/e as the bd to budget rato (let s call t γ) tends to 0 (1 1/e s also the best copettve rato that any randozed algorth can acheve n the worst case) Devanur and Hayes [DH09] showed that n the rando perutaton odel the copettve rato tends to 1 as γ tends to 0 Ths result showed that copettve rato of algorths n stochastc odels could be uch better than that of algorths n the worst case The portant queston snce then has been to deterne the optal trade-off between γ and the copettve rato [DH09] showed how to get a 1- O() copettve rato when γ s at ost 3 where n s the nuber of advertsers and n log(n/) s the nuber of eywords Subsequently Agrawal Wang and Ye [AWY09] proved the bound on γ to The papers of 2 n log(n/) Feldan et al [FHK + 10] and Agrawal Wang and Ye [AWY09] have also shown that the technque of [DH09] can be extended to other onlne probles The frst an result n ths paper s the followng 3-fold proveent of prevous results: (Theores 2-4) 1 We gve an algorth whch proves the bound on γ to 2 /log(1/) 2 Ths s alost optal; we show a lower log(n)+log(1/) bound of 2 log(n) 2 The bound apples to a ore general odel of stochastc nput called the adversaral stochastc nput odel Ths s a generalzaton of the d odel wth unnown dstrbuton but s ncoparable to the rando perutaton odel 3 It apples to a ore general class of onlne probles that we call the resource allocaton fraewor A foral defnton of the fraewor s presented n Secton 22 and a dscusson of any nterestng specal cases s presented n Secton 7 Regardng the bound on γ the reoval of the factor of n s sgnfcant Consder for nstance the Adwords proble and suppose that the bds are all n [01] The earler bound ples that the budgets need to be of the order of n/ 2 n order to get a 1 copettve algorth where n s the nuber of advertsers Wth realstc values for these paraeters t sees unlely that ths condton would be et Whle wth the proved bounds presented n ths paper we only need the budget to be of the order of log n/ 2 and ths condton s et for reasonable values of the paraeters We note here that so far all the algorths for the d odel (wth unnown dstrbuton) were actually desgned for the rando perutaton odel It sees that any algorth that wors for one should also wor for the other However we can only show that our algorth wors n the d odel so the natural queston s f our algorth wors for the rando perutaton odel It would be very surprsng f t ddn t One drawbac of the stochastc odels consdered so far s that they are te nvarant that s the nput dstrbuton does not change over te The adversaral stochastc nput odel allows the nput dstrbuton to change over te The odel s as follows: n every step the adversary pcs a dstrbuton possbly adaptvely dependng on what the algorth has done so far and the actual eyword n that step s drawn fro ths dstrbuton The copettve rato s defned wth respect to the optu fractonal soluton for an offlne nstance of the proble called the dstrbuton nstance whch s defned by the dstrbuton (see Secton 22) In Secton 22 where we defne the dstrbuton nstance we also prove that the optal fractonal soluton for the dstrbuton nstance s at least as good as the coonly used benchar of expected value of optal fractonal soluton where the expectaton s wth respect to the dstrbuton A detaled descrpton of ths odel how the adversary s constraned to pc ts dstrbutons and how t dffers fro the worst-case odel s presented n Secton 22 Second Result Another portant open proble s to prove the copettve rato for the Adwords proble when there s no bound on γ The best copettve rato nown for ths proble s 1/2 n the worst case Nothng better was nown even n the stochastc odels (For the specal case of onlne bpartte atchng n the case of d nput wth a nown dstrbuton recent seres of results acheve a rato of better than 1-1/e for nstance by Feldan et al [FMMM09] and Bahan and Kapralov [BK10] The best rato so far s 702 by Manshad Gharan and Saber [MGS11]) The second result n ths paper s that for the Adwords proble n the adversaral stochastc nput odel wth no assupton on γ the greedy algorth gets a copettve rato of 1 1/e aganst the optal fractonal soluton to the dstrbuton nstance (Theore 5) The greedy algorth s partcularly nterestng snce t s a natural algorth that s used wdely for ts splcty Because of ts wde use prevously the perforance of the greedy algorth has been analyzed by Goel and Mehta [GM08] who showed that n the rando perutaton and the d odels t has a copettve rato of 1 1/e wth an assupton whch s essentally that γ tends to 0 Thrd Result Charles et al [CCD + 10] consdered the followng (offlne) proble: gven a lopsded bpartte graph G = (L R E) that s a bpartte graph where = L R = n does there exst an assgnent M : L R wth (j M(j)) E for all j L and such that for every vertex R M 1 () B for soe gven values B Even though ths s a classc proble n cobnatoral optzaton wth well nown polynoal te algorths the nstances of nterest are too large to use tradtonal approaches to solve ths proble (The value of n partcular s very large) The approach used by [CCD + 10] was to essentally desgn an onlne algorth n the d odel: choose vertces fro L unforly at rando and assgn the to vertces n R n an onlne fashon The onlne algorth s guaranteed to be close to optal as long as suffcently any saples are drawn Therefore t can be used to solve the orgnal proble (approxately): the onlne algorth gets an alost satsfyng assgnent f and only f the orgnal graph has a satsfyng assgnent (wth hgh probablty) The thrd result n ths paper s a generalzaton of ths result to get fast approxaton algorths for a wde class of probles n the resource allocaton fraewor (Theore 6) Probles n the resource allocaton fraewor where the nstances are too large to use tradtonal algorths occur farly often especally n the context of onlne advertsng Foral stateents and a ore detaled dscusson are presented n Secton 23 The underlyng dea used for all these results can be suarzed at a hgh level as thus: consder a hypothetcal algorth called 30
Pure-rando that nows the dstrbuton fro whch the nput s drawn and uses an optal soluton wrt ths dstrbuton Now suppose that we can analyze the perforance of Pure-rando by consderng a potental functon and showng that t decreases by a certan aount n each step Now we can desgn an algorth that does not now the dstrbuton as follows: consder the sae potental functon and n every step choose the opton that nzes the potental functon Snce the algorth nzes the potental n each step the decrease n the potental for ths algorth s better than that for Pure-rando and hence we obtan the sae guarantee as that for Pure-rando For nstance for the case where γ s sall the perforance of Pure-rando s analyzed usng Chernoff bounds The Chernoff bounds are proven by showng bounds on the expectaton of the oent generatng functon of a rando varable Thus the potental functon s the su of the oent generatng functons for all the rando varables that we apply the Chernoff bounds to The proof shows that n each step ths potental functon decreases by soe ultplcatve factor The algorth s then desgned to acheve the sae decrease n the potental functon A partcularly pleasng aspect about ths technque s that we obtan very sple proofs For nstance the proof of Theore 5 s extreely sple: the potental functon n ths case s sply the total aount of unused budgets and we show that ths aount (n expectaton) decreases by a factor of 1 1/ n each step where there are steps n all On the surface ths technque and the resultng algorths 1 bear a close reseblance to the algorths of Young [You95] for derandozng randozed roundng and the fast approxaton algorths for solvng coverng/pacng LPs of Plotn Shoys and Tardos [PST91] Garg and Köneann [GK98] and Flescher [Fle00] In fact Arora Hazan and Kale [AHK05] showed that all these algorths are related to the ultplcatve weghts update ethod for solvng the experts proble and especally hghlghted the slarty between the potental functon used n the analyss of the ultplcatve update ethod and the oent generatng functon used n the proof of Chernoff bounds and Young s algorths Hence t s no surprse that our algorth s also a ultplcatve update algorth It sees that our algorth s closer n sprt to Young s algorths than others It s possble that our algorth can also be nterpreted as an algorth for the experts proble In fact Mehta et al [MSVV05] ased f there s a 1 o(1) copettve algorth for Adwords n the d odel wth sall bd to budget rato and n partcular f the algorths for experts could be used They also conjectured that such an algorth would teratvely adjust a budget dscount factor based on the rate at whch the budget s spent Our algorths for resource allocaton proble when specalzed for Adwords loo exactly le that and wth the connectons to the experts fraewor we answer the questons n [MSVV05] n the postve Organzaton: The rest of the paper s organzed as follows In Secton 2 we defne the resource allocaton fraewor the adversaral stochastc odel and state our results forally as theores We also dscuss one specal case of the resource allocaton fraewor the adwords proble and forally state our results In Secton 3 we consder a splfed n-ax verson of the resource allocaton fraewor and present the proofs for ths verson The other results buld upon ths sple verson In Secton 4 we gve a fast approxaton algorth for the xed coverng-pacng proble (Theore 6) The 1 O() copettve onlne algorth for the resource allocaton fraewor wth stochastc nput (Theo- 1 For the case of sall γ It s not clear f ths dscusson apples to the case of large γ that s to Theore 5 re 2) s n Secton 5 The 1 1/e copettve algorth (Theore 5) for the Adwords proble s n Secton 6 Several specal cases of the resource allocaton fraewor are consdered n Secton 7 Secton 8 concludes wth soe open probles and drectons for future research 2 PRELIMINARIES & MAIN RESULTS 21 Resource allocaton fraewor We consder the followng fraewor of optzaton probles There are n resources wth resource havng a capacty of c There are requests; each request j can be satsfed by a vector x j that s constraned to be n a polytope P j (We refer to the vector x j as an opton to satsfy a request and the polytope P j as the set of optons) The vector x j consues a j x j aount of resource and gves a proft of w j x j Note that a j w j and x j are all vectors The objectve s to axze the total proft subject to the capacty constrants on the resources The followng LP descrbes the proble: axze X j w j x j st X a j x j c j jx j P j We assue that we have the followng oracle avalable to us: gven a request j and a vector v the oracle returns the vector x j that axzes vx j aong all vectors n P j Let γ = ax({ a jx j c } j { w jx j } W j) be the noton correspondng to the bd to budget rato for Adwords Here W s the optal offlne objectve to the dstrbuton nstance defned n Secton 22 The canoncal case s where each P j s a unt splex n R K e P j = {x j R K : P x j = 1} Ths captures the case where there are K dscrete optons each wth a gven proft and consupton Ths case captures ost of the applcatons we are nterested n whch are descrbed n Secton 7 All the proofs wll be presented for ths specal case for ease of exposton The co-ordnates of the vectors a j and w j wll be denoted by a( j ) and w j respectvely e the th opton consues a( j ) aount of resource and gves a proft of w j For an exaple of an applcaton that needs ore general polytopes see Secton 74 We consder two versons of the above proble The frst s an onlne verson wth stochastc nput: requests are drawn fro an unnown dstrbuton The second s when the nuber of requests s uch larger than the nuber of resources and our goal s to desgn a fast PTAS for the proble 22 Onlne Algorths wth Stochastc Input We now consder an onlne verson of the resource allocaton fraewor Here requests arrve onlne We consder the d odel where each request s drawn ndependently fro a gven dstrbuton The dstrbuton s unnown to the algorth The algorth nows the total nuber of requests The copettve ratos we gve for resource allocaton probles wth bounded γ are wth respect to an upper bound on the expected value of fractonal optal soluton naely the fractonal optal soluton of the dstrbuton nstance defned below Consder the followng dstrbuton nstance of the proble It s an offlne nstance defned for any gven dstrbuton over requests and the total nuber of requests The capactes of the resources n ths nstance are the sae as n the orgnal nstance Every request n the support of the dstrbuton s also a request n ths 31
nstance Suppose request j occurs wth probablty p j Assue wlog that p j 1/ (If p j > 1/ for soe request then repeat that request p j tes wth probablty 1/ and one ore te wth probablty p j p j / Ths breang up of a request j wth p j > 1 s done n order not to ncrease γ) The resource consupton of j n the dstrbuton nstance s gven by p ja j for all and the proft s p jw j The ntuton s that f the requests were drawn fro ths dstrbuton then the expected nuber of tes request j s seen s p j and ths s represented n the dstrbuton nstance by scalng the consupton and the proft vectors by p j To suarze the dstrbuton nstance s as follows X axze p jw jx j st j n the support X p ja jx j c j jx j P j We now prove that the fractonal optal soluton to the dstrbuton nstance s an upper bound on the expectaton of OPT where OPT s the offlne fractonal optu of the actual sequence of requests LEMMA 1 OPT[Dstrbuton nstance] E[OPT] PROOF The average of optal solutons for all possble sequences of requests should gve a feasble soluton to the dstrbuton nstance wth a proft equal to E[OPT] Thus the optal proft for the dstrbuton nstance could only be larger The copettve rato of an algorth n the d odel s defned as the rato of the expected proft of the algorth to the fractonal optal proft for the dstrbuton nstance The an result s that as γ tends to zero the copettve rato tends to 1 In fact we gve the alost optal trade-off THEOREM 2 For any > 0 we gve an algorth such that f γ = 2 / log(1/) 2 then the copettve rato of the algorth s log(n)+log(1/) 1 O() THEOREM 3 There exst nstances wth γ = 2 such that log(n) no algorth can get a copettve rato of 1 o() 2 Also our algorth wors when the polytope P j s obtaned as an LP relaxaton of the actual proble 3 To be precse suppose that the set of optons that could be used to satsfy a gven request corresponds to soe set of vectors say I j Let the polytope P j I j be an α approxate relaxaton of I j f for the proft vector w j and for all x j P j there s an oracle that returns a y j I j such that w jy j αw jx j Gven such an oracle our algorth acheves a copettve rato of α O() THEOREM 4 Gven a resource allocaton proble wth an α approxate relaxaton and for any > 0 we gve an algorth such that f γ = 2 /log(1/) 2 then the copettve rato of the log(n)+log(1/) algorth s α O() We prove Theore 4 n the full verson of the paper In fact our results hold for the followng ore general odel the adversaral stochastc nput odel In each step the adversary 2 The proof of ths theore s obtaned by a odfcaton of a slar theore for rando perutatons presented n [AWY09] 3 There ay be trval ways of defnng P j such that ts vertces correspond to the actual optons The otvaton for allowng nontrval relaxatons s coputatonal: recall that we need to be able to optze lnear functons over P j adaptvely chooses a dstrbuton fro whch the request n that step s drawn The adversary s constraned to pc the dstrbutons n one of the followng two ways In the frst case we assue that a target objectve value OPT T s gven to the algorth and that the adversary s constraned to pc dstrbutons such that the fractonal optu soluton of each of the correspondng dstrbuton nstances s at least OPT T (or at ost OPT T for nzaton probles) The copettve rato s defned wth respect to OPT T In the second case we are not gven a target but the adversary s constraned to pc dstrbutons so that the fractonal optu of each of the correspondng dstrbuton nstances s the sae whch s the benchar wth respect to whch the copettve rato s defned Note that whle the d odel can be reduced to the rando perutaton odel these generalzatons are ncoparable to the rando perutaton odel as they allow the nput to vary over te Also the constrant that each of the dstrbuton nstances has a large optu value dstngushes ths fro the worst-case odel Ths constrant n general ples that the dstrbuton ust contan suffcently rch varety of requests n order for the correspondng dstrbuton nstance to have a hgh optu To truly sulate the worst-case odel n every step the adversary would chose a deternstc dstrbuton that s a dstrbuton supported on a sngle request Then the dstrbuton nstance wll sply have copes of ths sngle request and hence wll not have a hgh optu For nstance consder onlne bpartte b-atchng where each resource s a node on one sde of a bpartte graph wth the capacty c denotng the nuber of nodes t can be atched to and the requests are nodes on the other sde of the graph and can be atched to at ost one node A deternstc dstrbuton n ths case corresponds to a sngle onlne node and f that node s repeated tes then the optu for that nstance s just the weghted (by c ) degree of that node If the adversary only pcs such deternstc dstrbutons then he s constraned to pc nodes of very hgh degree thus ang t easy for the algorth to atch the We refer the reader to Secton 7 for a dscusson on several probles that are specal cases of the resource allocaton fraewor and have been prevously consdered Here we dscuss one specal case the adwords proble 221 The Adwords proble In the d Adwords proble there are n bdders and each bdder has a daly budget of B dollars Keywords arrve onlne wth eyword j havng an (unnown) probablty p j of arrvng n any gven step For every eyword j each bdder subts a bd b j whch s the proft obtaned by the algorth on allocatng eyword j to bdder The objectve s to axze the proft subject to the constrant that no bdder s charged ore than hs budget Here the resources are the daly budgets of the bdders the requests are the eywords and the optons are once agan the bdders The aount of resource consued and the proft are both b j For ths proble wth no bounds on γ we show that the greedy algorth has a copettve rato of 1 1/e For our results for the adwords proble wth bounded γ see Secton 71 THEOREM 5 The greedy algorth acheves a copettve rato of 1 1/e for the Adwords proble n the adversaral stochastc nput odel wth no assuptons on the bd to budget rato We note here that the copettve rato of 1 1/e s tght for the greedy algorth [GM08] It s however not nown to be tght for an arbtrary algorth 23 Fast algorths for very large LPs Charles et al [CCD + 10] consder the followng proble: gven a bpartte graph G = (L R E) where = L R = n does 32
there exst an assgnent M : L R wth (j M(j)) E for all j L and such that for every vertex R M 1 () B for soe gven values B They gave an algorth that runs n te lnear 4 n the nuber of edges of an nduced subgraph obtaned by tang a rando saple fro R of sze O log n n {B } 2 for a gapverson of the proble wth gap When n {B } s reasonably large such an algorth s very useful n a varety of applcatons nvolvng ad assgnent for onlne advertsng We consder a generalzaton of the above proble (that corresponds to the resource allocaton fraewor) In fact we consder the followng xed coverng-pacng proble Suppose that there are n 1 pacng constrants one for each {1n 1} of the for P j=1 ajxj c and n2 coverng constrants one for each {1n 2} of the for P j=1 bjxj d Each xj s constraned to be n P j Does there exsts a feasble soluton to ths syste of constrants? The gap-verson of ths proble s as follows Dstngush between the two cases: YES: There s a feasble soluton NO: There s no feasble soluton even f all of the c s are ultpled by 1 + and all of the d s s ultpled by 1 We note that solvng (offlne) an optzaton proble n the resource allocaton fraewor can be reduced to the above proble through a bnary search on the objectve functon value Suppose as n [CCD + 10] that s uch larger than n Assue that solvng the followng costs unt te: gven j and v fnd x j P j that axzes vx j Let γ = ax{ [n 1] j [] : a j x j c } { [n 2] j [] : b jx j d } THEOREM 6 There s an algorth that solves the gap verson of the xed coverng-pacng proble wth a runnng te of O ` log n 2 Applcatons to onlne advertsng: The atchng proble ntroduced by [CCD + 10] was otvated by the proble of coputng the avalable nventory for dsplay ad allocaton (see the orgnal paper for detals) In fact the atchng proble was a splfed verson of the real proble whch fts nto the resource allocaton fraewor Moreover such algorths are used n ultple ways For nstance although the technque of Devanur and Hayes [DH09] was orgnally desgned to solve the purely onlne proble t can be used n the PAC odel where the algorth can ae use of a predcton of the future arrval of requests (see for nstance Vee Vasslvtsy and Shanugasundara [VVS10]) The ey technque s to forulate an LP relaxaton of the proble and learn the optal dual varables usng the predcton and these duals can then be used for the allocaton onlne Even f the predcton s not entrely accurate we note that such an approach has certan advantages Ths otvates the proble of fndng the optal duals We observe that our algorth can also be used to copute near optal duals whch can then be used to do the allocaton onlne Probles such as the Dsplay ad allocaton proble (please see full verson of the paper for detals) can beneft fro such an algorth A slar approach was consdered by Abras Mendelevtch and Toln [AMT07] for the followng proble otvated by sponsored search auctons: for each query j one can show an advertser n each of the K slots Each advertser bds a certan aount on each query j and has a daly budget However the cost to an advertser depends on the entre ordered set of advertsers shown (called 4 In fact the algorth aes a sngle pass through ths graph a slate) based on the rules of the aucton Gven the set of queres that arrve n a day (whch n practce s an estate of the queres expected rather than the actual queres) the goal s to schedule a slate of advertsers for each query such that the total cost to each advertser s wthn the budget and axze a gven objectve such as the total revenue or the socal welfare Ths proble s odeled as an LP and a colun-generaton approach s suggested to solve t Also any coproses are ade n ters of ltng the nuber of queres etc due to the dffcultes n solvng an LP of very large sze We observe that ths LP fts n the resource allocaton fraewor and thus can be solved qucly usng our algorth 3 MIN-MAX VERSION In ths secton we solve a slghtly splfed verson of the general onlne resource allocaton proble whch we call the n-ax verson In ths proble requests arrve onlne and each of the ust be served The objectve s to nze the axu fracton of any resource consued (There s no proft) The followng LP descrbes t forally nze λ st X j a( j )x j λc j X x j = 1 j x j 0 For ease of llustraton we assue that the requests arrve d (unnown dstrbuton) n the followng proof At the end of ths secton we show that the proof holds for the adversaral stochastc nput odel also The algorth proceeds n steps Let λ denote the fractonal optal objectve value of the dstrbuton nstance of ths proble Let X t be the rando varable ndcatng the aount of resource consued durng step t that s X t = a( j ) f n step t request j was chosen and was served usng opton Let S T be the total aount of resource consued n the frst T steps Let γ = ax j{ a(j) c } whch ples that for all j and a( j ) γc Let φ t = (1 + ) St /(γc) For the sae of convenence we let S 0 = 0 and φ 0 = 1 for all The algorth s as follows ALG Mn-ax In step t + 1 on recevng request j use opton ( ) X a( j )φ t arg n c = P T t=1 Xt LEMMA 7 The algorth ALG Mn-ax descrbed above approxates λ wthn a factor of (1 +) wth a probablty at least 1 δ where δ = n exp 2 λ 4γ We wll prove Lea 7 through a seres of leas naely Leas 8 9 and 10 Before we begn the proof we gve soe ntuton Consder a hypothetcal algorth call t Pure-rando that nows the dstrbuton Let x j denote the optal fractonal soluton to the dstrbuton nstance Pure-rando s a non-adaptve algorth whch uses x j to satsfy request j e t serves request j usng opton wth probablty x j Suppose we wanted to prove a bound on the perforance of Pure-rando that s show that wth hgh probablty Pure-rando s wthn 1 + O() of the optu 33
say λ Ths can be done usng Chernoff bounds: for each resource separately bound the probablty that the total consupton s ore than λ c (1 + O()) usng Chernoff bounds and tae a unon bound Note that the Chernoff bounds are shown by provng an upper bound on the expectaton of the oent generatng functon of the rando varables If we could show the sae bound for our algorth then we would be done Let φ t = P φt We wsh to upper bound the expectaton of φ Consder the state of the algorth after the frst t steps Let X t+1 denote the aount of resource consued n step t + 1 had we served the request at step t+1 usng the Pure-rando algorth nstead of our algorth Then we show that the expectaton of φ t+1 s upper bounded by P t+1 X φt 1 + γc «and the rest of the proof s along the lnes of the Chernoff bound proof LEMMA 8 For all t PROOF φ t+1 = X X φ t+1 X φ t+1 φ t φ t 1 +! t+1 X γc X t+1 γc = X φ t (1 + ) «1 + Xt+1 X γc φ t 1 +! t+1 X γc The frst nequalty s because the convexty of the functon (1+) x can be used to upper bound t by 1 + x for all x [0 1] and X t ax j a( j ) γc The second nequalty follows fro the defnton n of our algorth as t chooses the opton that P o nzes a(j)φ t c LEMMA 9 For all T E[φ T λ ] nexp T where λ s the optal soluton to the LP PROOF Fro Lea 8 t follows that E ˆφ "!# X t+1 φ t for all E φ t t+1 X 1 + γc X «φ t 1 + λ «= φ t 1 + λ «λ φ t exp and hence the lea follows snce φ 0 = n The second nequalty t+1 follows fro the fact that E[ X ] λ c for all Ths s because requests are drawn d and hence the optal value of the dstrbuton nstance s the sae for all te steps and s equal to λ We prove Lea 10 n the full verson of the paper LEMMA 10 j ff S T Pr ax c > T λ (1 + ) nexp 2 Tλ Substtutng T = n Lea 10 we get Lea 7 4 «Adversaral stochastc nput odel In the above leas we assued that the requests are drawn d e we used the fact that E[ X ] t λ c / for all t But n an adversaral stochastc odel snce the dstrbuton fro whch a request s drawn changes each step the optal objectve of the dstrbuton nstance also changes every step e t could be λ t at step t So n the proof Lea 9 where we proved that E ˆφ «λ t+1 φ t for all φ t exp we would nstead have E ˆφ t+1 φ t for all φ t exp λ t But gven a target λ we now the adversary s constraned to pc dstrbutons whose dstrbuton nstance has an optu objectve at ost λ (recall that ths s a nzaton proble) Therefore we can upper bound φ t exp λ t by φ t exp λ The rest of the steps n the proof rean the sae Thus the adversary s not constraned to pc requests fro the sae dstrbuton at every te step All we requre s that whatever dstrbuton t uses for drawng ts request the correspondng dstrbuton nstance has an optu objectve value at ost λ whch s the target value we a for In the followng sectons we llustrate all our proofs n the d odel wth unnown dstrbuton and t s easy to convert the to proofs for the adversaral stochastc nput odel 4 MIXED COVERING-PACKING AND ONLINE RESOURCE ALLOCATION 41 Mxed Coverng-Pacng In ths secton we consder the xed pacng-coverng proble stated n Secton 23 and prove Theore 6 We restate the LP for the xed coverng-pacng proble here X j X j a( j )x j c b( j )x j d j X x j = 1 j x j 0 The goal s to chec f there s a feasble soluton to ths LP We solve a gap verson of ths proble Dstngush between the two cases: YES: There s a feasble soluton NO: There s no feasble soluton even f all of the c s are ultpled by 1 + and all of the d s are ultpled by 1 For convenence of descrpton we refer to the quanttes ndexed by j as requests those ndexed by as resources and those ndexed by as optons As before the algorth proceeds n steps In each step the algorth saples a request d fro the total of possble requests ln(n/δ)) We wll prove that f the nuber of saples T Θ( ) then the algorth solves the gap verson wth probablty 2 at least (1 δ) Snce the te taen for servng any gven request s one (by tang the te consued by a sngle oracle call to be one) ths proves that the total run-te s O( Theore 6 f δ s a constant ln(n/δ)) 2 ) thus provng 34
Let X t Xt S t be as defned n Secton 3 Let Y t be the rando varable ndcatng the aount of deand satsfed durng step t that s Y t = b( j ) f n step t request j was chosen and was served usng opton Let Ỹ t denote the aount of deand satsfed durng step t by the optal algorth for the dstrbuton nstance of ths proble Let V T = P T t=1 Y t Let φ t = η c(1 + St ) 2 γc (1 + 2 )T t where η c = (1 + 2 ) (1+ 2 ) T Let ψ t = η d (1 V t ) γd (1 2 2 )T t where η d = (1 2 ) (1 2 ) T Let φ t = P φt let ψ t = P ψt and Φ t = φ t + ψ t As before we let S 0 = 0 and V 0 = 0 The algorth s as follows ALG Pacng-Coverng Gven request j n step t + 1 use the opton 8 1 X >< 1 + 2 arg n 1 T >: 1 2 X φ t a( j ) c ψ t b( j ) d At the end of T steps the algorth checs f ax S T c < V T ) and f n 9 >= >; > T (1 ) If true the algorth 2 (1 + 2 d answers YES Else t says NO We now proceed to prove that whenever the real answer s YES the algorth says YES wth a hgh probablty Leas 11 and 12 prove ths case We prove the n the full verson of the paper LEMMA 11 For a YES nstance E ˆΦ T Φ 0 LEMMA 12 For a YES nstance S T Pr ax > T c (1 + 2 ) V T + Pr n < T d (1 2 ) Φ 0 Observe that Φ 0 = η c(1 + 2 )T + η d (1 2 )T whch s upper bounded by nexp + nexp If T = 2 T 16 2 T 8 log(n/δ) O( ) we have the falure probablty to be at ost δ Thus Lea 2 12 proves that the algorth ALG Pacng-Coverng says YES wth a probablty at least 1 δ when the real answer s YES We now proceed to prove that when the real answer s NO our algorth says NO wth a probablty at least 1 δ e LEMMA 13 For a NO nstance f T Θ( S T Pr ax < T c (1 + 2 ) & n log(n) 2 V T d > T (1 2 ) ) then < δ The proof appears n the full verson of the paper Leas 11 12 and 13 prove that the gap-verson of the xed log(n) coverng-pacng proble can be solved n te O( ) thus provng Theore 6 2 REMARK 1 We show that a greedy-type algorth won t solve the gap verson of even just the coverng proble Ths answers an open queston fro [CCD + 10] regardng whether or not the greedy algorth solves the gap verson The precse algorth and why t fals are ncluded n the full verson of the paper 5 ONLINE ALGORITHMS WITH STOCHASTIC INPUT In ths secton we use the potental functon based algorth to solve the onlne verson of the resource allocaton proble ntroduced n Secton 22 The followng LP descrbes the resource allocaton proble axze X j X j w j x j st (1) a( j )x j c j X x j 1 j x j 0 Slar to the algorth n [AWY09] our algorth coputes ncreasngly better estates of the objectve value by coputng the optal soluton for the observed requests and uses t to gude future allocatons Through Leas 14 and 15 we show that our algorth acheves a copettve rato of 1 O() thus provng Theore 2 We assue that the nuber of requests s nown n advance Algorth 1 descrbes our algorth Algorth 1 : Algorth for stochastc onlne resource allocaton 1: Intalze t 0 : t 0 2: for r = 0 to l 1 do 3: for t = t r + 1 to t r+1 do 4: If the ncong request s j use the followng opton : arg n 8 c/γ X >< 1 + c >: φ t 1 a( j ) c o/wax 1 oz(r) w ax φ t 1 obj w j 9 >= >; 5: f t = t r + 1 then 6: For all S(r) t = X t and V t (r) = Y t 7: else 8: For all S(r) t = S t 1 (r) + X t and V t (r) = V t 1 (r) + Y t 9: end f 10: end for 11: end for The frst requests are not served but used just for coputatonal purposes After these frst requests the algorth proceeds n l stages naely 0 1 l 1 where l s such that 2 l = 1 and s a postve nuber between 0 and 1 that the algorth desgner gets to choose In stage r the algorth serves t r = 2 r requests Note that the stage r conssts of all steps t (t r t r+1] Let W denote the optal soluton to the dstrbuton nstance of the proble Let X t be as defned n Secton 3 Let Y t be the aount of proft earned durng step t e Y t = w j f n step t request j was served usng opton Instead of the usual S t we now defne S t (r) = P t u=t r+1 Xu whch s the su of X u s tll t for u s belongng to stage r alone e u (t r t r+1] Slarly V t (r) = P t u=t r+1 Y u Let w ax = ax j w j The potental functon for constrant n step t when t (t r t r+1] 35
s defned by φ t S t (r) = η c(r)(1 + c(r)) γc 1 + c(r) tr (1+c(r)) η c(r) = (1 + c(r)) c(r) = r 4 ln((n + 1)/δ) t r «tr+1 t Slarly the potental functon for objectve at step t s φ t obj = η obj(r)(1 o(r)) V t (r) wax 1 o(r)z(r) «tr+1 t w ax trz(r) (1 o(r)) η obj(r) = (1 o(r)) wax o(r) = s 2w axln((n + 1)/δ) t rz(r) Note that apart fro constants the only dfference between o(r) and c(r) s that nstead of γ o(r) has w ax/z(r) The value Z(r) as we defne below gets progressvely updated but wthn a sngle stage r reans the sae After stage r the algorth coputes the optal objectve value e r to the followng nstance I r: the nstance I r has the t r requests of stage-r and the capacty of resource s trc (1+ c(r)) e the capacty of resources are scaled down accordng to the nuber of requests but has a slght extra allowance by a factor of (1 + c(r)) It uses e r to copute the value Z(r + 1) to be used n the potental functon for objectve n stage r + 1 LEMMA 14 Wth probablty at least 1 δ(2 + 1 α r ) 4 c(r) α r t rw (1 c(r)) e r trw (1 c(r))(1 + α r) We prove Lea 14 n the full verson of the paper Usng these e r s we defne our Z(r + 1) as follows: Z(r + 1) = e r t r(1 c(r))(1 + α r) Usng the bounds on e r n Lea 14 we note that Z(r+1) W and that Z(r + 1) W 1+α r Thus wth probablty at least 1 δ(2 + 1 ) + 4 c(r) α r α r Z(r) satsfes these bounds for all r Gven the bounds on Z(r) we use Lea 12 to see that wth a probablty at least (1 δ) the objectve value acheved n stage r s at least trz(r) (1 o(r)) and the aount of resource consued n stage r s at ost trc (1 + c(r)) Hence these bounds are true for all r wth probablty at least 1 log(1/)δ snce the total nuber of stages l = log(1/) The total falure probablty s the su of the falure probablty durng estaton of Z(r) through e r gven by δ(2 + 1 ) + 4 c(r) α r α r and the falure probablty of our algorth n all stages together gven by log(1/)δ Thus the total falure probablty s gven by TFP = δ(2 + 1 ) + 4 c(r) + log(1/)δ α r α r δ = 3 log(1/)δ + + 4 c(r) α r α r Wth a probablty of 1 T F P the algorth obtans an objectve value of at least «t rw 1 o(r) 1 + α r and for each the aount of resource consued s at ost On settng γ = O t rc (1 + c(r)) 2 /(log(1/) 2 log(n)+log(1/)+log(1/δ ) for soe postve constant δ < 1 and settng α r = log(1/)c(r) and δ = δ the δ above equatons can be splfed to yeld the followng lea LEMMA 15 Wth probablty hgher than (1 8δ ) the value of the objectve functon acheved at the copleton of the algorth s at least W (1 O(/δ )) and no resource s consued ore than ts capacty For a fxed constant δ we have γ = O 2 /(log(1/) 2 Thus log(n)+log(1/) Lea 15 proves that our algorth acheves a copettve rato of (1 O()) and hence proves Theore 2 6 ADWORDS IN IID SETTING In ths secton we gve a sple proof of Theore 5: greedy algorth acheves a copettve rato of (1 1/e) n the adwords proble where the pressons coe fro an adversaral stochastc nput odel As before we llustrate our proofs for the d odel wth unnown dstrbuton below We now brefly descrbe the adwords settng Settng There are a total of n advertsers and queres arrve onlne fro soe pool of queres Let the (unnown) nuber of queres that arrve be The queres that appear each day are drawn d fro soe unnown dstrbuton Advertser bds an aount b j on query j Advertser has a budget B denotng the axu aount of oney that can be spent on a gven day The bd aounts b j are revealed onlne as the queres arrve The objectve s to axze the su of the bd aounts successfully allocated subject to budget constrants Whenever a query j arrves wth a bd aount b j > reanng budget of we are stll allowed to allot that query to advertser but we only earn a revenue of the reanng budget of and not the total value b j Goel and Mehta [GM08] prove that the greedy algorth gves a (1 1/e) approxaton to the adwords proble when the queres arrve n a rando perutaton or n d but under an assupton whch alost gets down to bds beng uch saller than budgets We gve a uch spler proof for a (1 1/e) approxaton by greedy algorth for the d unnown dstrbutons case and our proof wors rrespectve of the the relaton between the sze of the bds and the budgets nvolved Let p j be the probablty of query j appearng n any gven presson Let y j = p j Let x j denote the offlne fractonal 36
optal soluton for the dstrbuton nstance Let w (t) denote the aount of oney spent by advertser at te step t e for the t-th query n the greedy algorth Let f (0) = P j bjxjyj Let f (t) = f (0) P t r=1 w(r) Let f(t) = P n =1 f(t) Note that f (0) s the aount spent by n the offlne fractonal optal soluton to the dstrbuton nstance Consder the greedy algorth whch allocates the query j arrvng at te t to the advertser who has the axu effectve bd for that query e argax n{b j B P t 1 r=1 w(r)} We prove that ths algorth obtans a revenue of (1 1/e) P j bjxjyj and thus gves the desred 1 1/e copettve rato aganst the fractonal optal soluton to the dstrbuton nstance The proof s slar to the proof we presented n Lea 8 for the resource allocaton proble Consder a hypothetcal algorth that allocates queres to advertsers accordng to the x j s We prove that ths hypothetcal algorth obtans an expected revenue of (1 1/e) P j bjxjyj and argue that the greedy algorth only perfors better Let w h (t) and f h (t) denote the quanttes analogous to w (t) and f (t) for the hypothetcal algorth wth the ntal value f h (0) = f (0) = P j bjxjyj Let fh (t) = P n =1 fh (t) LEMMA 16 The hypothetcal algorth satsfes the followng: E[f h (t) f h (t 1)] f h (t 1)(1 1/) LEMMA 17 E[GREEDY] (1 1/e) P j bjxjyj We prove Lea 16 and 17 n the full verson of the paper Lea 17 proves Theore 5 7 APPLICATIONS We now lst the probles that are specal cases of the resource allocaton fraewor and have been prevously consdered See the full verson of the paper for ore specal cases 71 Adwords Proble Whle n Secton 221 we noted that we could get a 1 1/e approxaton to the adwords proble wth unbounded γ we note b here that γ s sall e ax j j B 2 /log(1/) 2 we get a log(n)+log(1/) 1 O() approxaton to the axu proft through the resource allocaton fraewor 72 Networ Routng and Load Balancng Consder a graph (ether undrected or drected) wth edge capactes Requests arrve onlne; a request j conssts of a source-sn par (s j t j) and a bandwdth ρ j In order to satsfy a request a capacty of ρ j ust be allocated to t on every edge along soe path fro s j to t j n the graph In the congeston nzaton verson all requests ust be satsfed and the objectve s to nze the axu (over all edges) congeston whch s the rato of the allocated bandwdth to the capacty of the edge In the throughput axzaton verson the objectve s to axze the nuber of satsfed requests whle not allocatng ore bandwdth than the avalable capacty for each edge (Dfferent requests could have dfferent values on the and one could also consder axzng the total value of the satsfed requests) Both the congeston nzaton verson and the throughput axzaton verson can be solved through our algorth 1 for resource allocaton fraewor Kaath Palon and Plotn [KPP96] consdered a varant of ths proble wth the requests arrvng accordng to a statonary Posson process and show a copettve rato that s very slar to ours 73 Cobnatoral Auctons Suppose we have n tes for sale wth c copes of te Bdders arrve onlne and bdder j has a utlty functon U j : 2 [n] R If we posted prces p for each te then bdder j buys a bundle S that axzes U j(s) P S p We assue that bdders can copute such a bundle The goal s to axze socal welfare the total utlty of all the bdders subject to the supply constrant that there are only c copes of te Frstly ncentve constrants asde ths proble can be wrtten as an LP n the resource allocaton fraewor The tes are the resources and agents arrvng onlne are the requests All the dfferent subsets of tes for the set of optons The utlty U j(s) represents the proft w js of servng agent j through opton S e subset S If an te S then a js = 1 for all j and zero otherwse Incentve constrants asde our algorth for resource allocaton at step t wll choose the opton (or equvalently the bundle S) as specfed n pont 4 of algorth 1 e nze the potental functon That can be equvalently wrtten as 8 9 o/w ax arg ax >< >: 1 oz(r) w ax φ t 1 obj w j c/γ X 1 + c φ t 1 a( j ) c Now axzng the above expresson s the sae as pcng the to axze w j P pa( j ) where p = c/γ 1+ c o/w ax φ t 1 1 oz(r) obj wax φt 1 c Thus f we post these prces p on tes agents wll do exactly what the algorth would have done otherwse Suppose that the bdders are d saples fro soe dstrbuton (or they arrve as n the adversaral stochastc nput odel) Here γ = 1/n {c } and we can use Theore 2 to get an ncentve copatble posted prce aucton 5 wth a copettve rato of 1 O() whenever n {c } log(n)+log(1/) 2 /log(1/) 2 Further f an analog of Theore 2 also holds n the rando perutaton odel then we get a slar result for cobnatoral auctons n the offlne case: we sply consder the bdders one by one n a rando order 74 Selectve Call-out Charaborty et al [CEDG + 11] forulated the followng proble that arses n the desgn of an ad-exchange An exchange gets ad-requests onlne; each ad-request ay have ultple slots wth dfferent qualtes Whenever the exchange gets an ad-request t calls out to a subset of ad-networs for a bd Gven the bds t then allocates the slots to the hghest bdders The ad-networs have constrants on how frequently they want to be called out In addton the followng assuptons are ade: the ad-requests are d saples fro an unnown dstrbuton and for every ad-networ ts values for all ad-requests of a certan type are d fro a dstrbuton that s nown to the exchange They consder varous objectve functons such as socal welfare revenue of a partcular aucton GSP wth reserve and so on They state ther results n the PAC odel where they use an ntal saple of pressons to tran ther algorth They gve a bound on the nuber of saples needed n order to get a 1 1/e copettve algorth We 5 Here we assue that each agent reveals hs true utlty functon after he aes hs purchase Ths nforaton s necessary to copute the prces to be charged for future agents >= >; 37
can use our algorths (wth an approxate relaxaton Theore 4) to prove ther results n the followng two ways Ether we are gven the target objectve value n whch case we acheve the sae copettve rato n the onlne settng wthout the need for an ntal saple If we are not gven the target objectve value then we need an ntal saple to estate that value The nuber of saples we need s less than what s requred by [CEDG + 11] by a factor of n Further our algorth would also wor n the adversaral stochastc nput odel 8 CONCLUSION AND FUTURE WORK Our wor rases the followng open questons As entoned n the ntroducton we can show that our algorth wors n the d odel so the natural queston s f our algorth wors for the rando perutaton odel Currently n our algorth for the onlne case we need to estate the optu objectve functon value perodcally For ths we need to solve (at least approxately) an offlne nstance of the proble repeatedly Is there an algorth that avods ths? Perhaps the holy gral for an onlne algorth for say Adwords s to get a guarantee of the followng for: f the nput s d fro soe gven dstrbuton then get a copettve rato that s close to 1 whle sultaneously gettng a copettve rato of 1 1/e f the nput s adversaral Our algorth for Adwords (or soe sple varant) could actually acheves ths At the very least can we get such a guarantee for onlne b-atchng wth dfferent budgets? Note that when all the budgets are the sae then our algorth for the nax verson s equvalent to a sple algorth called BAL- ANCE that acheves ths (Ths observaton follows fro the results n Kalyanasundara and Pruhs [KP00] and Motwan Pangrahy and Xu [MP06]) A hgh level goal s to coe up wth other reasonable defntons that go beyond worst case The otvaton for ths s to brdge the gap between the theory and practce of onlne algorths 9 REFERENCES [AHK05] [AMT07] [AWY09] [BJN07] [BK10] [CCD + 10] Sanjeev Arora Elad Hazan and Satyen Kale The ultplcatve weghts update ethod: a eta algorth and applcatons Techncal report 2005 Zoe Abras Ofer Mendelevtch and John Toln Optal delvery of sponsored search advertseents subject to budget constrants In EC 07: Proceedngs of the 8th ACM conference on Electronc coerce pages 272 278 New Yor NY USA 2007 ACM Shpra Agrawal Zzhuo Wang and Ynyu Ye A dynac near-optal algorth for onlne lnear prograng arxv:09112974v1 2009 Nv Buchbnder Kaal Jan and Joseph Seff Naor Onlne pral-dual algorths for axzng ad-auctons revenue In ESA 07: Proceedngs of the 15th annual European conference on Algorths pages 253 264 Berln Hedelberg 2007 Sprnger-Verlag Bahan Bahan and Mchael Kapralov Iproved bounds for onlne stochastc atchng In ESA pages 170 181 2010 Dens Charles Max Chcerng Nhl R Devanur Kaal Jan and Manan Sangh Fast algorths for fndng atchngs n lopsded bpartte graphs wth applcatons to dsplay ads In EC 10: Proceedngs of the 11th ACM conference on Electronc coerce pages 121 128 New Yor NY USA 2010 ACM [CEDG + 11] Tanoy Charaborty Eyal Even-Dar Sudpto Guha Yshay Mansour and S Muthurshnan Selectve call out and real te bddng In To appear n WINE 2011 [DH09] Nhl R Devanur and Thoas P Hayes The adwords proble: onlne eyword atchng wth budgeted bdders under rando perutatons In John Chuang Lance Fortnow and Pearl Pu edtors ACM Conference on Electronc Coerce pages 71 78 ACM 2009 [FHK + 10] Jon Feldan Mona Henznger Ntsh Korula Vahab S Mrron and Clfford Sten Onlne stochastc pacng appled to dsplay ad allocaton In ESA pages 182 194 2010 [Fle00] Lsa K Flescher Approxatng fractonal ultcoodty flow ndependent of the nuber of coodtes SIAM J Dscret Math 13(4):505 520 2000 [FMMM09] Jon Feldan Aranya Mehta Vahab Mrron and S Muthurshnan Onlne stochastc atchng: Beatng 1-1/e In FOCS 09: Proceedngs of the 2009 50th Annual IEEE Syposu on Foundatons of Coputer Scence pages 117 126 Washngton DC USA 2009 IEEE Coputer Socety [GK98] Naveen Garg and Jochen Koeneann Faster and spler algorths for ultcoodty flow and other fractonal pacng probles In FOCS 98: Proceedngs of the 39th Annual Syposu on Foundatons of Coputer Scence page 300 Washngton DC USA 1998 IEEE Coputer Socety [GM08] Gagan Goel and Aranya Mehta Onlne budgeted atchng n rando nput odels wth applcatons to adwords In SODA 08: Proceedngs of the nneteenth annual ACM-SIAM syposu on Dscrete algorths pages 982 991 Phladelpha PA USA 2008 Socety for Industral and Appled Matheatcs [KP00] Bala Kalyanasundara and Kr R Pruhs An optal deternstc algorth for onlne b-atchng Theor Coput Sc 233(1-2):319 325 2000 [KPP96] Anl Kaath Or Palon and Serge Plotn Routng and adsson control n general topology networs wth posson arrvals In SODA 96: Proceedngs of the seventh annual ACM-SIAM syposu on Dscrete algorths pages 269 278 Phladelpha PA USA 1996 Socety for Industral and Appled Matheatcs [MGS11] Vahdeh Manshad Shayan Gharan and An Saber Onlne stochastc atchng: Onlne actons based on offlne statstcs In To appear n SODA 2011 [MP06] Rajeev Motwan Rna Pangrahy and Yng Xu 0002 Fractonal atchng va balls-and-bns In APPROX-RANDOM pages 487 498 2006 [MSVV05] Aranya Mehta An Saber Uesh Vazran and Vjay Vazran Adwords and generalzed on-lne atchng In In FOCS 05: Proceedngs of the 46th Annual IEEE Syposu on Foundatons of Coputer Scence pages 264 273 IEEE Coputer Socety 2005 [PST91] Serge A Plotn Davd B Shoys and Éva Tardos Fast approxaton algorths for fractonal pacng and coverng probles In FOCS 91: Proceedngs of the 32nd annual syposu on Foundatons of coputer scence pages 495 504 Washngton DC USA 1991 IEEE Coputer Socety [VVS10] Er Vee Serge Vasslvts and Jayavel Shanugasundara Optal onlne assgnent wth forecasts In EC 10: Proceedngs of the 11th ACM conference on Electronc coerce pages 109 118 New Yor NY USA 2010 ACM [You95] Neal E Young Randozed roundng wthout solvng the lnear progra In SODA 95: Proceedngs of the sxth annual ACM-SIAM syposu on Dscrete algorths pages 170 178 Phladelpha PA USA 1995 Socety for Industral and Appled Matheatcs 38