Online Competitive Algorithms for Ad Allocation in Cellular Networks (AdCell)

Transcription

1 Online Competitive Algoithms fo Ad Allocation in Cellula Netwoks (AdCell) Saeed Alaei 1, Mohammad T. Hajiaghayi 12, Vahid Liaghat 1, Dan Pei 2, and Bana Saha 1 {saeed, hajiagha, vliaghat, bana}@cs.umd.edu, peidan@eseach.att.com 1 Univesity of Mayland, College Pak, MD, AT&T Labs - Reseach, 180 Pak Avenue, Floham Pak, NJ Abstact. With moe than fou billion usage of cellula phones woldwide, mobile advetising has become an attactive altenative to online advetisements. In this pape, we popose a new tageted advetising policy fo Wieless Sevice Povides (WSPs) via SMS o MMS- namely AdCell. In ou model, a WSP chages the advetises fo showing thei ads. Each advetise has a valuation fo specific types of customes in vaious times and locations and has a limit on the maximum available budget. Each quey is in the fom of time and location and is associated with one individual custome. In ode to achieve a non-intusive delivey, only a limited numbe of ads can be sent to each custome. Recently, new sevices have been intoduced that offe location-based advetising ove cellula netwok that fit in ou model (e.g., ShopAlets by AT&T). We conside both online and offline vesion of the AdCell poblem and develop appoximation algoithms with constant competitive atio. Fo the online vesion, we assume that the appeaances of the queies follow a stochastic distibution and thus conside a Bayesian setting. Futhemoe, queies ae not i.i.d. and they may come fom diffeent distibutions on diffeent times. This model genealizes seveal pevious advetising models such as online secetay poblem [8], online bipatite matching [12,6] and AdWods [19]. Since ou poblem genealizes the well-known secetay poblem, no non-tivial appoximation can be guaanteed in the online setting without stochastic assumptions. We popose an online algoithm that is simple, intuitive and easily implementable in pactice. It is based on pe-computing a factional solution fo the expected scenaio and elies on a novel use of dynamic pogamming to compute the conditional expectations. We show that ou algoithms ae 1 and 1 1 -competitive when we have constaints on capacities and budgets espectively. We 2 e give tight lowe bounds on the appoximability of some vaiants of the poblem as well. In the offline setting we achieve nea-optimal bounds, matching the integality gap of the consideed linea pogam. We believe that ou poposed solutions can be used fo othe advetising settings whee pesonalized advetisement is citical. Keywods: Mobile Advetisement, AdCell, Online, Matching A peliminay vesion of this pape appeaed in Euopean Symposium on Algoithms, Saabücken, Gemany, Sep Suppoted in pat by NSF Gant CCF Suppoted in pat by NSF CAREER Awad and ONR Young Investigato Awad. Suppoted in pat by NSF CAREER Awad and Google Faculty Reseach Awad. Suppoted in pat by NSF Awad CCF , NSF Awad CCF

2 2 S. Alaei, M.T. Hajiaghayi, V. Liaghat, D. Pei and B. Saha 1 Intoduction In this pape, we popose a new mobile advetising concept called Adcell. Moe than 4 billion cellula phones ae in use wold-wide, and with the inceasing populaity of smat phones, mobile advetising holds the pospect of significant gowth in the nea futue. Some eseach fims [1] estimate mobile advetisements to each a business woth ove 10 billion US dollas by Given the built-in advetisement solutions fom popula smat phone OSes, such as iads fo Apple s ios, mobile advetising maket is poised with even faste gowth. In the mobile advetising ecosystem, wieless sevice povides (WSPs) ende the physical delivey infastuctue, but so fa WSPs have been moe o less left out fom pofiting via mobile advetising because of seveal challenges. Fist, unlike web, seach, application, and game povides, WSPs typically do not have uses application context, which makes it difficult to povide tageted advetisements. Deep Packet Inspection (DPI) techniques that examine packet taces in ode to undestand application context, is often not an option because of pivacy and legislation issues (i.e., Fedeal Wietap Act). Theefoe, a tageted advetising solution fo WSPs need to utilize only the infomation it is allowed to collect by govenment and by customes via opt-in mechanisms. Second, without the luxuy of application context, tageted ads fom WSPs equie non-intusive delivey methods. While uses ae familia with othe ad foms such as banne, seach, in-application, and in-game, push ads with no application context (e.g., via SMS) can be intusive and annoying if not done caefully. The numbe and fequency of ads both need to be well-contolled. Thid, tageted ads fom WSPs should be well pesonalized such that the uses have incentive to ead the advetisements and take puchasing actions, especially given the equiement that the numbe of ads that can be shown to a custome is limited. In this pape, we popose a new mobile tageted advetising stategy, AdCell, fo WSPs that deals with the above challenges. It takes advantage of the detailed eal-time location infomation of uses. Location can be tacked upon uses consent. This is aleady being done in some sevices offeed by WSPs, such as Spint s Family Location and AT&T s Family Map, thus thee is no associated pivacy o legal complications. To locate a cellula phone, it must emit a oaming signal to contact some neaby antenna towe, but the pocess does not equie an active call. GSM localization is then done by multi-lateation 3 based on the signal stength to neaby antenna masts [21]. Location-based advetisement is not completely new. Fousquae mobile application allows uses to explicitly check in at places such as bas and estauants, and the shops can advetise accodingly. Similaly thee ae also automatic poximity-based advetisements using GPS o bluetooth. Fo example, some GPS models fom Gamin display ads fo the neaby business based on the GPS locations [22]. ShopAlets by AT&T 4 is anothe application along the same line. On the advetise side, popula stoes such as Stabucks ae epoted to have attacted significant footfalls via mobile coupons. Most of the existing mobile advetising models ae On-Demand, howeve, AdCell sends the ads via SMS, MMS, o simila methods without any pio notice. Thus to deal with the non-intusive delivey challenge, we popose use subsciption to advetising sevices that delive only a fixed numbe of ads pe month to its subscibes (as it is the case in AT&T ShopAlets). The constaint of deliveing limited numbe of ads to each custome adds the main algoithmic challenge in the AdCell model (details in Section 1.1). In ode to ovecome the incentive challenge, the WSP can pay uses to ead ads and puchase based on them though a ewad pogam in the fom of cedit fo monthly wieless bill. To begin with, both customes and advetises should sign-up fo the AdCellsevice povided by the WSP (e.g., cuently thee ae 9 chain-companies paticipating in ShopAlets). Customes enolled fo the sevice should sign an ageement that thei location infomation will be tacked; but solely fo the advetisement pupose. Advetises (e.g., stoes) povide thei advetisements and a maximum chageable budget to the WSP. The WSP selects pope ads (these, fo example, may depend on time and distance of a custome fom a stoe) and sends them (via SMS) to the customes. The WSP chages the advetises fo showing thei ads and also fo successful ads. An ad is deemed successful if a custome visits the advetised stoe. Depending on the sevice plan, customes ae entitled to eceive diffeent numbe of advetisements pe month. Seveal logistics need to be employed to impove AdCell expeience and enthuse customes into paticipation. We povide moe details about these logistics in Appendix A. 1.1 AdCell Model & Poblem Fomulation In the AdCell model, advetises bid fo individual customes based on thei location and time. The tiple (k, l, t) whee k is a custome, l is a neighbohood (location) and t is a time foms a quey and thee is a bid amount (possibly zeo) associated with each quey fo each advetise. This definition of quey allows advetises to customize thei bids based on customes, neighbohoods and time. We assume a custome can only be in one neighbohood 3 The pocess of locating an object by accuately computing the time diffeence of aival of a signal emitted fom that object to thee o moe eceives. 4

3 AdCell 3 at any paticula time and thus at any time t and fo each custome k, the queies (k, l 1, t) and (k, l 2, t) ae mutually exclusive, fo all distinct l 1, l 2. Neighbohoods ae places of inteest such as shopping malls, aipots, etc. We assume that queies ae geneated at cetain times (e.g., evey half hou) and only if a custome stays within a neighbohood fo a specified minimum amount of time. The fomal poblem definition of AdCell Allocation is as follows: AdCell Allocation Thee ae m advetises, n queies and s customes. Advetise i has a total budget b i and bids u ij fo each quey j. Futhemoe, fo each custome k [s], let S k denote the queies coesponding to custome k and c k denote the maximum numbe of ads which can be sent to custome k. The capacity c k is associated with custome k and is dictated by the AdCell plan the custome has signed up fo. Advetise i pays u ij if his advetisement is shown fo quey j and if his budget is not exceeded. That is, if x ij is an indicato vaiable set to 1, when advetisement fo advetise i is shown on quey j, then advetise i pays a total amount of min( j x iju ij, b i ). The goal of AdCell Allocation is to specify an advetisement allocation plan such that the total payment i min( j x iju ij, b i ) is maximized. The AdCell poblem is a genealization of the budgeted AdWods allocation poblem [4,20] with capacity constaint on each custome and thus is NP-had. In this pape, we mainly concentate on the online vesion whee queies aive online and a decision to assign a quey to an advetise has to be done ight away. With abitay queies/bids and optimizing fo the wost case, one cannot obtain any appoximation algoithm with atio bette than 1 n. This follows fom the obsevation that online AdCell poblem also genealizes the secetay poblem fo which no deteministic o andomized online algoithm can get appoximation atio bette than 1 n in the wost case. 5. Theefoe, we conside a stochastic setting. Fo the online AdCell poblem, we assume that each quey j aives with pobability p j. Upon aival, each quey has to be eithe allocated o discaded ight away. We note that each quey encodes a custome id, a location id and a time stamp. Also associated with each quey, thee is a pobability, and a vecto consisting of the bids fo all advetises fo that quey. Futhemoe, we assume that all queies with diffeent aival times o fom diffeent customes ae independent, howeve queies fom the same custome with the same aival time ae mutually exclusive (i.e., a custome cannot be in multiple locations at the same time). Note that if a custome is at a specific location, it is moe likely that in the next peiod of time he is at a neighboing location. We cuently do not conside such coelated queies, and leave this as a futue wok. Thee is a cucial diffeence between the AdCell model and the pevious models studied in the liteatue fo AdWods allocation. In the adwod allocation poblem the aival of adwods (queies) is i.i.d., e.g., we assume we may see the adwod lunch in the moning with the same pobability that we may see it at noon. Howeve, in AdCell the queies ae in the fom of the tiple (custome, location, and time) and thus a quey (k, l, t) can only be evealed at time t. This is a moe geneal and ealistic assumption since fo example fo a location school we may have diffeent pobability distibutions fo diffeent times, e.g., fo a student k the pobability of the quey (k, school, midnight) is zeo but the pobability of the quey (k, school, noon) is high, hence the biddes can also customize thei bid values based on time. This shows that AdCell is a genealization of Pophet Inequality ove bipatite matchings. The topic of Pophet Inequality has been studied in optimal stopping theoy since the 1970s [14,15,13] and moe ecently in compute science [9]. In pophet inequality setting, given the distibution of a sequence of andom vaiables x 1,, x n an onlooke has to choose fom the succession of these values, whee x t is evealed to us at time t. The onlooke can only choose a cetain numbe of values (called he capacity) and cannot choose a past value. The onlooke s goal is to maximize he evenue. The inequality has been intepeted as meaning that a pophet with complete foesight has only a bounded advantage ove an onlooke who obseves the andom vaiables one by one, and this explains the name Pophet Inequality. We note that the pophet inequality poblem is a special case of ou AdCell model in which all the budgets ae infinity and thee is only one custome. The capacity of the custome would coespond to the numbe of values the onlooke can choose in the pophet inequality poblem. 1.2 Ou Results and Techniques Hee we povide a summay of ou esults and techniques. We conside pimaily the online vesion of the poblem. We only know the aival pobabilities of queies (i.e., p 1,, p m ). 5 The eduction of the secetay poblem to AdCell poblem is as follows: conside a single advetise with lage enough budget and a single custome with a capacity of 1. The queies coespond to secetaies and the bids coespond to the values of the secetaies. So we can only allocate one quey to the advetise.

4 4 S. Alaei, M.T. Hajiaghayi, V. Liaghat, D. Pei and B. Saha We can wite the AdCell poblem as the following andom intege pogam in which I j is the indicato andom vaiable which is 1 if quey j aives and 0 othewise: maximize. min( X ij u ij, b i ) (IP BC ) i j j [n] : X ij I j (F ) k [s] : i X ij c k j S k i X ij {0, 1} In the above intege pogam, X ij is set to 1 if quey j aives and advetise i is allocated quey j. (C) denote the capacity constaints. We efe to the vaiant of the poblem explained above as IP BC. We also conside vaiants in which thee ae eithe budget constaints o capacity constaints but not both. We efe to these vaiants as IP B and IP C espectively. The above intege pogam can be elaxed to obtain a linea pogam LP BC, whee we j X iju ij with the constaints (F ), (C) and additional budget constaint j X iju ij b i which maximize i we efe to by (B). We elax X ij {0, 1} to X ij [0, 1]. We also efe to the vaiant of this linea pogam with only eithe constaints of type (B) o constaints of type (C) as LP B and LP C. In the online vesion, we assume to know the E[I j ] in advance and we lean the actual value of I j online. We note a cucial diffeence between ou model and the i.i.d model. In i.i.d model the pobability of the aival of a quey is independent of the time, i.e., queies aive fom the same distibution on each time. Howeve, in AdCell model a quey encodes time (in addition to location and custome id), hence we may have a diffeent distibution on each time. This implies a pophet inequality setting in which on each time, an onlooke has to decide accoding to a given value whee this value may come fom a diffeent distibution on diffeent times (e.g. see [14,9]). In the online vesion, we compae the expected evenue of ou solution with the expected evenue of the optimal offline algoithm. We obtain the following esults. ( ) We povide a e -appoximation algoithm fo the online vesion with both capacity and budget constaints, that is fo the AdCell poblem (Section 3). We povide a ( ) 1 1 e -appoximation algoithm when thee is only budget constaints (Section 3). We povide a 1 2-appoximation algoithm when thee is only capacity constaints. (Section 3). We should emphasis that we make no assumptions about bid to budget atios (e.g., bids could be as lage as budgets). To daw a compaison to the esults known fo the Pophet Inequality, note that the famous Pophet Inequality by Kengel, Sucheston, and Galing in 1970 s concens the case in which the values ae chosen independently fom known distibutions (but not necessay identical) and the onlooke can only choose one value. Using a vey simple example, they showed no online algoithm can be bette than 1 2-competitive [14]. Theefoe one may conside ou 1 2 -appoximation algoithm when only capacity constaints ae pesent as a genealization of the Pophet Inequality settings of Kengel, Sucheston, and Galing. We have a moe complex setting whee we have many customes, and additionally, budget constaints ove the advetises. We now biefly descibe ou main techniques. Beaking into smalle sub-poblems that can be optimally solved using conditional expectation. Theoetically, ignoing the computational issues, any online stochastic optimization poblem can be solved optimally using conditional expectation as follows: At any time a decision needs to be made, compute the total expected objective conditioned on each possible decision, then chose the one with the highest total expectation. These conditional expectations can be computed by backwad induction, possibly using a dynamic pogam. Howeve fo most poblems, including the AdCell poblem, the size of this dynamic pogam is exponential which makes it impactical. We avoid this issue by using a andomized stategy to beak the poblem into smalle subpoblems such that each subpoblem can be solved by a quadatic dynamic pogam. Using an LP to analyze the pefomance of an optimal online algoithm against an optimal offline factional solution. Note that we compae the expected objective value of ou algoithm against the expected objective value of the optimal offline factional solution. Theefoe fo each subpoblem, even though we use an optimal online algoithm, we still need to compae its expected objective value against the expected objective value of the optimal offline solution fo that subpoblem. Basically, we need to compae the expected objective of an stochastic online algoithm, which woks by maximizing conditional expectation at each step, against the expected objective value (C)

5 AdCell 5 of its optimal offline solution. To do this, we ceate a minimization linea pogam that encodes the dynamic pogam and whose optimal objective is the minimum atio of the expected objective value of the online algoithm to the expected objective value of the optimal offline solution. We then pove a lowe bound of 1 2 on the objective value of this linea pogam by constucting a feasible solution fo its dual obtaining an objective value of Related Wok Online advetising alongside seach esults is a multi-billion dolla business [16] and is a majo souce of evenue fo seach engines like Google, Yahoo and Bing. A elated ad allocation poblem is the AdWods assignment poblem [19] that was motivated by sponsoed seach auctions. When modeled as an online bipatite assignment poblem, each edge has a weight, and thee is a budget on each advetise epesenting the uppe bound on the total weight of edges that might be assigned to it. In the offline setting, this poblem is NP-Had, and seveal appoximations have been poposed [3,2,4,20]. Fo the online setting, it is typical to assume that edge weights (i.e., bids) ae much smalle than the budgets, in which case thee exists a (1 1/e)-competitive online algoithm [19]. Recently, Devanu and Hayes [5] impoved the competitive atio to (1 ɛ) in the stochastic case whee the sequence of aivals is a andom pemutation. Anothe elated poblem is the online bipatite matching poblem which is intoduced by Kap, Vaziani, and Vaziani [12]. They poved that a simple andomized online algoithm achieves a (1 1/e)-competitive atio and this facto is the best possible. Online bipatite matching has been consideed unde stochastic assumptions in [7,6,18], whee impovements ove (1 1/e) appoximation facto have been shown. The most ecent of of them is the wok of Manshadi et al. [18] that pesents an online algoithm with a competitive atio of They also show that no online algoithm can achieve a competitive atio bette than Moe ecently, Mahdian et al.[17] and Mehta et al.[11] impoved the competitive atio to fo unknown distibutions. 3 Online Setting In this section, we pesent thee online algoithms fo the thee vaiants of the poblem mentioned in the pevious section (i.e., IP B, IP C and IP BC ). Fist, we pesent the following lemma which povides a means of computing an uppe bound on the expected evenue of any algoithm fo the AdCell poblem. Lemma 1 (Expectation Linea Pogam). Conside a geneal andom linea pogam in which b is a vecto of andom vaiables: (Random LP) maximize. c T x s.t. Ax b; x 0 Let OP T (b) denote the optimal value of this pogam as a function of the andom vaiables. Now conside the following linea pogam: (Expectation LP) maximize. c T x s.t. Ax E[b]; x 0 We efe to this as the Expectation Linea Pogam coesponding to the Random Linea Pogam. Let OP T denote the optimal value of this pogam. Assuming that the oiginal linea pogam is feasible fo all possible daws of the andom vaiables, it always holds that E[OP T (b)] OP T. Poof. Let x (b) denote the optimal assignment as a function of b. Since the andom LP is feasible fo all ealizations of b, we have Ax (b) b. Taking the expectation fom both sides, we get AE[x (b)] E[b]. So, by setting x = E[x (b)] we get a feasible solution fo the expectation LP. Futhemoe, the objective value esulting fom this assignment is equal to the expected optimal value of the andom LP. The optimal value of the expectation LP might howeve be highe so its optimal value is an uppe bound on the expected optimal value of andom LP. As we will see next, not only does the expectation LP povide an uppe bound on the expected evenue of the optimal offline allocation, it also leads to a good appoximate algoithm fo the online allocation. We adopt the notation of using an oveline to denote the expectation linea pogam coesponding to a andom linea pogam (e.g. LP BC fo LP BC ). Next we pesent an online algoithm fo the vaiant of the poblem in which thee ae only budget constains but not capacity constaints.

6 6 S. Alaei, M.T. Hajiaghayi, V. Liaghat, D. Pei and B. Saha 3.1 Online Algoithm with Only Budget Constaints Algoithm 1 (STOCHASTIC ONLINE ALLOCATOR FOR IP B ) Compute an optimal assignment fo the coesponding expectation LP (i.e. LP B ). Let x ij denote this assignment. Note that x ij might be a factional assignment. If quey j aives, pick an advetise i [m] with pobability x ij p j, and allocate the quey to it. Theoem 1. The expected evenue of Algoithm 1 is at least 1 1 e of the optimal value of the expectation LP (i.e., LP B ) which implies that the expected evenue of 1 it is at least 1 1 e of the expected evenue of the optimal offline allocation too. Note that this esult holds even if u ij s ae not small compaed to b i. Futhemoe, the esult of Theoem 1 holds even if we elax the independence equiement among the queies in the oiginal poblem and equie negative coelation instead. Two andom vaiables X, Y ae called negatively coelated, if cov(x, Y) := E[XY]E[X]E[Y] 0. The following definition fom [10] is a natual genealization to the case of negative coelation to the case of n andom vaiables. Definition 1 (Negative Coelation). The andom vaiables X = (X 1, X 2,..., X n ) ae negatively coelated (associated) if fo evey index set I [n], cov(f(x i, i I), g(x i, i Ī)) 0, that is, E[f(X i, i I), g(x i, i Ī)] E[f(X i, i I)]E[g(X i, i Ī)] fo all non-deceasing functions f : R I R, and g : R n I R. (A function h : R k R is said to be non-deceasing, if h(x) h(y) wheneve x y in the component-wise odeing of R k.) Allowing negative coelation instead of independence makes the above model much moe geneal. Fo example, suppose thee is a custome, location tuple that may aive at seveal diffeent times but may only aive at most once o only a limited numbe of times, we can model this by ceating a new quey fo each possible time instance the custome and location pai can aive. These queies ae negatively coelated. Remak 1. It is woth mentioning that thee is an integality gap of 1 1 e between the optimal value of the integal allocation and the optimal value of the expectation LP. So the lowe bound of Theoem 1 is tight. To see this, conside a single advetise and n queies. Suppose p j = 1 n and u 1j = 1 fo all j. The optimal value of LP B is 1 but even the expected optimal evenue of the offline optimal allocation is 1 1 e when n because with pobability (1 1 n )n no quey aives. To pove Theoem 1, we use the following theoem: Theoem 2. Let C be an abitay positive numbe and let X 1,, X n be independent andom vaiables (o negatively coelated) such that X i [0, C]. Let µ = E[ i X i]. Then: E[min( i X i, C)] (1 1 e µ/c )C Futhemoe, if µ C then the ight hand side is at least (1 1 e )µ. Poof ( Theoem 2). Define the andom vaiables R i = max(r i 1 X i, 0) and R 0 = C. Obseve that fo each i, R i = max(c i j=1 X j, 0); so min( i j=1 X j, C) + R i = C. Theefoe E[min( i j=1 X j, C)] + E[R i ] = C and to pove the theoem it is enough to show that E[R n ] 1 C. To show this we will pove the following e µ/c inequality: E[R i ] (1 E[X i] C )E[R i 1] (U) Assuming that (U) is tue, we can conclude the following which poves the claim. n E[R n ] C (1 E[X i] C ) i=1 n C e E[Xi]/C = Ce n i=1 E[Xi]/C = C i=1 1 e µ/c

7 i AdCell 7 The second inequality follows fom the fact that 1 x e x and the last equality follows by obseving that E[X i] C = µ C. Futhemoe, to pove the second claim, that is if µ C then (1 1 ) (1 1 e µ/c e )µ, we can use the fact that fo a 1, (1 e a ) = (1 (1 a + a 2 /2! a 3 /3!...) = a(1 a/2! + a 2 /3!...) = a((1 a) + a(1 1/2! + 1/3!...)) = a((1 a)+a(1 1 e )) = a(1 a 1 e ) a(1 1/e). Theefoe, (1 ) C (1 1 e µ/c e ) µ C C = (1 1 e )µ wheneve µ C. Now it only emains to pove the inequality (U): E[R i ] = E[max(R i 1 X i, 0)] E[max(R i 1 X i R i 1 C, 0)] = E[R i 1 X i R i 1 C ] = E[R i 1 ] 1 C E[X ir i 1 ] Suppose E[X i R i 1 ] E[X i ]E[R i 1 ] then: E[R i 1 ] 1 C E[X i]e[r i 1 ] = (1 E[X i] C )E[R i 1] This gives us the desied esult. Theefoe, we ae only left with poving E[X i R i 1 ] E[X i ]E[R i 1 ]. This clealy holds when X 1, X 2,..., X n ae independent andom vaiables. Now conside the case when they ae negatively coelated. We have R i 1 = max(c i 1 j=1 X j, 0). Hence if the values of X 1, X 2,..., X i 1 inceases, the value of R i 1 does not incease. Theefoe, R i 1 is a non-deceasing function of X 1, X 2,..., X i 1. Theefoe, by the definition of negative coelation (Definition 1) E[ R i 1 X i ] E[ R i 1 ]E[X i ], and hence E[R i 1 X i ] E[R i 1 ]E[X i ]. This completes the poof. Now we pove Theoem 1 using the above theoem. Poof (Theoem 1). We apply Theoem 2 to each advetise i sepaately. Fom the pespective of advetise i, each quey is allocated to he with pobability x ij and by constaint (B) we can ague that have µ = j x ij u ij b i = C so µ C and by Theoem 2, the expected evenue fom advetise i is at least (1 1 e )( j x ij u ij). Theefoe, oveall, we achieve at least 1 1 e of the optimal value of the expectation LP and that completes the poof. Next we pesent an online algoithm fo the vaiant of the poblem in which thee ae only capacity constains but not budget constaints. 3.2 Online Algoithm with Only Capacity Constaints The main diffeence between Algoithm 1 and Algoithm 2 is that in the fome wheneve we choose an advetise at andom, we always allocate the quey to that advetise (assuming they have enough budget). Howeve, in the latte, we un a dynamic pogam fo each custome k and once an advetise is picked at andom, the quey is allocated to this advetise only if doing so inceases the expected evenue associated with custome k. Below we give details of the algoithm, and how the dynamic pogam is un on each custome. Algoithm 2 (STOCHASTIC ONLINE ALLOCATOR FOR IP C ) Compute an optimal assignment fo the coesponding expectation LP (i.e. LP C ). Let x ij denote this assignment. Note that x ij might be a factional assignment. Patition the items to sets T 1,, T u in inceasing ode of thei aival time and such that all of the items in the same set have the same aival time. Fo each k [s], t [u], [c k ], let Ek,t denote the expected evenue of the algoithm fom queies in S k (i.e., associated with custome k) that aive at o afte T t and assuming that the emaining capacity of custome k is. We fomally define Ek,t late.

8 8 S. Alaei, M.T. Hajiaghayi, V. Liaghat, D. Pei and B. Saha If quey j aives then choose one of the advetises at andom with advetise i chosen with a pobability of x ij p j. Let k and T t be espectively the custome and the patition which quey j belongs to. Also, let be the emaining capacity of custome k (i.e. is c k minus the numbe of queies fom custome k that have been allocated so fa). If u ij + E 1 k,t+1 E k,t+1 then allocate quey j to advetise i othewise discad quey j. We can now define Ek,t ecusively as follows: Ek,t = x ij max(u ij + E 1 k,t+1, E k,t+1) j T t i [m] + (1 x ij)ek,t+1 (EXP k ) j T t i [m] Also define Ek,t 0 = 0 and E k,u+1 = 0. Note that we can efficiently compute E k,t using dynamic pogamming. Theoem 3. The expected evenue of Algoithm 2 is at least 1 2 of the optimal value of the expectation LP (i.e., LP C ) which implies that the expected evenue of Algoithm 2 is at least 1 2 of the expected evenue of the optimal offline allocation fo IP C too. Remak 2. The appoximation atio of 2 is tight. Thee is no online algoithm that can achieve in expectation bette than 1 2 of the evenue of the optimal offline allocation without making futhe assumptions. We show this by poviding a simple example. Conside an advetise with a lage enough budget and a single custome with a capacity of 1 and two queies. The queies aive independently with pobabilities p 1 = 1 ɛ and p 2 = ɛ with the fist quey having an ealie aival time. The advetise has submitted the bids b 11 = 1 and b 12 = 1 ɛ ɛ. Obseve that no online algoithm can get a evenue bette than (1 ɛ) 1 + ɛ 2 1 ɛ ɛ 1 in expectation because at the time quey 1 aives, the online algoithm does not know whethe o not the second quey is going to aive and the expected evenue fom the second quey is just 1 ɛ. Howeve, the optimal offline solution would allocate the second quey if it aives and othewise would allocate the fist quey so its evenue is ɛ 1 ɛ ɛ + (1 ɛ) in expectation. Befoe we pove Theoem 3, we fist define a simple stochastic knapsack poblem which will be used as a building block fo the poof. Definition 2 (Stochastic Unifom Knapsack). Thee is a knapsack of capacity C and a sequence of n possible items. Each item j is of size 1, has a value of v j and aives with pobability p j. Let I j denote the indicato andom vaiable indicating the aival of item j. We assume that items can be patitioned into sets T 1,, T u based on thei aival times such that all the items in the same patition have the same aival time and ae mutually exclusive (i.e. at most one of them aives) and items fom diffeent patitions ae independent. Futhemoe, we assume that j [n] p j C. The following algoithm based on conditional expectation computes the optimal online allocation fo this poblem: Algoithm 3 (STOCHASTIC UNIFORM KNAPSACK - OPTIMAL ONLINE ALLOCATOR) Conside a stochastic unifom knapsack poblem as defined in Definition 2. Fo each t [u] and [C], let E t denote the expected evenue of the algoithm fom queies that aive at o afte time t (i.e. T t,, T u ) and assuming that the emaining capacity of the knapsack is. We fomally define E t late. If item j aives do the following. Let t be the index of the patition which j belongs to and let be the emaining capacity of the knapsack. Put item j in the knapsack if v j + E 1 t+1 E t+1. E t can be defined ecusively as follows and can be efficiently computed using dynamic pogamming: E t = j T t p j max(v j + E 1 t+1, E t+1) + (1 j T t p j )E t+1 (EXP) Also define E 0 t = 0 and E u+1 = 0. Clealy the above algoithm achieves the best evenue that any online algoithm can achieve in expectation fo the stochastic unifom knapsack. Howeve, we need a stonge esult since we need to compae its evenue against the optimal value of the expectation LP.

9 AdCell 9 Lemma 2. Conside the stochastic unifom knapsack poblem as defined in Definition 2. Let O o denote the andom vaiable epesenting the expected evenue of Algoithm 3 fo this poblem (i.e. O o = E C 1, since C is the capacity of the knapsack). Also define O e = j p jv j. Assuming that j p j C, the following always holds: 1 2 O e E[O o ] O e Poof (Theoem 3). We apply Lemma 2 to the subset of queies associated with each custome k (i.e. S k ) sepaately. We may think of this as having a knapsack of capacity c k fo custome k. Each pai of advetise/quey, (i, j) is a knapsack item with value u ij. All knapsack items of the fom (i, j) with the same j ae mutually exclusive (because at most one advetise is chosen at andom) and they all have the same aival time. Theefoe, by applying Lemma 2, fom the knapsack of each custome k we get at least 1 2 ( j S k i x ij u ij) in expectation. So oveall, we get 1 2 of the optimal value of the expectation LP and that completes the poof. 3.3 Online Algoithm with both Capacity and Budget Constaints Next, we show that an algoithm simila to the one with only capacity constaints can be used when thee ae both budget and capacity constaints, that is fo the AdCell poblem. Algoithm 4 (STOCHASTIC ONLINE ALLOCATOR FOR IP BC ) Run the same algoithm as in 2 except that now x ij is a factional solution of LP BC instead of LP C. Theoem 4. The expected evenue of Algoithm 4 is at least e of the optimal value of the expectation LP (i.e., LP BC ) which implies that the expected evenue of Algoithm 4 is at least 1 2 (1 1 e ) of the expected evenue of the optimal offline allocation too. Poof. The poof is essentially the same as the poof of Theoem 3. The only diffeence is that we may also lose at most a facto of 1 e fom each advetise due to going ove the budget limit. Fom the point of view of advetise i, on aival of quey j, he is chosen with pobability x ij, and is allocated quey j with pobability (say) yij if thee is no budget constaints. Fom Theoem 3, i j y ij u ij 1 2 i j x ij u ij. Now µ i = j y ij j x ij b i by budget constaint. Now apply Theoem 2 as in Theoem 1 to each advetise sepaately. Hence, the expected evenue fom advetise i is at least (1 1 e ) j y ij. Theefoe, the total evenue is at least (1 1 e ) i j y ij 1 2 (1 1 e ) i j x ij. So, oveall, we get at least 1 2 (1 1 e ) of the optimal value of the expectation LP. Note that this is a goss oveestimation because using conditional expectation on each custome may esult in discading some of the queies which would make it less likely fo advetises to hit thei budget limit. 3.4 Poof of Lemma 2 Hee we pove Lemma 2. We ecall the statement of Lemma 2. Lemma 2. Conside the stochastic unifom knapsack poblem as defined in Definition 2. Let O o denote the andom vaiable epesenting the expected evenue of Algoithm 3 fo this poblem (i.e. O o = E C 1, since C is the capacity of the knapsack)). Also define O e = j p jv j. Assuming that j p j C, the following always holds: 1 2 O e E[O o ] O e Poof ( Lemma 2). The uppe bound is tivial. Clealy, no algoithm (offline o online) can get moe than p j v j evenue in expectation fom each item j. So the total expected evenue is uppe bounded by O e = j p jv j. Next we pove the lowe bound. To pove the lowe bound we fist naow down the instances that would give the smallest E[Oo] O e. The plan of the poof is as follows. Fist, we show that fo each t if we eplace all the items aiving at time t (i.e. all items in set T t ) with a single item with pobability p t = j T t p j and value v t = p j T t v j j p t, we may only decease E[O o ] but O e does not change. So this eplacement may only decease E[Oo] O e and afte the eplacement, each patition only contains one item. So, WLOG, we only need to pove the lowe bound fo instances in which each patition contains one item. Next, we ague that if we scale all v j s by a constant, both E[O o ] and O e ae scaled

10 10 S. Alaei, M.T. Hajiaghayi, V. Liaghat, D. Pei and B. Saha by the same constant. So, WLOG, we assume that O e = 1. Theefoe, we only need to pove a lowe bound on the following pogam: minimize. E[O o ] s.t. O e 1 We then conside a linea elaxation of the above pogam and pove a lowe bound of 1 2 on this elaxation which also implies a lowe bound of 1 2 fo the oiginal pogam. We pove this by constucting a feasible solution fo the dual of this linea pogam that achieve a value of 1 2. In what follows, we explain each step of the poof in moe detail: Fist of all, we claim that if we eplace all of the items aiving at time t (i.e., all items in T t ) with a single item with pobability p t = j T t p j and value v t = p j T t v j j p t, then O o may decease but O e is not affected. Let E[O o] and O e espectively denote the esult of making this eplacement. The fact that O e is not affected is tivial because v t p t = j T t p j v j so O e = O e. Let E t denote the expectation afte eplacing all items in T t with a single item as explained. Fo all values of t > t nothing is affected so E k = Ek. Conside what happens at t = t when we make the eplacement: Fom (EXP) we have: E t = j T t p j max(v j + E 1 t+1, E t+1) + (1 j T t p j )E t+1 fo any convex function f( ) and nonnegative α i s with i α i = 1 it always holds that i α if(x i ) f( i α ix i ) (the Jensens inequality) and max(x + a, b) is a convex function of x so: E t = p t j T t p j max(v j + Et+1 1 p, E t+1) + (1 p t )Et+1 t p t max( j T t p j p t v j + E 1 t+1, E t+1) + (1 p t )E t+1 = p t max(v t + E 1 t+1, E t+1) + (1 p t )E t+1 = E t So we poved that E t Et fo t = t and fo all [1, C]. Futhemoe, notice that accoding to equation (EXP), fo each t, Et 1 is an inceasing function of Et and Et 1, since j T t p j 1 (fom mutual exclusivity of items aiving at time t). So if Et and/o Et 1 decease then Et 1 may only decease. Theefoe fo all values of t t we can ague that E t Et and in paticula E[O o] = E C 1 E1 C = E[O o ]. That means the eplacement may only decease the expected evenue of ou algoithm. So if we eplace all the items in each T t with a single item as explained above one by one we get an instance in which each patition only contains one item and with a possibly lowe expected evenue fom ou algoithm. Theefoe, WLOG, it is enough to pove a lowe bound fo the case whee each patition contains one item. Since scaling all v j s by a constant scales both E[O o ] and O e by the same constant, we can scale all v j s so that O e = 1. So, WLOG, we only need to pove the lowe bound fo cases whee O e = 1. Now, we ague that the optimal value of the following pogam gives a lowe bound on E[O o ]. Theefoe, we only need to pove the optimal value of this pogam is bounded below by 1 2. minimize. E[O o ] s.t. O e 1 We now ewite the pevious pogam as the following linea pogam with vaiables E t and v t (with t [u] and [C] by using the definition of E t fom (EXP). Note that E[O o ] = E C 1. Also note that, in the following, we addess each item by the index of the patition to which it belongs, since thee is only one item in each patition.

11 AdCell 11 minimize. E1 C t [1, u 1], [1, C] : Et p t (v t + Et+1 1 t)et+1 t [1, u 1], [1, C] : Et Et+1 [1, C] : Eu p u v u p t v t 1 [1, C] : v t 0, E t 0 t [1, u] E 0 t = 0 t Notice that any feasible assignment fo the oiginal pogam satisfies all the constaints of the above pogam and hence is also a feasible assignment fo the above pogam. Theefoe its optimal value is a lowe bound fo the optimal value of the oiginal pogam. The above linea pogam is still not quite easy to analyze, so we conside a loose elaxation as we explain next. Fist, it is not had to show that Et as defined in (EXP) has deceasing maginal value in which implies E 1 t 1 E t (This can be poved by induction on t with the base case being t = u and then poving fo smalle t s. We will pove this fomally late). Combining this with the definition of Et fom (EXP), we get the following inequality: E t = p t max(v t + E 1 t+1, E t+1) + (1 p t )E t+1 = max(p t v t + p t E 1 t+1 + (1 p t)e t+1, E t+1) max(p t (v t + 1 E t+1) + (1 p t )Et+1, Et+1) = max(p t v t + (1 p t )E t+1, Et+1) Using the above, we wite a moe elaxed linea pogam. minimize. E C 1 t [1, u 1], [1, C] : Et p t v t (1 p t )E t+1 0 (3.1) [1, C] : Eu p u v u 0 (3.2) t [1, u 1], [1, C] : Et Et+1 0 (3.3) u p t v t 1 (3.4) t=1 t [1, u], [1, C] : v t 0, E t 0 (3.5) t [1, u] E 0 t = 0 (3.6) We now futhe elax the above linea pogam, and only keep the constaints fo = C. minimize. E C 1 t [u 1] : E C t p t v t (1 p t )EC t+1 0 (α t ) E C u p u v u 0 (α u ) t [u 1] : Et C Et+1 C 0 (β t ) u p t v t 1 (γ) t=1 v t 0, E C t 0 Next, we show that the optimal value of the above pogam is bounded below by 1 2 which implies that the optimal value of the oiginal pogam is also bounded below by 1 2 and that completes the poof. To do this, we

12 12 S. Alaei, M.T. Hajiaghayi, V. Liaghat, D. Pei and B. Saha pesent a feasible assignment fo the dual pogam that obtains an objective value of at least 1 2. Note that the objective value of any feasible assignment fo the dual pogam gives a lowe bound on the optimal value of the pimal pogam. The following is the dual pogam: maximize. γ t [u] : γp t α t p t 0 (v t ) t [2 u 1] : α 1 + β 1 1 (E C 1 ) α t + β t (1 p t 1 C )α t 1 β t 1 0 (E C t ) α u (1 p u 1 C )α u 1 β u 1 0 (E C u ) α t 0, β t 0, γ 0 Now, suppose we set all β t = β t 1 pt 1 C γ fo all t except β 1 = 1 γ. And, we set all α t = γ except α u = β u + γ. Fom this assignment, we get β t = 1 γ γ t 1 p k k=1 C. Obseve that we get a feasible solution as long as all β t s esulting fom this assignment ae non-negative. Futhemoe, it is easy to see that β t > u k=1 1 γ γ p k C = 1 2γ. Theefoe, fo γ = 1 2, all β t s ae non-negative and we always get a feasible solution fo the dual with an objective value of 1 2 which completes the main poof. Next, we pesent the poof of ou ealie claim that Et 1 1 E t. We now pove that Et +1 E+1 t by induction on t with the base case being t = u which is tivially tue because Eu = p u v u fo all 1. Next we assume that ou claim holds fo t + 1 and all values of. We then pove it fo t and all values of as follows: E t = p t max(v t + E 1 t+1, E t+1) + (1 p t )E t+1 = max(p t (v t + E 1 t+1 ) + (1 p t)e t+1, E t+1) Obseve that max(a, b) max((1 ɛ)a + ɛb, b) fo all ɛ [0, 1] so: Et max((1 ɛ)[p t (v t + Et+1 1 ) + (1 p t)et+1] + ɛet+1, Et+1) = max((1 ɛ)p t (v t + Et ɛ 1 ɛ E t+1) + (1 p t )Et+1, Et+1) Now by applying the induction hypothesis on E 1 t+1 and E t+1 and setting ɛ = 1 +1 : So we poved that E t Refeences Et max( + 1 p t[v t + 1 Et E t+1] + (1 p t ) + 1 E+1 t+1, + 1 E+1 t+1 ) = + 1 max(p t[v t + Et+1] + (1 p t )Et+1 +1, E+1 t+1 ) = + 1 E+1 t +1 E+1 t and that completes the poof. 1. M. Agawal. Oveview of mobile advetising /11/oveview-of-mobile-advetising. 2. N. Andelman and Y. Mansou. Auctions with budget constaints. In In 9th Scandinavian Wokshop on Algoithm Theoy (SWAT), Y. Aza, B. Binbaum, A. R. Kalin, C. Mathieu, and C. T. Nguyen. Impoved appoximation algoithms fo budgeted allocations. ICALP , 2008.

13 AdCell D. Chakabaty and G. Goel. On the appoximability of budgeted allocations and impoved lowe bounds fo submodula welfae maximization and gap. SIAM J. Comp. p2189, N. R. Devenu and T. P. Hayes. The adwods poblem: online keywod matching with budgeted biddes unde andom pemutations. EC , J. Feldman, A. Mehta, V. Miokni, and S. Muthukishnan. Online stochastic matching: Beating 1-1/e. FOCS , G. Goel and A. Mehta. Online budgeted matching in andom input models with applications to adwods. SODA , M. T. Hajiaghayi, R. Kleinbeg, and D. C. Pakes. Adaptive limited-supply online auctions. EC , M. T. Hajiaghayi, R. D. Kleinbeg, and T. Sandholm. Automated online mechanism design and pophet inequalities. In AAAI, K. Joag-Dev and F. Poschan. Negative association of andom vaiables with applications. The Annals of Statistics, pages , C. Kaande, A. Mehta, and P. Tipathi. Online bipatite matching with unknown distibutions. In STOC, R. M. Kap, U. V. Vaziani, and V. V. Vaziani. An optimal algoithm fo on-line bipatite matching. STOC , D. P. Kennedy. Pophet-type inequalities fo multi-choice optimal stopping. In Stoch. Poc. Applic., U. Kengel and L. Sucheston. Semiamats and finite values. Bull. Am. Math. Soc., U. Kengel and L. Sucheston. On semiamats, amats, and pocesses with finite value. In Kuelbs, J., ed., Pobability on Banach Spaces., S. Lahaie. An analysis of altenative slot auction designs fo sponsoed seach. EC , M. Mahdian and Q. Yan. Online bipatite matching with andom aivals: An appoach based on stongly facto-evealing lps. In STOC, V. H. Manshadi, S. O. Ghaan, and A. Sabei. Online stochastic matching: Online actions based on offline statistics. SODA 11, A. Mehta, A. Sabei, U. Vaziani, and V. Vaziani. Adwods and genealized online matching. J. ACM, 54, A. Sinivasan. Budgeted allocations in the full-infomation setting. APPROX , S. Wang, J. Min, and B. K. Yi. Location based sevices fo mobiles: Technologies and standads. ICC 08, F. Zahadnik. Gamin to offe fee, advetising-suppoted taffic detection and avoidance. A AdCell Policies AdCell stategy pesents new modeling and algoithmic challenges fo making it wok efficiently. We ae intoducing AdCell as a new concept in Intenet Maketing/Advetising via Cell Phones. Conside a WSP which povides mobile sevices to cell phone uses. The WSP can easily use the location of the use to taget the advetisements by demogaphic, inteests, Designated Maket Aea 6, mobile device and caie which impoves the ad expeience of customes and advetising of Advetise Companies. In this section we discuss the possible mechanisms fo impoving advetising and motivating customes to paticipate. A.1 Customes Policy The WSP may povide diffeent AdCell plans. Each plan offes discounts fo ageeing to eceive advetisement via SMS and based on the custome s geogaphical location. Fo example 5% discount fo eceiving 50 ads evey month o 10% discount fo eceiving 100 ads evey month. In addition we may have a pofile fom fo each custome (e.g. as an online fom). Customes ae encouaged to fill this fom and based on this infomation, we would be able to design custome types which can povide additional infomation fo pedicting if an ad would be useful fo that peson. Fo example we may ask about gende, age, what kind of food shops they like, etc in the fom. This may help us impove the ad expeience of each individual peson, specially compaing to the simple AdWods Advetising in seach engines which ae indiffeent to the custome who is seaching the keywod. A.2 Scoing Policy Clealy the wieless povide should encouage the customes fo paticipating in AdCell sevice. Othe than the discount fo singing in the plans, we may impove the atio of successful ads by implementing a scoing policy such that a custome with high scoe may get exta discounts. The idea is that when a custome visits a stoe - 6 O Media Maket is a egion whee the population can eceive the same (o simila) television and adio station offeings and may also include othe types of media.

14 14 S. Alaei, M.T. Hajiaghayi, V. Liaghat, D. Pei and B. Saha 3. Repoting the Puchase with Ad# Via SMS 1. Advetisement with Ad# 4.2. Confimation # & Acquied Points Wieless Povide Via SMS 4.1. Confimation# Custome 5.Confimation # Stoe 2. Puchase with Ad# Fig. A.1. A Puchase Scenaio. Numbes show the sequence of events. due to an advetisement - she should get some points (and since the ad was successful, the WSP would chage the stoe an additional fee). The acquied point may depend on the plan the custome selects o how much she spends at that paticipating stoe. In ode fo the custome to get the points, the stoe whee she visits must epot it to the WSP. If a stoe epots a visit then that stoe will have to pay an additional fee. Thus to be able to enfoce the scoing policy, each advetisement should contain an ad numbe that encodes the use id, date (o the ange ove which the ad is valid), and stoe id. A scenaio fo a puchase based on AdCell ads is shown in Figue A.1. The custome must show this one-time-usable ad numbe to the stoe, the stoe calls the WSP (o does it by Intenet) and povides this ad numbe. The WSP upon eceiving the ad numbe will confim the call by the stoe, by poviding a confimation numbe and the stoe will give this numbe to the custome. Also the WSP will send the same confimation numbe and a notification about the acquied points via SMS to the custome. Specific pats of the confimation numbe may show a custome-specific id (only povided and known to the custome) and thus the custome can veify the confimation numbe by checking this specific pat; o by compaing it to the numbe in the confimation SMS fom WSP. In case of a faud, customes can always submit a complain poviding the ad numbe, eceipt fom the stoe and the appoximate time of the visit. The WSP can check the location and the eceipt to veify that the custome is telling the tuth and then take an appopiate action. Stoes get a scoe based on thei eputation. Fauds will decease the eputation significantly and the atio of successful ads will incease the eputation. If an stoe s scoe becomes low, the company eithe stops giving them advetisement slots o chages them much moe just fo showing the ad.