Fair Allocation with Succinct Representation

Transcription

1 Far Allocaton wth Succnct Representaton Saeed Alae Rav Kumar Dept. of Computer Scence Unversty of Maryland College Park, MD 074. {saeed, Azarakhsh Malekan Erk Vee Yahoo! Research 701 Frst Ave Sunnyvale, CA {ravkumar, ABSTRACT Motvated by applcatons n guaranteed delvery n computatonal advertsng, we consder the general problem of far allocaton n a bpartte supply-demand settng. Our formulaton captures the noton of devaton from farness by a convex penalty functon. Whle ths formulaton admts a convex programmng soluton, we strve for more tme- and space-effcent algorthms. For the case of L 1 penalty functons we obtan a smple combnatoral algorthm based on mn-cost flow and show how to precompute a lnear amount of nformaton such that the allocaton along any edge can be approxmated n constant tme. We then extend our combnatoral soluton to any convex functon by solvng a convex cost flow. Our methods may have applcatons n other contexts stpulatng far allocaton. We study the performance of our algorthms on a large real-world dataset and show that they are both effcent and effectve n practce. Categores and Subect Descrptors. H.3.m [Informaton Storage and Retreval]: Mscellaneous General Terms. Algorthms, Expermentaton, Theory Keywords. Far allocaton, maxmum flow, convex flow 1. INTRODUCTION In the guaranteed delvery settng, advertsers and users are medated by the publsher (e.g., a search engne, an onlne newspaper). The advertser buys a contract for a certan number of mpressons (user vsts to the publsher s page) and declares nterest n a subset of user populaton called buckets The goal of the publsher s to satsfy the demands by placng an ad from the advertser on the web page vsted by a user, f the user (.e., the mpresson) belongs to the advertser s bucket. Part of ths work was done whle the authors were vstng Yahoo! Research. Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal 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 permsson and/or a fee. Copyrght 00X ACM X-XXXXX-XX-X/XX/XX...$5.00. Motvaton. Consder the followng settng n the context of advertsers and search engnes. Each search engne user has three attrbutes (gender, age, locaton) and there are four advertsers who have bought contracts to target varous user subsets; each advertser specfes ts bucket of nterest by specfyng approprate user attrbutes. Advertser A s bucket s young females (gender=female, age=young), B s bucket s all males (gender=male), C s bucket s senor Flordans (age=old, locaton=fl), and D s bucket s all Calfornans (locaton=ca). If the search engne only needs to satsfy the contract s requrement, assgnng a suffcent number of users to an advertser as long as they belong to the advertser s bucket s an apparent feasble soluton. Unfortunately, such an assgnment can be unfar and unrewardng to the advertser for the followng two man reasons. (1) Whle each advertser s bucket has some of the user attrbutes specfed explctly, the unspecfed attrbutes are subect to nterpretaton. Most often, the advertser s equally nterested n all the users who belong to the bucket. For nstance, t wll be undesrable f B were a sports car dealer and the search engne assgns mostly mddle-aged men to B. Lkewse, t s undesrable f a dsproportonate number of old women from Florda were assgned to C, who happens to be Florda real-estate agent. Thus, there s a tact assumpton by each advertser that the users assgned to t are as balanced and far as possble from the set of avalable users. () There can be a large number of attrbutes and each at dfferent levels of granularty (locaton=cty, state, country, etc.) that t mght never be fully possble for any advertser to specfy the desred buckets to the fnest concevable detal. For nstance, A could be a toy store who faled to specfy an explct age group; lkewse, D could be an earthquake nsurance agent prmarly targetng young homeowners near the fault-lne. In ether case, t s more desrable for the search engne to assgn a balanced set of users rather than blame the advertser for under-specfyng ts bucket. In ths paper we address ths problem. Gven a set of mpressons (.e., the supply) and contracts (wth demands and buckets), how to fnd a feasble assgnment of mpressons to contracts that s as far as possble? Answerng ths queston nvolves formulatng what farness precsely means n ths context. And, gven the large number of advertsers (typcally, n the hundreds of thousands) and the astronomcal number of mpressons (typcally, n the hundreds of mllons) n an onlne settng, we nsst on a soluton that s effcent, n both tme and space, and that yelds nsghts nto the structure of the allocaton problem tself. In partc-

2 ular, we desre an allocaton algorthm that s practcal and combnatoral and whose allocaton can be stored succnctly, deally, usng space lnear n the number of mpressons and contracts as opposed to the nave storage that s lnear n the number of edges (whch can be quadratc). Of course, ths succnct representaton should let us reconstruct the allocaton along every edge n a tme-effcent manner. Our contrbutons. We consder the general problem of far allocaton n a bpartte supply-demand settng. Our formulaton, nspred by [13], s combnatoral and captures the noton of devaton from farness by a natural and general form of a penalty functon. Whle ths formulaton admts a convex programmng soluton (assumng the penalty functon s convex), t s undesrable n practce because of effcency consderatons and therefore we seek more effcent solutons. For the case of L 1 penalty functons we obtan a smple combnatoral algorthm for the far allocaton problem. Our soluton s based on solvng a mn-cost flow problem on bpartte graphs, whch can be done very effcently. By usng a powerful dual formulaton stemmng from our combnatoral treatment of allocaton and constranng the flow to be unque n a certan way, we also show how to precompute and store a lnear amount of nformaton such that the allocaton along any edge n the bpartte graph can be approxmately answered n constant tme, under mld assumptons on the nput nstances. Ths space-effcent reconstructon method mght be of ndependent nterest n contexts beyond far allocaton. We also prove two addtonal propertes of our formulaton. Frst s robustness, where we show how to upper bound the performance loss when the supply estmates are only approxmately known. Second s extensblty, where we show an even smpler greedy approxmate algorthm when some of the demand constrants are allowed to be volated. We perform an expermental evaluaton of our algorthm on a large real-world dataset obtaned from the Yahoo! s dsplay advertsng system. Our experments demonstrate the effcency and the effcacy of our mn-cost based algorthm. Furthermore, t llustrates the space savngs enabled by the reconstructon procedure. Fnally, we extend our combnatoral soluton to any convex functon. Ths nvolves solvng a convex cost flow, whch once agan s more effcent than solve a general convex program. Related work. The related work falls nto three classes. The frst s work on onlne allocaton problems. The second s the ever-ncreasng body of lterature n the area of computatonal advertsng. The thrd s network flow problems and the role of prmal-dual methods n computatonal advertsng. Vee, Vasslvtsk, and Shanmugasundaram frst studed the onlne allocaton wth forecast problem, where gven an approxmaton of the onlne supply, the goal s to create an effcently reconstructble plan for performng some form of far allocaton [14]. They focus on the effcency and samplng aspect of the problem and consder only the strctly convex verson, whch makes t amenable to usng fxed pont crtera such as KKT condtons for non-lnear optmzaton. Ghosh et al. [9] studed the problem of representatve allocaton for dsplay advertsng when there are both spot markets and guaranteed contracts; they propose a soluton where guaranteed contracts are mplemented by randomzed bddng n spot markets. Onlne advertsng s one of the most proftable resources for the large search engne companes such as Yahoo! and Google and publshng stes such as cnn.com, nytmes.com. Two popular methods used for onlne advertsng are slot ad aucton and dsplay advertsement. Most of the recent lterature for onlne advertsement are focused on studyng slot ad aucton from the game theoretc perspectve [7]. There have been some recent work on dsplay advertsement and guaranteed delvery. In [8], Fege et al. studed the guaranteed delvery for dsplay advertsement wth penaltes. In the guaranteed delvery model, advertsers act as contractors. Each advertser requests some number of mpresson. If ths request s accepted by the search engne, t would be called a contract. In ths model, for each accepted contract, ether the whole demand requested n the contract should be satsfed or the search engne wll pay extra penaltes for the non-satsfed porton of the demand. They showed that there s no constant approxmaton for ther problem and present a bcrtera algorthm. Also they proved a structural approxmaton result for the adaptve greedy algorthm. The problem of advanced bookng wth costly cancellaton also have been studed n [5] and [3] from a game-theoretc pont of vew. Our soluton s manly based on the network flow problem and ts dual. There s a large amount of lterature on the network flow problem (e.g., []). The closest work to our method s the push-relabel algorthm of Goldberg and Taran [11]; they ntroduced a method for computng the maxmum flow problem wthout usng augmentng paths. The reconstructon of the mn-cost flow nstance s based on the dual varables of the mn-cost flow soluton. Prmal-dual methods have been largely used as a tool to fnd approxmaton algorthms for varous problems (e.g., [4, 1]). Recently, Devanur et al. [6] and Jan and Vazran [1] used prmaldual methods and KKT condtons for solvng market equlbra problems. Organzaton. In Secton, we formally descrbe the far allocaton problem and set up necessary notaton. In Secton 3, we consder the far allocaton obectve wth an L 1 penalty functon and formulate t as a lnear programmng problem and show how to solve ths problem combnatorally. In Secton 4, we descrbe how to preprocess ths soluton so that by keepng track of very few varables, one can compute the optmal allocaton effcently. In Secton 5, we analyze a smple greedy algorthm for soft demand constrants. Secton 6 presents our expermental results. In Secton 7 we present a combnatoral soluton to general convex penalty functons. Fnally, Secton 8 contans concludng thoughts.. PRELIMINARIES Suppose we are gven a set I of mpressons and a set J of contracts. Each mpresson I has a supply s > 0. Each contract J has () a weght W > 0 that captures ts mportance, () a desred bucket mp() I of mpressons, and () a demand d > 0, denotng the number of mpressons that need to be allocated to ths contract. For an mpresson, let con() J denote the set of contracts that desre.

3 As stated earler, the goal s to fnd the most far allocaton of mpressons to contracts. Let y be the number of mpressons that are assgned to contract n a gven allocaton. Let δ = s d mp() s, (d, 0) (d δ, w ) ( δ, 0) (s, 0) be quantty that captures the deal far allocaton of mpresson to contract,.e. a perfectly balanced number of mpressons from mpresson are assgned to contract. Let w = W /d. The goal s to mnmze w (y, δ ), mp() where (, ) s the penalty functon that penalzes devaton from the deal far allocaton, subect to the supply and demand constrants. Dfferent norm/dstance functons can be used for (, ); for example, = L 1, = L, = KL, and so on [13]. If (, ) s not restrcted to be convex, then the problem becomes NP-hard to even approxmate to wthn a constant factor (see Appendx A). We defne the noton of ɛ-robust nput. Defnton 1 (ɛ-robust nput). An nput nstance to our problem s ɛ-robust f there s a feasble assgnment of mpressons to contracts when we scale up all the demands by a factor of 1 + ɛ. Henceforth, we wll assume that our nput nstances are ɛ-robust for a sutable ɛ; ths s a mld techncal assumpton that typcally holds n practce. Also, a superscrpt wll always denote a contract and a subscrpt wll always denote an mpresson. 3. FAIR ALLOCATION WITH L 1 PENALTY The far allocaton problem wth the L 1 penalty functon can be formulated as a lnear programmng (LP) problem. Our man result s that we can solve ths LP by nstead solvng a mn-cost flow problem,.e., there s a combnatoral soluton to the far allocaton problem wth L 1 penalty. In Secton 4, we show that n fact by preprocessng the network flow soluton, we can fnd a succnct representaton that only stores O( J + I ) values and can reconstruct the asymptotcally optmal soluton n O(1) tme. Later n Secton 5, we obtan an approxmate soluton (along wth effcent reconstructon) when the demand constrants are soft. We frst consder an LP formulaton of the problem: mn w y δ, (1) J I subect to J, I, mp() con() y = d (demand) y s (supply) To smplfy the descrpton, for a gven allocaton A, we defne unfar(a) = w A δ. The flow network, wth capacty cap(, ) and cost cost(, ) on each edge (, ), s constructed as a four layer graph G (Fgure 1). The frst and the last layers are the source s and snk t respectvely. The second layer represents the set J of contracts, and the thrd layer stands for the set I of mpressons. source s contracts mpressons Fgure 1: The network constructon wth (capacty, cost) on the edges for L 1. t snk Source s has an edge (s, ) to each contract J n the second layer, wth cap(s, ) = d and cost(s, ) = 0. Contract J n the second layer s connected to the mpresson I n the thrd layer ff mp(). In ths case, there are two edges a top edge and a bottom edge between and. The top edge has capacty d δ and cost w and the bottom edge has capacty δ and cost zero. Fnally each mpresson I n the thrd layer s connected to the snk t by an edge (, t) wth cap(, t) = s and cost(, t) = 0. Let E denote the set of edges between the second and the thrd layers n Fgure 1. Theorem. The mn-cost s-t flow on G s the soluton to (1). Proof. We need to show the followng: () a maxmum s-t flow n G s a feasble soluton to (1), provded (1) has a feasble soluton; () any feasble soluton to (1) s a feasble s-t flow n G; and () mnmzng the cost of the maxmum s-t flow n G s equvalent to mnmzng the obectve n (1). It s easy to see that f (1) has a feasble soluton, then a maxmum s-t flow n G s a feasble soluton. Snce we look for the maxmum flow, f there s any feasble soluton to the LP that can satsfy all the demands and the supply constrant, then that soluton would be selected as the maxmum flow as well. Ths means f (1) has a feasble soluton, then any feasble maxmum flow n our network s a feasble soluton to (1). Second, t s easy to see that any feasble soluton to (1) s also a feasble s-t flow n G. And last, we argue that the cost of the optmal soluton to (1) s the same as the mnmum cost of the maxmum flow n G. In other words, we need to show that the costs of the flow are computed correctly n G. Note that the total contrbuton to the cost of the pars of mpressons and contracts that are over-assgned,.e., y > δ, s half of the total cost. So t s enough to compute twce ths cost. Now consder an mpresson and a contract. If y > δ, then we can route at most δ flow through the edge wth cost 0 and route y δ through the edge wth cost w, whch s equal to w (y δ ), whch s exactly twce the cost of ths par. So we showed that our mn-cost flow soluton has exactly

4 the same value as (1). Next, we show that n fact, we can preprocess the mn-cost flow soluton so that by keepng track of ust O( J + I ) values, we can reconstruct the complete flow. 4. RECONSTRUCTION FOR L 1 Even though the mn-cost flow formulaton obtans the optmal soluton, as we mentoned earler, t s not feasble to store the entre allocaton nformaton. Navely storng the soluton representaton s expensve snce t uses O( E ) space. (As we wll see n Secton 6, supply demand bpartte graphs that occur n practce tend to have hgh average degree,.e., E s much larger compared to J + I.) Ideally, we wsh to store ust O( J + I ) nformaton (.e., amortzed O(1) per node) that wll let us reconstruct the flow along every edge; for practcal reasons, we also requre such a reconstructon to be tme-effcent. We consder the L 1 formulaton and show an approxmate reconstructon va dual varables for the nodes. Frst, we wrte the LP correspondng to the mn-cost flow; from Theorem, ths LP s equvalent to (1). subect to mn J, (, ) E, I, mp() mp() con() w x () (x + x ) = d x δ (x + x ) s (, ) E, x, x 0 Here, x denotes the flow along the top edge and x denotes the flow along the bottom edge from to. The dual of the above LP looks as follows. max Y d Z s A δ (3) J I subect to (,) E (, ) E, Y Z A 0 (, ) E, Y Z w (, ) E, A, x 0 Here, A denotes the allocaton, where A s the allocaton amount from mpresson to contract. Snce we want to maxmze the dual functon and the coeffcent of each A n the obectve functon s negatve, we would lke to set the A s as small as possble. From the constrants n the dual, ths means A = max(0, Y Z ) and hence we do not need to keep track of A s. Next, we show how to reconstruct y s by only keepng Y s and Z s and then we show that n fact t s enough ust to keep track of the Y s. Snce we are consderng an optmal soluton, because of the complementary slackness, we have three cases for each edge between and. Frst we consder the bottom edges. () Y Z < 0: n ths case, we have x = 0. () Y Z > 0: n ths case, x s fully saturated. () Y Z = 0: n ths case, we have Y = Z. Ths s the only case that we have an edge between (, ), wth the only constrant beng the edge capacty constrant. We have the same scenaro for the top edges as well. In the thrd case, the dual varables do not gve us any nformaton on the value of the prmal. We call the edges belongng to the thrd class slack edges. Next, we argue that any feasble assgnment of flow to slack edges would be a soluton. To see ths, frst notce that the cost of any slack edge (, ) s exactly to Y Z. Further, any path from to consstng of only slack edges have the same cost. Also, any cycle consstng of slack edges has a cost of 0. Therefore, any feasble maxmum flow on the slack edges would consttute a soluton. However, ths means we have to be able to reconstruct a maxmum flow on the slack edges. Thus, n the worst case, reconstructng an arbtrary maxmum flow usng the dual s no easer than fndng a maxmum flow from scratch! Therefore, we need to store extra nformaton to be able to reconstruct a maxmum flow on the slack edges effcently. Next, we provde a combnatoral soluton n whch we show how to compute a representaton for approxmate reconstructon (Secton 4.1) and how to use ths representaton to do the actual reconstructon (Secton 4.). In Secton 4.3 we dscuss the generc effect of supply and/or demand scalng, whch s of nterest when supply forecasts, for nstance, are not avalable precsely. Remark. At a frst glance, a basc feasble soluton to the prmal LP of the maxmum flow mght appear to requre only lnear storage; ths s not the case, however. The prmal LP of a maxmum flow for a graph wth n nodes and m edges has m varables and m + n constrants. If a basc feasble soluton to ths LP has k non-zero varables, there should be k lnearly ndependent tght constrants. Among those constrants, at most n of them could be node constrants and the remanng k n should be edge capacty constrants. However, a tght edge capacty constrant unquely dentfes the flow on t respectve edge so we do not need to store the flow value for these edges. So we need to store the flow explctly for at most n edges. Ths argument however requres that we already know whch edges have non-zero flow. The number of edges wth non-zero flow could be of O(m) whch would requre more than lnear space to store. To llustrate ths wth an example, consder a maxmum flow on a graph wth three nodes s, v, and t. Suppose there s one edge of capacty k from s to v and k + 1 edges from v to t, each of capacty 1. Among the k + 1 edges from v to t, at most k edges can be tght and we need to store whch k of them are non-zero. Note that there s no way to dentfy those k edges from the dual LP. 4.1 Computng the representaton As descrbed earler, by computng the dual varables of the mn-cost flow, we can decde whch edges are saturated and whch ones are empty. For the remanng (.e., slack) edges, the problem s reduced to computng a maxmum flow n the new subgraph. Here, we present a way to fnd a specfc maxmum flow soluton that s easy to reconstruct. We start by developng some defntons. For a gven node v V and a gven flow functon flow : E R + on the edges, let n(v) = flow(u, v), out(v) = u:(u,v) E u:(v,u) E flow(v, u),

5 1: Intalze heght(s) = J + and heght(v) = 0 for v s : repeat 3: For the current functon heght, fnd the node v wth largest xheght(v) 4: Set heght(v) = heght(v) + xheght(v) and update the excess flow values and xheght( ) for the rest of the nodes 5: untl xheght(v) µ for all v V Fgure : Computng heght( ). excess(v) = n(v) out(v). Defnton 3 (ɛ-feasble flow). A flow functon flow(x, y) : E R + s called ɛ-feasble f and only f for any contract, 0 excess() ɛ out() or equvalently out() n() (1 + ɛ) out(). Suppose we have a functon heght : V R + such that the flow on each edge e = (, ) s gven by flow(, ) = mn(1, max(heght() heght(), 0)) cap(, ). (4) Before we show the exstence of such a functon, we defne another functon xheght(v) as follows. For a gven heght( ), f by changng heght(v) to heght(v) + δ we get excess(v) = 0, then we defne xheght(v) = δ. Intutvely, xheght(v) ndcates how much we need to ncrease the heght of v (whle keepng other nodes fxed) to reduce the excess flow of v down to 0. It s easy to see that xheght(v) can be computed usng a bnary search. (To see ths, frst assume heght(v ) s fxed for all v v and wlog heght(v 1) heght(v I + J 1 ). If we assume that when excess(v) = 0, heght(v ) heght(v) heght(v +1), then computng xheght(v) wll be reduced to solvng a lnear equaton. Now to fnd, we can perform a bnary search over all the nodes at the gven heghts.) Next we show that there exsts a functon heght(v) for whch the correspondng flow s ɛ-feasble and t s greater or equal to maxmum feasble flow on the slack edges. Note that for a gven functon heght(v), computng the correspondng flow s trval ust from ts defnton usng (4). The method we use for computng the heght functon s as ɛ I (1+ɛ). follows. Let µ = Note that the above algorthm mght appear smlar to the push-relabel algorthm of Goldberg and Taran [11]. However, the requrement that the amount of flow gong through an edge s a functon of the heght dfference of ts endponts and the non-ntegralty of the heghts makes our settng dfferent from thers. We now show that the correspondng flow for heght( ) after the termnaton of the algorthm s ɛ-feasble and s at least as large as the maxmum feasble flow. Frst we show some propertes of the algorthm. Lemma 4. After the termnaton of the algorthm, the set of edges between s and contract nodes are all fully saturated or n other words for any contract node, heght() heght(s) 1. Proof. Frst we partton the nodes nto two sets X and Y based on whether they are reachable from s on the resdual flow graph. We frst clam that Y and t Y. Ths follows snce there can be no smple path from s to t n the resdual graph. Any smple path from s to t s of length at most J + 1 and snce heght(s) = J +, t should be the case that for some edge (u, v) on the path, heght(u) > heght(v) + 1 and therefore flow(u, v) = cap(u, v), whch means (u, v) s not n the resdual graph. Thus, s and t are dsconnected. Snce X and Y are non-empty and dsont, (X, Y ) s a cut. Notce that the sze of ths cut s less than or equal to the cut whch conssts of edges between s and the contract nodes, because we may have some excess flow on nodes n X. We can conclude that f the cut (X, Y ) s not the same as the cut between s and the contract nodes then the demands of the contracts were not satsfable to begn wth whch contradcts our assumpton. Therefore, X = {s} and Y = X, so for any contract node, heght() heght(s) 1. Lemma 5. For all v V, excess(v) s always non-negatve durng runnng the algorthm. Proof. Frst, we can see that at the ntalzaton step, excess(v) 0. We show that after runnng each step, the property stll holds. Suppose the algorthm selected v at a round and relabeled t so after the relabel step excess(v) = 0. For all nodes that has an outgong edge to v, ther outgong flow s only decreased so ther excess wll ncrease. And for all nodes wth ncomng edges from v, ther ncomng flows only ncrease so ther excess wll ncrease as well. For the rest of the nodes the excess wll not change. Theorem 6. The flow computed by Algorthm s ɛ-feasble. Proof. Consder the tme when the algorthm has termnated. We know that for each v V, f we ncrease ts heghts by xheght(v) < µ then excess(v) becomes 0. Consder a node from the contract layer at the end of the algorthm. Snce we know that all the contracts are satsfable, the set of edges from s to the contract nodes s the mnmum cut and heght() heght(s). (Note that heght(s) = J + and heght(t) = 0 and any smple path between each contract and t has a length less than or equal to J.) Therefore, flow(s, ) = cap(s, ) and rasng by µ wll not change ts ncomng flow. Suppose out max () s the total capacty of edges gong out of. We know that after the algorthm termnates, xheght() µ. Hence, n() out() = excess() µ out max (). Also note that there can be at most I edges gong out of, each one wth a capacty of less than the ncomng capacty of whch was cap(s, ). Ths mples out max () I cap(s, ) and therefore n() out() µ I cap(s, ). On the other hand snce n() = flow(s, ) = cap(s, ), we can conclude that n() out() µ I n(). By rearrangng the terms, we get n() (1 + ɛ) out(). Next, we bound the runnng tme of the algorthm that computes the representaton. Lemma 7. The algorthm termnates after O(( J + I ) I J /ɛ) teratons. Proof. Frst of all, at each teraton, we ncrease the heght(v) for some node v by at least µ. Notce that no node wll ever go hgher than the source s so each node can be relabeled at most ( J + )/µ tmes so the total number of teratons (for all nodes) n the worst case s O(( J + I ) J /µ) whch s O(( J + I ) I J /ɛ).

6 1: Buld the mn-cost flow graph G, run the mn-cost flow and compute the dual varables {Z } I and {Y } J : Remove the edges that are forced to be saturated or empty, update the supples and demands; let G = (I, J, E ) be the new graph 3: Scale all the demands by (1 + ɛ) 4: v I J, compute the heght(v) usng Algorthm Fgure 3: Preprocessng. 1: {(, ) s the gven edge} : f Y Z < cost(, ), then let flow(, ) = 0 3: f Y Z > cost(, ), then let flow(, ) = cap(, ) 4: f Y Z = cost(, ), then let flow(, ) = mn(1, max(heght() heght(), 0)) cap(, )/(1 + ɛ) Fgure 4: Reconstructon (, ). 4. Reconstructon usng the representaton We now descrbe how to reconstruct the flow usng ust heght(v). For now we assume that after the termnaton of the algorthm there s no excess flow on any node,.e., v, excess(v) = 0. Obvously, because of the way we constructed the heght functon, the flow of every edge (u, v) s flow(u, v) = mn(1, max(heght(u) heght(v), 0)) cap(u, v). Ths would work perfectly well f there was no excess flow on any node. However because of the excess flows, the soluton may not be feasble. To fx that we frst tweak the demands before computng the heght functon and show that f the nput nstance s (ɛ + ɛ )-robust, then we can reconstruct a feasble soluton. Consder the followng modfcaton of our method. () Scale up all the demands by a factor of (1+ɛ), compute the heght functon heght( ) as explaned n Algorthm, and then set the demands back to ther orgnal values. () At reconstructon tme, for each contract and mpresson, reconstruct the flow on (, ) usng heght() as before but then scale t down by a factor of (1 + ɛ). Theorem 8. Suppose that the gven nput nstance s (ɛ+ ɛ )-robust. Then, the reconstructed soluton accordng to the above modfcaton s feasble. Furthermore, t may assgn mpressons to contracts up to (1 + ɛ) tmes ther demand. Proof. Frst, notce that snce the nput s (ɛ + ɛ )- robust, we can stll satsfy all the demands whch means the set of all demand edges s stll a mnmum cut. After Algorthm termnates, snce the soluton s ɛ-feasble, for any contract we have out() n()/(1+ɛ). But notce that we scaled up all the demands by (1 + ɛ) at the begnnng out heght computaton algorthm, so n() = (1 + ɛ) d where d s the orgnal demand of the contract. Therefore, out() (1 + ɛ)d and clearly f we scale down the flow that we reconstruct for the edges gong out of by (1+ɛ), stll the outgong flow of s at least as much as d whch means the demand constrants are satsfed. Usng a smlar argument for the supply sde we can show that for any supply node the ncomng flow of cannot exceed ts supply s. A summary of the whole process s gven n Algorthm 3 and Algorthm Effect of supply scalng Notce n applyng Theorem 8, we scale up the demands before runnng the heght algorthm but we use the same supply. There mght be also other reasons for scalng up the demands. For example, suppose we do not know the exact supply of the supply nodes (.e., the s values), but we may have an estmate of each supply node whch we call s. For example suppose we know that wth hgh probablty, s s. Under, such a crcumstance we may want to scale 1+ɛ down all the supply estmate s by some factor 1 + α (or equvalently scale up all the demand constrants, compute the flow and then scale the flow down by the same factor) to make sure that wth hgh probablty we can always meet the the supply constrants. Scalng the demands (or supples) may affect the value of our obectve functon. The next result gves an upper bound on the change of the obectve functon when we scale the demands by an arbtrary factor. Theorem 9. For a gven nput nstance whch s α-robust, and wth the optmal obectve functon value Opt, f we scale the demands by 1 + α, then the new optmal value of the obectve functon Opt s at most J α max J d away from Opt and that s tght. Proof. Consder the flow correspondng to the optmal allocaton of the orgnal nput (before scalng up the demands). Now, for each contract one by one, we scale cap(s, ) by 1 + α. Snce the nput nstance s α-robust, we should be able to augment the flow by αd whch s the amount of ncrease n the capacty of (s, ). By applyng the augmentaton we may change the flow of each of the other contract nodes by at most αd whch means the value of the obectve functon may ncrease at most by αd. Snce there are J augmentatons and each augmentaton may affect the flow of all the other contract nodes, n the worst case the total change n the obectve functon value s upper bounded by J α max J d. A tght example s shown n Appendx B. 5. A GREEDY SOLUTION FOR L 1 ALLO- CATION WITH SOFT DEMAND In ths secton we present a smple greedy approach for a slghtly generalzed verson of the L 1 penalty functon. In ths verson, we assume the demand constrants are soft, meanng, t s possble to satsfy a contract partally. (The search engne, however, should pay extra amount per unsatsfed demand, smlar to the model used n [8]; we wll capture ths by a parameter β.) We also show how to preprocess and then reconstruct the greedy soluton usng O( J + I ) space for storng the preprocessed nformaton and O(max I con() ) tme to recompute the allocaton. We assume each contract has ts own weght w = W d and to mplement the soft demand constrant, we assume an amount of β w s pad for each mpresson that cannot be allocated to the contract, where the factor β 1. We now present a greedy algorthm and prove that the total cost of ts soluton s at most (1 + β)/ that of the optmal soluton. The greedy algorthm proceeds as follows. We now show that ths algorthm obtans an approxmaton to the optmum. Lemma 10. Algorthm 5 s a (1 + β)/-approxmaton for the L 1 penalty functon for the soft demand case wth factor β.

7 1: repeat : Let be the next contract wth the largest w 3: Gve the most far allocaton possble to contract 4: untl all contracts are consdered Fgure 5: Greedy allocaton Proof. The proof s based on chargng. We start by defnng some notaton. As usual, for a gven allocaton A, let A be the number of allocatons from mpresson to contract. In an allocaton A, we call a contract on mpresson as under-represented f A < δ ; let under (A) = max(0, δ A ). Smlarly, we call over-represented on f A > δ and let over (A) = max(0, A δ ). Let Opt denote the optmal allocaton and Greedy denote the greedy allocaton. Now we make the followng clam: n any allocaton A, we have unfar(a) I, J under (A). The clam holds because for any contract, I over (A) I under (A), where the nequalty changes to equalty when d s completely satsfed n A. In addton, unfar(a) = (under (A) + over (A)). I, J Ths means that t s enough to consder the under-represented contracts and lose only a factor of two n Opt. Also, snce the allocaton s greedy, the amount of under-farness on each mpresson n greedy s lower than any other allocaton. I.e., I, J under (Greedy) I, J under (A) for any allocaton A ncludng Opt. Now, let us consder the total cost of under-farness on mpresson n the greedy soluton, under (Greedy) = under (Greedy). con() Even f we do not accommodate these mpressons n any other mpresson and pay the β factor nstead of allocatng them, the total amount would be at most β under (Greedy). Now the total value of the obectve functon for the greedy soluton s at most (β + 1) I under(greedy). From the earler argument, we know that I under(greedy) Opt/. These mply that the greedy soluton s a (β + 1)/ approxmaton for the L 1 penalty wth soft demand. 5.1 Reconstructng the greedy soluton Next, we show how we can reconstruct the greedy soluton by storng only O( J ) preprocessed nformaton. The runnng tme for reconstructng the allocaton based on stored nformaton s O(max I con() ). In the preprocessng phase, we compute the greedy allocaton as descrbed n 5. The stored nformaton for each contract s mp() over (Greedy). The reconstructon s as follows: As descrbed earler, n ths method we need to keep track of only one varable for each contract. Also at each mpresson arrval, n the worst case we have O(max I con() ) processng tme. Wth smlar argument gven n Lemma 10, we can show that the computed soluton here s also 1+β approxmaton for the far allocaton wth soft demands. 6. EXPERIMENTS In ths secton we perform an expermental evaluaton of our algorthm on a large real-world dataset. The goal of the 1: {A new mpresson from bucket } : For all contracts, set over = mp() over (Greedy) 3: J = { con() s under-represented} 4: f J then 5: Assgn the new mpresson to arg max J w 6: else 7: J = { con() over > 0} 8: Assgn the new mpresson to arg max J w 9: over = over 1. Fgure 6: Greedy reconstructon () I 13,414 J 14,880 E 1,113,776 (1/ I ) I con() (1/ J ) J mp() I s J d Table 1: Impresson contract graph G: characterstcs. experment s three-fold: () to demonstrate the practcal feasblty and the effcacy of our mn-cost based algorthm, () to compare the performance of our algorthm to that of baselnes such as Greedy (Secton 5) and ts heurstc varants, and () to understand the space savngs enabled by the reconstructon procedure. Data and mplementaton. The dataset conssts of a subset of actual mpressons buckets and advertser contracts from Yahoo! s dsplay advertsng system. The basc characterstcs of ths graph s shown n Table 6. We can see that the graph s reasonably large and the average degree of both the mpresson and contractor nodes s farly hgh. The supples and demands are ntegral and n our experments, we treat all contracts equally,.e., w = 1 for all J. Frst, we ensure that ths nstance s n fact feasble by checkng that the maxmum flow n the nstance s at least the sum of the demands of the contracts. Next, we mplement the mn-cost based algorthm for mnmzng the L 1 penalty. To do ths, we use Goldberg s algorthm [10] for mn-cost flow, obtaned from avglab.com/andrew/soft.html. Snce ths mplementaton requres ntegral capactes and costs, we round the costs and capactes when constructng the graph n Fgure 1. Addtonally, snce parallel edges are not supported n the mplementaton, we splt each bottom edge n the mddle layer of Fgure 1 by ntroducng a new node; the costs and capactes of the newly created edges are the same as the orgnal bottom edge. The program was run on a sngle CPU 1.8GHz Lnux machne wth 16G memory. The mn-cost based algorthm took 178 seconds to run on our nput nstance. Besdes provdng nsghts nto the problem tself, ths s more tme-effcent than applyng a black-box LP solver. Performance. The performance of ths algorthm s compared to that of the Greedy algorthm. Snce the contracts are unweghted n our nstance, we try two addtonal varants of Greedy: the contracts are ordered by decreasng

8 Algorthm L 1 L % unsatsfed penalty penalty ( 10 8 ) demands Theorem Greedy Greedy < Greedy > Table : Performance of varous algorthms. demands (denoted Greedy <) and ncreasng demands (denoted Greedy >). The results are shown n Table 6. The second column reports the L 1 penalty as n (1). For comparson purposes, we also compute the L penalty as n (5) and report t n the thrd column. The fourth column contans the fracton of contracts that are unsatsfed by the allocaton; notce that the mn-cost based algorthm always satsfes all the demands, as long as the nstance s feasble. As we see, the L 1 penalty of the mn-cost based algorthm s more than the frst two versons of Greedy. Interestngly, the thrd verson Greedy > does better n terms of both L 1 and L penaltes, but unfortunately does not satsfy all the demands. If however the demands are soft,.e., there s a cost assocated wth not fully satsfyng a demand, we can apply Lemma 10 to comment on the performance of Greedy. To understand the soluton produced by the mn-cost algorthm better, we consder the dstrbuton of the L 1 penaltes per contract (.e., for a contract, the penalty value w I y δ ). Fgure 7 shows ths dstrbuton: the x-axs represents the L 1 penalty values and the y-axs represents the fracton of contracts wth ths penalty value. As we see, the penaltes of the mn-cost flow-based algorthm s sandwched between the greedy counterparts. Ths shows that n order to satsfy all the contracts, one has to ncur farly hgh L 1 penaltes at least for some contracts. (Greedy and ts varants do not take ths nto account and hence mght end up not satsfyng some contracts.) 1 L1 penalty dstrbuton for contracts Mn-cost based algorthm Greedy> algorthm Greedy< algorthm saturated or empty for more than 99% of the edges n the graph. Thus, the reconstructon becomes an effcent proposton snce we only need to address the remanng 0.36% of the edges. Condton % edges on (, ) Y < Z Y > Z Y = Z Table 3: Dstrbuton of the dual varables. The graph G n Algorthm 3 for our nstance s farly small; Table 4 shows ts propertes. From the data, we can store ( J + I + J + I ) values (nstead of E ) and can reconstruct the soluton. Ths s 3.1% smaller by almost two orders of magntude of the space wthout the reconstructon! The tme for reconstructng the flow for each edge s essentally neglgble by applyng Algorthm 4. I 001 J 4160 E 5407 (1/ I ) I con().70 (1/ J ) J mp() 1.30 Table 4: Reconstructon graph G : characterstcs. 7. CONVEX PENALTY FUNCTIONS In ths secton we descrbe the combnatoral soluton for a general convex penalty functon. Our soluton s based on fndng a convex cost flow. For smplcty of exposton, we descrbe the method for the L functon; the general method, however, works for any convex dstance functon. For L penalty functon, the quadratc programmng formulaton of the problem s as follows. mn w (y δ ), (5) 0.1 subect to fracton of contracts e-04, y = d (demand), y s (supply), z s con() y 1e penalty Fgure 7: Dstrbuton of penaltes. Effcacy of reconstructon. Fnally, we look at the effcacy of the reconstructon. To study the space savngs, we examne the values of the dual varables n (3). Table 3 shows the values. From ths t s clear that the flow s ether We show how to solve (5) usng a convex cost flow. The flow network, wth capacty and cost on each edge, s constructed n the followng manner. We create a node for each contract and a node for each mpresson. Now compute the completely far allocaton for each contract. In ths tentatve allocaton, some of the mpressons could be overfull and some of them could have excess supply. The goal s to reallocate the contracts to mpressons n the least expensve way (accordng to the L penalty) such that none of the mpressons s overfull. We represent the excess of mpresson

9 by e = s con() δ. If e 0, then we consder the mpresson node as a supply node and set ts supply equal to e. If e 0, then we set the mpresson node as a demand node and set ts demand equal to e. Also we connect each supply mpresson node to each contract that s nterested n that mpresson, settng cap(, ) = δ and cost x W, where x s the flow on that edge. See Fgure 8. (nf, x w ) contracts ( δ, x w ) mpressons e (nf, r ) Fgure 8: The network constructon wth (capacty, cost) on the edges for L. From our constructon, t s easy to see the followng (proof smlar to that of Theorem ). Lemma 11. The soluton to network descrbed above s equvalent to the soluton to (5). 7.1 Effcent reconstructon for L In [14], Vee et al. showed that the soluton to the L can be reconstructed usng the Lagrange multplers of the quadratc program. As part of ther method, they show that, assumng that the Lagrange multpler for supply constrant s p and the Lagrange multpler for demand constrant s α, the best allocaton that mnmzes L norm and s a feasble soluton satsfes y = max(0, g (α p )), where g (x) = d s mp() s (1 x d ). (6) and store the values, we can reconstruct the allocaton. They also showed that the soluton to L s unque, whch means that the returned soluton by the convex cost functon also fts n ther argument. Earler n ths secton, we showed how to compute y values. Here, we gve a soluton sketch on how we can compute Lagrange multplers combnatorally as well, gven the prmal soluton. For smplcty, we denote φ = d s and σ =. Rewrtng (6), mp() s s mp() s we have y = φ σ (α p ). Thus f we compute y, we can compute α p as well. So, for all pars (, ) wth y > 0, we frst compute d = α p. We now construct a graph wth one node for each Lagrange multpler (n total I + J nodes) and place an edge between two of them f there s a correspondng y > 0 for them; let the length of the edge be d. Frst of all t s easy to see that n each connected component, f we know the Lagrange multpler for one of the nodes (for example p ), the Lagrange multpler for the rest of the nodes n the same component can be computed as well. So for each connected component, we assgn some varable p to one of the nodes and compute the dstance of all the nodes from that node. Ths way, the value of the Lagrange multplers n each component can be represented by p+d for whch d s already computed and the only varable s p. So the only remanng queston s how to choose p values n each component. Lookng back at the Lagrangan for the quadratc program, and replacng y s and α and p by ther computed values, the obectve can be wrtten as a lnear functon n terms of the selected representng values (p values) for each component. It s enough now to compute the best value for each of them assumng that we want to mnmze the Lagrange obectve functon. We omt the detals n ths verson. 8. CONCLUSIONS In ths paper we consdered the problem of far allocaton n a bpartte supply-demand settng. We worked wth a smple formulaton and crcumvented the need for mathematcal programmng solutons by takng a more drect and combnatoral look at the formulaton. We obtaned a flowbased algorthm for the L 1 penalty case, the nsghts from whch allowed us to precompute and store a lnear amount of nformaton such that the allocaton can be reconstructed n constant tme per edge. We extended the flow-based algorthm to the general convex functon case. Interestng drectons for future work nclude the followng. Is there a non-flow based soluton to the L 1 penalty case, assumng certan structure on the contracts and/or mpressons? Can samplng be used to approxmate the L 1 penalty case and obtan an even more effcent soluton? Is there a regular (.e., not convex) flow soluton to the L penalty case? Acknowledgments We thank Andre Broder, Flavo Cherchett, Kshore Papnen, Preston McAfee, Prabhakar Raghavan, Jayavel Shanmugasundaram, and Serge Vasslvtsk for many useful dscussons. 9. REFERENCES [1] A. Agrawal, P. N. Klen, and R. Rav. When trees collde: An approxmaton algorthm for the generalzed Stener problem on networks. In Proc. 3rd Annual ACM Symposum on Theory of Computng, pages , [] R. K. Ahua, T. L. Magnant, and J. B. Orln. Network Flows: Theory, Algorthms and Applcatons. Prentce Hall, [3] M. Babaoff, J. Hartlne, and R. Klenberg. Sellng banner ads: Onlne algorthms wth buyback. In Proc. 4th Workshop on Ad Auctons, 008. [4] R. Bar-Yehuda and S. Even. A lnear-tme approxmaton algorthm for the weghted vertex cover problem. J. Algorthms, ():198 03, [5] F. Constantn, J. Feldman, S. Muthukrshnan, and M. Pál. An onlne mechansm for ad slot reservatons wth cancellatons. In Proc. 19th Annual ACM-SIAM Symposum on Dscrete Algorthms, pages , 009.

10 [6] N. R. Devanur, C. H. Papadmtrou, A. Saber, and V. V. Vazran. Market equlbrum va a prmal-dual algorthm for a convex program. J. ACM, 55(5), 008. [7] B. Edelman, M. Ostrovsky, and M. Schwarz. Internet advertsng and the generalzed second prce aucton: Sellng bllons of dollars worth of keywords. Amercan Economc Revew, 97(1):4 59, 007. [8] U. Fege, N. Immorlca, V. Mrrokn, and H. Nazerzadeh. A combnatoral allocaton mechansm wth penaltes for banner advertsng. In Proc. 17th Internatonal Conference on World Wde Web, pages , 008. [9] A. Ghosh, P. McAfee, K. Papnen, and S. Vasslvtsk. Bddng for representatve allocatons for dsplay advertsng. In Proc. 5th Workshop on Internet and Network Economcs, 009. [10] A. V. Goldberg. An effcent mplementaton of a scalng mnmum-cost flow algorthm. J. Algorthms, :1 9, [11] A. V. Goldberg and R. E. Taran. A new approach to the maxmum-flow problem. J. ACM, 35(4):91 940, [1] K. Jan and V. V. Vazran. Esenberg-Gale markets: Algorthms and structural propertes. In Proc. 39th Annual ACM Symposum on Theory of Computng, pages , 007. [13] P. McAfee and K. Papnen. Maxmally representatve allocaton for guaranteed delvery advertsng campagns, 008. Manuscrpt. [14] E. Vee, S. Vasslvtsk, and J. Shanmugasundaram. Onlne allocaton wth a compact plan, 008. Manuscrpt. APPENDIX A. NON-CONVEX PENALTIES: HARDNESS In ths secton we show that f we allow the cost functons to be arbtrary (as opposed to beng convex), then the problem becomes NP-hard. To show that, we reduce from a varant of the subset sum problem. Suppose we are gven a multset S of n numbers: S = {a 1,..., a n}. The problem s to fnd a subset A S such that x A x = m, for some gven m. We reduce ths problem to an nstance of far allocaton problem wth non-convex costs. There s a sngle type of mpresson wth supply m. For each a S we create a contract wth demand a. All the contracts desre the sngle type of mpresson that was defned. We defne the cost functon c (x) for each edge from contract as follows: Now f we can fnd an allocaton of mpressons to contracts wth total cost of 0, then we have found a subset of S that adds up to m. Note that the cost functon as defned above s postve everywhere n [0, a ] except at 0 and a. Therefore for the total cost to be 0, every contract should be allocated ether 0 or a mpressons. Those contracts that are allocated a non-zero number of mpressons correspond to the subset of S that add up to m. Also notce that f there was a constant approxmaton for the far allocaton problem, t would stll have to return 0 f the optmal soluton was 0 and therefore we could use t to solve the subset sum problem. B. SUPPLY SCALING FOR L 1: TIGHTNESS We prove the tghtness by constructng an example shown n Fgure 9. Suppose we have n contracts c 1,..., c n and n + 1 mpresson types b 1,..., b n+1. Each c s nterested n mpressons b and b +1. Also suppose the orgnal demands of all c are and the supply of b 1 s 1 and supples of b,..., b n+1 are. Now n the optmal soluton each contract c wll receve 1 unt of supply from b and 1 unt from b +1. Now, suppose we scale up the demand of each contract by 1 + α. Notce that for any 1 t < n, the total demand of the frst t contracts mnus the total supply of the frst t mpressons s exactly equal to 1 whch s equal to the amount of supply that c k receves from b k+1. After scalng the demand, that total s ncreased by α. It s easy to see that the total change n the obectve functon value n ths case s roughly n n(n+1) =1 α = α whch completes the argument. demand supply Fgure 9: Tght example for Theorem 9. 1 { x c (x) = a x x < a a < x < a