Online Stochastic Matching: New Algorithms and Bounds

Transcription

1 Online Stochastic Matching: New Algorithms and Bonds Brian Brbach, Karthik A. Sankararaman, Araind Sriniasan, and Pan X Department of Compter Science, Uniersity of Maryland, College Park, MD 20742, USA Original Version: May 206 This Version: Noember 207 Abstract Online matching has receied significant attention oer the last 5 years de to its close connection to Internet adertising. As the seminal work of Karp, Vazirani, and Vazirani has an optimal ( /e) competitie ratio in the standard adersarial online model, mch effort has gone into deeloping sefl online models that incorporate some stochasticity in the arrial process. One sch poplar model is the known I.I.D. model where different cstomer-types arrie online from a known distribtion. We deelop algorithms with improed competitie ratios for some basic ariants of this model with integral arrial rates, inclding: (a) the case of general weighted edges, where we improe the best-known ratio of de to Haepler, Mirrokni and Zadimoghaddam [2] to 0.705; and (b) the ertex-weighted case, where we improe the ratio of Jaillet and L [3] to We also consider an extension of stochastic rewards, a ariant where each edge has an independent probability of being present. For the setting of stochastic rewards with non-integral arrial rates, we present a simple optimal non-adaptie algorithm with a ratio of /e. For the special case where each edge is nweighted and has a niform constant probability of being present, we improe pon /e by proposing a strengthened LP benchmark. One of the key ingredients of or improement is the following (offline) approach to bipartitematching polytopes with additional constraints. We first add seeral alid constraints in order to get a good fractional soltion f; howeer, these gie s less control oer the strctre of f. We next remoe all these additional constraints and randomly moe from f to a feasible point on the matching polytope with all coordinates being from the set {0, /k, 2/k,..., } for a chosen integer k. The strctre of this soltion is inspired by Jaillet and L (Mathematics of Operations Research, 203) and is a tractable strctre for algorithm design and analysis. The appropriate random moe preseres many of the remoed constraints (approximately with high probability and exactly in expectation). This nderlies some of or improements and cold be of independent interest. A preliminary ersion of this appeared in the Eropean Symposim on Algorithms (ESA), bbrbach@cs.md.ed Spported in part by NSF Awards CNS and CCF kabina@cs.md.ed Spported in part by NSF Awards CNS and CCF srin@cs.md.ed Spported in part by NSF Awards CNS and CCF , and by research awards from Adobe, Inc. panx@cs.md.ed Spported in part by NSF Awards CNS and CCF

2 Introdction. Applications to Internet adertising hae drien the stdy of online matching problems in recent years [20]. In these problems, we consider a bipartite graph G = (U, V, E) in which the set of ertices U is aailable offline while the set of ertices in V arrie online. Wheneer some ertex arries, it mst be matched immediately (and irreocably) to (at most) one ertex in U. Each offline ertex can be matched to at most one. In the context of Internet adertising, U is the set of adertisers and V is the set of impressions. The edges E define the impressions that interest a particlar adertiser. When an impression arries, we mst choose an aailable adertiser (if any) to match with it. WLOG we consider the case where V can be matched at most once. Since adertising forms the key sorce of reene for many large Internet companies, finding good matching algorithms and obtaining een small performance gains can hae high impact. In the stochastic known I.I.D. model of arrial, we are gien a bipartite graph G = (U, V, E) and a finite online time horizon T (here we assme T = n). In each rond, a ertex is sampled with replacement from a known distribtion oer V. The sampling distribtions are independent and identical oer all of the T online ronds. This captres the fact that we often hae historical data abot the impressions and can predict the freqency with which each type of impression will arrie. Edge-weighted matching [9] is a general model in the context of adertising: eery adertiser gains a gien reene for being matched to a particlar type of impression. Here, a type of impression refers to a class of sers (e.g., a demographic grop) who are interested in the same sbset of adertisements. A special case of this model is ertex-weighted matching [], where weights are associated only with the adertisers. In other words, a gien adertiser has the same reene generated for matching any of the ser types interested in it. In some modern bsiness models, reene is not generated pon matching adertisements, bt only when a ser clicks on the adertisement: this is the pay-per-click model. From historical data, one can assign the probability of a particlar adertisement being clicked by a type of ser. Works inclding [2, 22] captre this notion of stochastic rewards by assigning a probability to each edge. One nifying theme in most of or approaches is the se of an LP benchmark with additional alid constraints that hold for the respectie stochastic-arrial models. We se the optimal soltion of this LP to gide or online actions. To do so, we se arios modifications of dependent randomized ronding. 2 Preliminaries and technical challenges. In the Unweighted Online Known I.I.D. Stochastic Bipartite Matching problem, we are gien a bipartite graph G = (U, V, E). The set U is aailable offline while the ertices arrie online and are drawn with replacement from an I.I.D. distribtion on V. For each V, we are gien an arrial rate r, which is the expected nmber of times will arrie. With the exception of Sections 5, this paper will focs on the integral arrial rates setting where all r Z +. For reasons described in [2], we can frther assme WLOG that each has r = nder the assmption of integral arrial rates. In this case, we hae that V = n where n is the total nmber of online ronds. In the ertex-weighted ariant, eery ertex U has a weight w and we seek a maximm weight matching. In the edge-weighted ariant, eery edge e E has a weight w e and we again seek a maximm weight matching. In the stochastic rewards ariant, each edge has both a 2

3 weight w e and a probability p e of being present once we probe edge e and we seek to maximize the expected weight of the matching. Asymptotic assmption and notation. We will always assme n is large and analyze algorithms as n goes to infinity: e.g., if x ( 2/n) n, we will jst write this as x /e 2 instead of the more-accrate x /e 2 + o(). These sppressed o() terms will sbtract at most o() from or competitie ratios. Another fact to note is that the competitie ratio is defined slightly differently than sal for this set of problems (Similar to notation sed in [20]). In particlar, it is defined as E[ALG] E[OPT]. Algorithms can be adaptie or non-adaptie. When arries, an adaptie algorithm can modify its online actions based on the realization of the online ertices ths far, bt a non-adaptie algorithm has to specify all of its actions before the start of the online phase. Throghot, we se WS to refer to the worst case instance for arios algorithms. 2. LP benchmark for deterministic rewards. As in prior work (e.g, see [20]), we se the following LP to pper bond the optimal offline expected performance and also se it to gide or algorithm in the case where rewards are deterministic. For the case of stochastic rewards, we se slightly modified LPs, whose definitions we defer ntil Sections 5 and 6. We first show an LP for the nweighted ariant, then describe the changes for the ertex-weighted and edge-weighted settings. As sal, we hae a ariable f e for each edge. Let (w) be the set of edges adjacent to a ertex w U V and let f w = e (w) f e. maximize sbject to f e () e E e () e () f e U (2) f e V (3) 0 f e /e e E (4) f e + f e /e 2 e, e (), U (5) Variants: The objectie fnction is to maximize U e () f ew in the ertex-weighted ariant and maximize e E f ew e in the edge-weighted ariant (here w e refers to w (,) ). Constraint 2 is the matching constraint for ertices in U. Constraint 3 is alid becase each ertex in V has an arrial rate of. Constraint 4 is sed in [9] and [2]. It captres the fact that the expected nmber of matches for any edge is at most /e. This is alid for large n becase the probability that a gien ertex does not arrie dring n ronds is /e. Constraint 5 is similar to the preios one, bt for pairs of edges. For any two neighbors of a gien U, the probability that neither of them arrie is /e 2. Therefore, the sm of ariables for any two distinct edges in () cannot exceed /e 2. Notice that constraints 4 and 5 redces the gap between the optimal LP soltion and the performance of the optimal online algorithm. In fact, withot constraint 4, we cannot in general achiee a competitie ratio better than /e. The edge realization process is independent for different edges. At each step, the algorithm probes an edge. With probability p e the edge e exists and with the remaining probability it does not. Once realization of an edge is determined, it does not affect the random realizations for the rest of the edges. 3

4 Note that the work of [9] does not se an LP to pper-bond the optimal ale of the offline instance. Instead they se Monte-Carlo simlations wherein they simlate the arrial seqence and compte the ector f by approximating (ia Monte-Carlo simlation) the probability of matching an edge e in the offline optimal soltion. We do not se a similar approach for or problems for a few reasons. () For the weighted ariants, namely the edge and ertex-weighted ersions, the nmber of samples depends on the maximm ale of the weight, making it expensie. (2) In the nweighted ersion, the rnning time of the sampling based algorithm is O( E 2 n 4 ); on the other hand, we show in Section 2.6 that the LP based algorithm can be soled mch faster, Õ( E 2 ) time in the worst case and een faster than that in practice. (3) For the stochastic rewards setting, the offline problem is not known to be polynomial-time solable. The paper [4] shows nder the assmption of constant p and OPT = ω(/p), they can obtain a ( ɛ) approximation to the optimal soltion. Howeer, these assmptions are too strong to be sed in or setting. Finally, for the stochastic rewards setting, one might be tempted to se an LP to achiee the same property obtained from Monte-Carlo simlation ia adding extra constraints. In the context of niform stochastic rewards where each edge e is associated with a niform constant probability p, what we really need is: S (), f e exp( S p) (6) p e S To garantee this ia the LP, a straightforward approach is to add this family of constraints into the LP. Howeer, the nmber of sch constraints is exponential and there seems to be no obios separation oracle. We oercome this challenge by showing it sffices to ensre that Ineqality (6) aboe holds for all S with S 2/p, which is a constant and ths the resltant LP is polynomial solable. 2.2 Oeriew of ertex-weighted algorithm and contribtions. A key challenge encontered by [3] was that their special LP cold lead to length for cycles of type C shown in Figre. In fact, they sed this cycle to show that no algorithm cold perform better than 2/e sing their LP. They mentioned that tighter LP constraints sch as 4 and 5 in the LP from Section 2 cold aoid this bottleneck, bt they did not propose a techniqe to se them. Note that the {0, /3, } soltion prodced by their LP was an essential component of their Random List algorithm. We show a randomized ronding algorithm to constrct a similar, simplified {0, /3, } ector from the soltion of a stricter benchmark LP. This allows for the inclsion of additional constraints, most importantly constraint 5. Using this ronding algorithm combined with tighter constraints, we will pper bond the probability of a ertex appearing in the cycle C from Figre at 2 3/e (See Lemma 4) Additionally, we show how to deterministically break all other length for cycles which are not of type C withot creating any new cycles of type C. Finally, we describe an algorithm which tilizes these techniqes to improe preios reslts in both the ertex-weighted and nweighted settings. For this algorithm, we first sole the LP in Section 2 on the inpt graph. In Section 4, we show how to se the techniqe in sb-section 2.7 to obtain a sparse fractional ector. We then present a randomized online algorithm (similar to the one in [3]) which ses the sparse fractional ector as a gide to achiee a competitie ratio of Preiosly, there was gap between the best nweighted algorithm with a ratio of 2e 2 de to [3] and the negatie reslt of e 2 4

5 Figre : This cycle is the sorce of the negatie reslt described by Jaillet and L [3]. Thick edges hae f e = while thin edges hae f e = /3. (C ) 2 2 de to [9]. We take a step towards closing that gap by showing that an algorithm can achiee > 2e 2 for both the nweighted and ertex-weighted ariants with integral arrial rates. In doing so, we make progess on Open Qestions 3 and 4 from the book [20] Oeriew of edge-weighted algorithm and contribtions. A challenge that arises in applying the power of two choices to this setting is when the same edge (, ) is inclded in both matchings M and M 2. In this case, the copy of (, ) in M 2 can offer no benefit and a second arrial of is wasted. To se an example from related work, Haepler et al. [2] choose two matchings in the following way. M is attained by soling an LP with constraints 2, 3 and 4 and ronding to an integral soltion. M 2 is constrcted by finding a maximm weight matching and remoing any edges which hae already been inclded in M. A key element of their proof is showing that the probability of an edge being remoed from M 2 is at most /e The approach in this paper is to constrct two or three matchings together in a correlated manner to redce the probability that some edge is inclded in all matchings. We show a general techniqe to constrct an ordered set of k matchings where k is an easily adjstable parameter. For k = 2, we show that the probability of an edge appearing in both M and M 2 is at most 2/e For the algorithms presented, we first sole an LP on the inpt graph. We then rond the LP soltion ector to a sparse integral ector and se this ector to constrct a randomly ordered set of matchings which will gide or algorithm dring the online phase. We begin Section 3 with a simple warm-p algorithm which ses a set of two matchings as a gide to achiee a competitie ratio, improing the best known reslt for this problem. We follow it p with a slight ariation that improes the ratio to 0.7 and a more complex competitie algorithm which relies on a conex combination of a 3-matching algorithm and a separate psedo-matching algorithm. 2.4 Oeriew of stochastic rewards and contribtions. This algorithm (Algorithm 9) is presented in Section 5 and 6. We beliee the known I.I.D. model with stochastic rewards is an interesting new direction motiated by the work of [2] and [22] in the adersarial model. We introdce a new, more general LP (see LP (9)) specifically for this setting and show that a simple algorithm sing the LP soltion directly can achiee a competitie ratio of /e, which is proed to be optimal among all non-adaptie algorithms. In [22], it is shown that no randomized algorithm can achiee a ratio better than 0.62 < /e in the adersarial 2 Open Qestions 3 and 4 state the following: In general, close the gap between the pper and lower bonds. In some sense, the ratio of 2e 2 achieed in [3] for the integral case, is a nice rond nmber, and one may sspect that it is the correct answer. 5

6 Table : Smmary of Contribtions Problem Preios Work This Paper Edge-Weighted (Section 3) [2] Vertex-Weighted (Section 4) [3] Unweighted 2/e 2 [3] (> 2/e 2 ) Stochastic Rewards (Section 5 and 6) N/A e for general ersion for the restricted ersion model. Hence, achieing a /e for the i.i.d. model shows that this lower bond does not extend to this model. Frther, the paper [5] shows that sing LP (9) one cannot achiee a ratio better than /e. We discss some challenges relating to why the techniqes sed in prior work do not directly extend to this model. Finally, we consider a restricted ersion of the problem where each edge is nweighted and has a niform constant probability p (0, ] nder integral arrial rates. By proposing a family of alid constraints, we are able to show that in this restricted setting, one can indeed beat /e. 2.5 Smmary of or contribtions Theorem. For ertex-weighted online stochastic matching with integral arrial rates, online algorithm VW achiees a competitie ratio of at least Theorem 2. For edge-weighted online stochastic matching with integral arrial rates, there exists an algorithm which achiees a competitie ratio of at least 0.7 and algorithm EW[q] with q = achiees a competitie ratio of at least Theorem 3. For edge-weighted online stochastic matching with arbitrary arrial rates and stochastic rewards, online algorithm SM (9) achiees a competitie ratio of /e, which is optimal all among all non-adaptie algorithms. Theorem 4. For nweighted online stochastic matching with integral arrial rates and niform stochastic rewards, there exists an adaptie algorithm which achiees a competitie ratio of at least Rntime of algorithm. In this section, we discss the implementation details of or algorithm. All of or algorithms sole an LP in the pre-processing step. The dimension of the LP is determined by the constraint matrix which consists of O( E + U + V ) rows and O( E ) colmns. Howeer, note that the nmber of non-zero entries in this matrix is of the order O(( U + V ) E ). Some recent work (e.g., [7]) shows that sch sparse programs can be soled in time Õ( E 2 ) sing interior point methods (which are 6

7 known to perform ery well in practice). This sparsity in the LP is the reason we can sole ery large instances of the problem. The second critical step in pre-processing is to perform randomized ronding. Note that we hae O( E ) ariables and in each step of the randomized ronding de to [], they incr a rnning time of O( E ). Hence the total rnning time to obtain a ronded soltion is of the order O( E 2 ). Finally, recall that both these operations are part of the pre-processing step. Hence in the online phase the algorithm incrs a per-time-step rnning time of at most O( U ) for the stochastic rewards case (in fact, a smarter implementation sing binary search rns as fast as O(log U )) and O() for the edge-weighted and the ertex-weighted algorithms in Section 3 and LP ronding techniqe DR[f, k]. For the algorithms presented, we first sole the benchmark LP in sb-section 2. for the inpt instance to get a fractional soltion ector f. We then rond f to an integral soltion F sing a two step process we call DR[f, k]. The first step is to mltiply f by k. The second step is to apply the dependent ronding techniqes of Gandhi, Khller, Parthasarathy, and Sriniasan [] to this new ector. In this paper, we will always choose k to be 2 or 3. This is becase a ertex in V may appear more than once, bt probably not more than two or three times. While dependent ronding is typically applied to ales between 0 and, the sefl properties extend natrally to or case in which kf e may be greater than for some edge e. To nderstand this process, it is easiest to imagine splitting each kf e into two edges with the integer ale f e = kf e and fractional ale f e = kf e kf e. The former will remain nchanged by the dependent ronding since it is already an integer while the latter will be ronded to with probability f e and 0 otherwise. Or final ale F e wold be the sm of those two ronded ales. The two properties of dependent ronding we se are:. Marginal distribtion: For eery edge e, let p e = kf e kf e. Then, Pr[F e = kf e ] = p e and Pr[F e = kf e ] = p e. 2. Degree-preseration: For any ertex w U V, let its fractional degree kf w be e (w) kf e and integral degree be the random ariable F w = e (w) F e. Then F w { kf w, kf w }. 2.8 Related work. The stdy of online matching began with the seminal work of Karp, Vazirani, Vazirani [4], where they gae an optimal online algorithm for a ersion of the nweighted bipartite matching problem in which ertices arrie in adersarial order. Following that, a series of works hae stdied arios related models. The book by Mehta [20] gies a detailed oeriew. The ertex-weighted ersion of this problem was introdced by Aggarwal, Goel and Karande [], where they gie an optimal ( e ) ratio for the adersarial arrial model. The edge-weighted setting has been stdied in the adersarial model by Feldman, Korla, Mirrokni and Mthkrishnan [9], where they consider an additional relaxation of free-disposal. In addition to the adersarial and known I.I.D. models, online matching is also stdied nder seeral other ariants sch as random arrial order, nknown distribtions, and known adersarial distribtions. In the setting of random arrial order, the arrial seqence is assmed to be a random permtation oer all online ertices, see e.g., [6, 5, 6, 8]. In the case of nknown distribtions, in each rond an item is sampled from a fixed bt nknown distribtion. If the sampling distribtions are reqired to be the same dring each rond, it is called nknown I.I.D. ([7, 8]); otherwise, it is 7

8 called adersarial stochastic inpt ([7]). As for known adersarial distribtions, in each rond an item is sampled from a known distribtion, which is allowed to change oer time ([2, 3]). Another ariant of this problem is when the edges hae stochastic rewards. Models with stochastic rewards hae been preiosly stdied by [2, 22] among others, bt not in the known I.I.D. model. Related Work in the Vertex-Weighted and Unweighted Settings. The ertex-weighted and nweighted settings hae many reslts starting with Feldman, Mehta, Mirrokni and Mthkrishnan [0] who were the first to beat /e with a competitie ratio of 0.67 for the nweighted problem. This was improed by Manshadi, Gharan, and Saberi [9] to with an adaptie algorithm. In addition, they showed that een in the nweighted ariant with integral arrial rates, no algorithm can achiee a ratio better than e Finally, Jaillet and L [3] presented an adaptie algorithm which sed a cleer LP to achiee and 2e for the ertex-weighted and nweighted problems, respectiely. Related Work in the Edge-Weighted Setting. For this model, Haepler, Mirrokni, Zadimoghaddam [2] were the first to beat /e by achieing a competitie ratio of They se a disconted LP with tighter constraints than the basic matching LP (a similar LP can be seen in 2.) and they employ the power of two choices by constrcting two matchings offline to gide their online algorithm. Other Related Work. Deanr et al [8] gae an algorithm which achiees a ratio of / 2πk for the Adwords problem in the Unknown I.I.D. arrial model with knowledge of the optimal bdget tilization and when the bid-to-bdget ratios are at most /k. Alaei et al. [2] considered the Prophet-Ineqality Matching problem, in which arries from a distinct (known) distribtion D t, in each rond t. They gae a / k + 3 competitie algorithm, where k is the minimm capacity of. 3 Edge-weighted matching with integral arrial rates 3. A simple competitie algorithm. As a warm-p, we describe a simple algorithm which achiees a competitie ratio of and introdces key ideas in or approach. We begin by soling the LP in sb-section 2. to get a fractional soltion ector f and applying DR[f, 2] as described in Sbsection 2.7 to get an integral ector F. We constrct a bipartite graph G F with F e copies of each edge e. Note that G F will hae max degree 2 since for all w U V, F w 2f w 2 and ths we can decompose it into two matchings sing Hall s Theorem. Finally, we randomly permte the two matchings into an ordered pair of matchings, [M, M 2 ]. These matchings sere as a gide for the online phase of the algorithm, similar to [2]. The entire warm-p algorithm for the edge-weighted model, denoted by EW 0, is smmarized in Algorithm. 3.. Analysis of algorithm EW 0. We will show that EW 0 (Algorithm ) achiees a competitie ratio of Let [M, M 2 ] be or randomly ordered pair of matchings. Note that there might exist some edge e which appears in both matchings if and only if f e > /2. Therefore, we consider three types of edges. We say an 8

9 Algorithm : [EW 0 ] Constrct and sole the benchmark LP in sb-section 2. for the inpt instance. 2 Let f be an optimal fractional soltion ector. Call DR[f, 2] to get an integral ector F. 3 Create the graph G F with F e copies of each edge e E and decompose it into two matchings. 4 Randomly permte the matchings to get a random ordered pair of matchings, say [M, M 2 ]. 5 When a ertex arries for the first time, try to assign to some if (, ) M ; when arries for the second time, try to assign to some 2 if ( 2, ) M 2. 6 When a ertex arries for the third time or more, do nothing in that step. edge e is of type ψ, denoted by e ψ, if and only if e appears only in M. Similarly e ψ 2, if and only if e appears only in M 2. Finally, e ψ b, if and only if e appears in both M and M 2. Let P, P 2, and P b be the probabilities of getting matched for e ψ, e ψ 2, and e ψ b, respectiely. According to the reslt in Haepler et al. [2], Lemma bonds these probabilities. Lemma. (Proof details in Section 3 of [2]) Gien M and M 2, in the worst case () P = ; (2) P 2 = and (3) P b = We can se Lemma to proe that the warm-p algorithm EW 0 achiees a ratio of by examining the probability that a gien edge becomes type ψ, ψ 2, or ψ b. Proof. (Analysis for EW 0 ) Consider the following two cases. Case : 0 f e /2: By the marginal distribtion property of dependent ronding, there can be at most one copy of e in G F and the probability of inclding e in G F is 2f e. Since an edge in G F can appear in either M or M 2 with eqal probability /2, we hae Pr[e ψ ] = Pr[e ψ 2 ] = f e. Ths, the ratio is (f e P + f e P 2 )/f e = P + P 2 = Case 2: /2 f e /e: Similarly, by marginal distribtion, Pr[e ψ b ] = Pr[F e = 2f e ] = 2f e 2f e = 2f e. It follows that Pr[e ψ ] = Pr[e ψ 2 ] = (/2)( (2f e )) = f e. Ths, the ratio is (noting that the first term is from case while the second term is from case 2) (( f e )(P + P 2 ) + (2f e )P b )/f e 0.688, where the WS is for an edge e with f e = /e. 3.2 A 0.7-competitie algorithm. In this section, we describe an improement pon the preios warm-p algorithm to get a competitie ratio of 0.7. We start by making an obseration abot the performance of the warm-p algorithm. After soling the LP, let edges with f e > /2 be called large and edges with f e /2 be called small. Let L and S, be the sets of large and small edges, respectiely. Notice that in the preios analysis, small edges achieed a mch higher competitie ratio of erss for large edges. This is primarily de to the fact that we may get two copies of a large edge in G F. In 9

10 this case, the copy in M has a better chance of being matched, since there is no edge which can block it (i.e. an edge with the same offline neighbor that gets matched first), bt the copy that is in M 2 has no chance of being matched. To correct this imbalance, we make an additional modification to the f e ales before applying DR[f, k]. The rest of the algorithm is exactly the same. Let η be a parameter to be optimized later. For all large edges l L sch that f l > /2, we set f l = f l + η. For all small edges s S which are adjacent to some large edge, let l L be the largest edge adjacent to s sch that f l > /2. Note that it is ( possible for s to hae two large neighbors, bt we only care abot the largest one. We set f s = f (fl +η) s f l ). In other words, we increase the ales of large edges while ensring that for all w U V, f w by redcing the ales of neighboring small edges proportional to their original ales. Note that it is not possible for two large edges to be adjacent since they mst both hae f e > /2. For all other small edges which are not adjacent to any large edges, we leae their ales nchanged. We then apply DR[f, 2] to this new ector, mltiplying by 2 and applying dependent ronding as before Analysis. We can now proe Theorem 2. Proof. As in the warm-p analysis, we ll consider large and small edges separately 0 f s 2 : Here we hae two cases Case : s is not adjacent to any large edges. In this case, the analysis is the same as the warm-p algorithm and we still get a competitie ratio for these edges. Case 2: s is adjacent to some large edge l. For this case, let f l be the ale of the largest neighboring edge in the original LP soltion. Then s achiees a ratio of f s ( (fl + η) f l ) ( )/f s = ( (fl + η) f l ) ( ) This follows from Lemma ; in particlar, the first two terms are the reslt of how we set f s in the algorithm, while the two nmbers, and , are the probabilities that s is matched when it is in M 2 and M, respectiely. Note that for f l [0, ) this is a decreasing fnction with respect to f l. So the worst case is f l = /e (de to constraint 4 in LP 2.) and we hae a ratio of 2 < f l e : ( ) ( /e + η) ( ) = ( /e) ( /e η /e ) ( ) Here, the ratio is (( (f l + η))(p + P 2 ) + (2(f l + η) )P b )/f l, where the WS is for an edge e with f l = /e. This follows becase of the fact that it is a decreasing fnction with respect to f l. To see that it is a decreasing fnction, note that it can be rearranged as (P + P 2 P b + η(2p b P P 2 ) + f l (2P b P P 2 ))/f l. Sbstitting the appropriate ales, 0

11 we hae that the ale of (2P b P P 2 ) = Hence, the expression can be written as (c f l )/f l. If we show that c 0, we are done. The optimal ale of η we choose trns ot to be Hence we hae c = P + P 2 P b = > 0. Choosing the optimal ale of η = 0.042, yields an oerall competitie ratio of 0.7 for this new algorithm. We now need to show that this ale of η ensres that both f l and f s are less than after modification. Since f l /e we hae that f l + η /e Note that ( f l /2. ) Hence, the modified ale of f s is always less than or eqal to the original ale, since (fl +η) f l is decreasing in the range f l [/2, /e] and has a ale less than 0.98 at f l = / A competitie algorithm. In the next few sbsections, we describe or final edge-weighted algorithm with all of the attenation factors. To keep it modlar, we gie the following gide to the reader. We note that the definition of large and small edges gien below in Sbsection 3.3. is different from the definition in the preios sbsection. Section 3.3. describes the main algorithm which internally inokes two algorithms, EW and EW 2, which are described in sections and 3.3.3, respectiely. Theorem 2 proes the final competitie ratio. This proof depends on the performance garantees of EW and EW 2, which are gien by Lemmas 2 and 3, respectiely. The proof of Lemma 2 depends on claims 4, 5, and 6 (Fond in the Appendix). Each of those claims is a carefl case-by-case analysis. Intitiely, 4 refers to the case where the offline ertex is incident to one large edge and one small edge (here the analysis is for the large edge), 5 refers to the case where is incident to three small edges and 6 refers to the case where is incident to a small edge and large edge (here the analysis is for the small edge). The proof of Lemma 3 depends on claims 7 and 8 (Fond in the Appendix). Again, both of those claims are proen by a carefl case-by-case analysis. Since there are many cases, we hae gien a diagram of the cases when we proe them A competitie algorithm. In this section, we describe an algorithm EW (Algorithm 2), that achiees a competitie ratio of The algorithm first soles the benchmark LP in sbsection 2. and obtains a fractional optimal soltion f. By inoking DR[f, 3], it obtains a random integral soltion F. Notice that from LP constraint 4 we see f e /e. Therefore after DR[f, 3], each F e {0,, 2}. Consider the graph G F where each edge e is associated with the ale of F e. We say an edge e is large if F e = 2 and small if F e = (note that this differs from the definition of large and small in Sbsection 3.2). We design two non-adaptie algorithms, denoted by EW and EW 2, which take the sparse graph G F as inpt. The difference between the two algorithms EW and EW 2 is that EW faors the small edges while EW 2 faors the large edges. The final algorithm is to take a conex combination of EW and EW 2 i.e. rn EW with probability q and EW 2 with probability q. The details of algorithm EW and EW 2 and the proof of Theorem 2 are presented in the following sections.

12 Algorithm 2: EW[q] Sole the benchmark LP in sb-section 2. for the inpt. Let f be the optimal soltion ector. 2 Inoke DR[f, 3] to obtain the ector F. 3 Independently rn EW and EW 2 with probabilities q and q respectiely on G F Algorithm EW. In this section, we describe the randomized algorithm EW (Algorithm 3). Sppose we iew the graph of G F in another way where each edge has F e copies. Let PM[F, 3] refer to the process of constrcting the graph G F with F e copies of each edge, decomposing it into three matchings, and randomly permting the matchings. EW first inokes PM[F, 3] to obtain a random ordered triple of matchings, say [M, M 2, M 3 ]. Notice that from the LP constraint 4 and the properties of DR[f, 3] and PM[F, 3], an edge will appear in at most two of the three matchings. For a small edge e = (, ) in G F, we say e is of type Γ if has two other neighbors and 2 in G F with F (, ) = F (,2 )=. We say e is of type Γ 2 if has exactly one other neighbor with F (, ) = 2. WLOG we can assme that for eery, F = e () F e = 3; otherwise, we can add a dmmy node to the neighborhood of. Note, we se the terminology, assign to to denote that edge (, ) is matched by the algorithm if is not matched ntil that step. Algorithm 3: EW [h] Inoke PM[F, 3] to obtain a random ordered triple matchings, say [M, M 2, M 3 ]. 2 When a ertex comes for the first time, assign to some with (, ) M. 3 When comes for the second time, assign to some 2 with ( 2, ) M 2. 4 When comes for the third time, if e is either a large edge or a small edge of type Γ then assign to some 3 with e = ( 3, ) M 3. Howeer, if e is a small edge of type Γ 2 then with probability h, assign to some 3 with e = ( 3, ) M 3 ; otherwise, do nothing. 5 When comes for the forth or more time, do nothing in that step. Here, h is a parameter we will fix at the end of analysis. Let R[EW, /3] and R[EW, ] be the competitie ratio for a small edge and large edge respectiely. Lemma 2. For h = , EW achiees a competitie ratio R[EW, ] = , R[EW, ] = for a large and small edge respectiely. Proof. In case of the large edge e, we diide the analysis into three cases where each case corresponds to e being in one of the three matchings. And we combine these conditional probabilities sing Bayes theorem to get the final competitie ratio for e. For each of the two types of small edges, we similarly condition them based on the matching they can appear in, and combine them sing Bayes theorem. A complete ersion of proof can be fond in Section A.. of Appendix Algorithm EW 2. Algorithm EW 2 (Algorithm 5) is a non-adaptie algorithm which takes G F as inpt and performs well on the large edges. Recall that the preios algorithm, EW, first inokes PM[F, 3] to obtain a 2

13 random ordered triple of matchings. In contrast, EW 2 will inoke a rotine, denoted by PM [F, 2] (Algorithm 4), to generate a (random ordered) pair of psedo-matchings from F. Recall that F is an integral soltion ector where e we hae F e {0,, 2}. WLOG, we can assme that F = for eery in G F ; otherwise we can dmmy ertex to ensre this is the case. Algorithm 4: PM [F, 2][y, y 2 ] Sppose has two neighbors in G F, say, 2, with e = (, ) being a large edge while e 2 = ( 2, ) being a small edge. Add e to the primary matching M and e 2 to the secondary matching M 2. 2 Sppose has three neighbors in G F and the incident edges are () = (e, e 2, e 3 ). Take a random permtation of (), say (π, π 2, π 3 ) Π( ()). Add π to M with probability y and π 2 to M 2 with probability y 2. Here 0 y, y 2 are parameters which will be fixed after the analysis. Algorithm 5 describes EW 2. Algorithm 5: [EW 2 ][y, y 2 ] Inoke PM [F, 2][y, y 2 ] to generate a random ordered pair of psedo-matchings, say [M, M 2 ]. 2 When a ertex comes for the first time, assign to some if (, ) M ; When comes for the second time, try to assign to some 2 if ( 2, ) M 2. 3 When a ertex comes for the third or more time, do nothing in that step. Let R[EW 2, /3] and R[EW 2, ] be the competitie ratios for small edges and large edges, respectiely. Lemma 3. For y = and y 2 =, EW 2 [y, y 2 ] achiees a competitie ratio of R[EW 2, ] = and R[EW 2, /3] = for a large and small edge respectiely. Proof. We analyze this on a case-by-case basis by considering the local neighborhood of the edge. A large edge can hae two possible cases in its neighborhood, while a small edge can hae eight possible cases. This is becase of the fact that a large edge can hae only small edges in its neighborhood while a small edge can hae both large and small edges in its neighborhood. Choosing the worst case among the two for large edge and the worst case among the eight for the small edge, we proe the claim. Complete details of the proof can be fond in section A..2 of Appendix Conex combination of EW and EW 2. In this section, we proe theorem 2. Proof. Let (a, b ) be the competitie ratios achieed by EW for large and small edges, respectiely. Similarly, let (a 2, b 2 ) denote the same for EW 2. We hae the following two cases. 0 f e 3 : By marginal distribtion property of DR[f, 3], we know that Pr[F e = ] = 3f e. Ths, the final ratio is 3f e (qb /3 + ( q)b 2 /3)/f e = qb + ( q)b 2 3

14 /3 f e /e: By the same properties of DR[f, 3], we know that Pr[F e = 2] = 3f e and Pr[F e = ] = 2 3f e. Ths, the final ratio is ( ) (3f e )(2qa /3 + 2( q)a 2 /3) + (2 3f e )(qb /3 + ( q)b 2 /3) /f e The competitie ratio of the conex combination is maximized at q = with a ale of Vertex-weighted stochastic I.I.D. matching with integral arrial rates. In this section, we consider ertex-weighted online stochastic matching on a bipartite graph G nder the known I.I.D. model with integral arrial rates. We present an algorithm in which each offline ertex has a competitie ratio of at least > 2e 2. Recall that after inoking DR[f, 3], we can obtain a (random) integral ector F with F e {0,, 2}. Define H = F/3 and let G H be the graph indced by H. Each edge e in G H ths takes a ale H e {0, /3, }. Notice that for each, H. = e () H e, which implies that has at most 3 neighbors in G H (we ignore all edges e with H e = 0). In this section, we focs on the sparse graph G H. The main idea of or algorithm is as follows.. Sole the ertex-weighted benchmark LP in Section 2.. Let f be an optimal soltion ector. 2. Inoke DR[f, 3] to obtain an integral ector F and a fractional ector H with H = F/3. 3. Apply a series of modifications to H and transform it to another soltion H (See Section 4.2). 4. Rn the Randomized List Algorithm [3] based on H, denoted by RLA[H ], on the graph G H. We first briefly describe how we oercome the bottleneck case for the algorithm in [3] and then explain the algorithm in fll detail. The WS for the ertex-weighted case in [3] is shown in Figre 2, which happens at a node with a competitie ratio of (recall that [3] analyze their algorithm by considering cases for arios neighborhood strctre at a gien offline ertex). From the analysis of [3], we hae that the node (in Figre 2) has a competitie ratio of at least Hence, we can boost the performance of at the cost of. Specifically, we increase the ale of H (,) and decrease the ale H (, ). Cases (0) and () in Figre 4 illstrate this. After this modification, the new WS for ertex-weighted is now the C cycle shown in both Figres and 2. In fact, this is the WS for the nweighted case in [3] as well. Howeer, Lemma 4 implies that C cycles can be aoided with probability at least 3/e. This helps s improe the ratio een for the nweighted case in [3]. Lemma 4 describes this formally. Lemma 4. For any gien U, appears in a C cycle after DR[f, 3] with probability at most 2 3/e. 4

15 Proof. Consider the graph G H obtained after DR[f, 3]. Notice that for some ertex to appear in a C cycle, it mst hae a neighboring edge with H e =. Now we try to bond the probability of this eent. It is easy to see that for some e () with f e /3, F e after DR[f, 3], and hence H e = F e /3 /3. Ths only those edges e () with f e > /3 will possibly be ronded to H e =. Note that, there can be at most two sch edges in (), since e () f e. Hence, we hae the following two cases. Case : () contains only one edge e with f e > /3. Let q = Pr[H e = /3] and q 2 = Pr[H e = ] after DR[f, 3]. By DR[f, 3], we know that E[H e ] = E[F e ]/3 = q 2 () + q (/3) = f e. Notice that q + q 2 = and hence q 2 = 3f e. Since this is an increasing fnction of f e and f e /e from LP constraint 4, we hae q 2 3( /e) = 2 3/e. Case 2: () contains two edges e and e 2 with f e > /3 and f e2 > /3. Let q 2 be the probability that after DR[f, 3], either H e = or H e2 =. Note that, these two eents are mtally exclsie since H. Using the analysis from case, it follows that q 2 = (3f e ) + (3f e2 ) = 3(f e + f e2 ) 2. From LP constraint 5, we know that f e +f e2 /e 2, and hence q 2 3( /e 2 ) 2 < 2 3/e. Now we present the details of RLA based on a gien H in Section 4. and then discss the two modifications transforming H to H in Section 4.2. We gie a formal statement of or algorithm in Section 4.3 and the related analysis. 4. RLA algorithm. Now we discss how to apply RLA based on H to the sparse graph G H. Let δ H () be the set of neighboring nodes of in G H. Here we assme WLOG that H. = e () H e = and ths each has at least 2 neighbors in G H since each non-zero H e satisfies H e {/3, } (we are in a better sitation when H < ). Additionally, we will see in Section 4.2 that dring the two modifications, we hae the sm of all edge ales incident to (i.e., H ) nchanged and hence we hae H = H = for each. Each time when a ertex comes, RLA first generates a random list R, which is a permtation oer δ H (), based on H as follows. If δ H () = 2, say δ H () = (, 2 ), then sample a random list R sch that Pr[R = (, 2 )] = H (,), Pr[R = ( 2, )] = H ( 2,) (7) If δ H () = 3, say δ H () = (, 2, 3 ). Then we sample a permtation of (i, j, k) oer {, 2, 3} sch that H ( j,) Pr[R = ( i, j, k )] = H ( i,) H ( j,) + H ( k,) We can erify that the sampling distribtions described in Eqations (7) and (8) are alid since H = e () H e =. The fll details of the Random List Algorithm, RLA[H ], are shown in Algorithm 6. (8) 5

16 Algorithm 6: RLA[H ] (Random List Algorithm based on H ) When a ertex comes, generate a random list R satisfying Eqation (7) or (8) 2 If all in the list are matched, then drop the ertex ; otherwise, assign to the first nmatched in the list. Algorithm 7: [Cycle breaking algorithm] Offline Phase While there is some cycle of type C 2 or C 3, Do: 2 Break all cycles of type C 2. 3 Break one cycle of type C 3 and retrn to the first step. 4.2 Two kinds of modifications to H. As stated earlier, we first modify H before rnning the RLA algorithm. In this section, we describe the modifications The first modification to H (WS) (C ) (C 2 ) (C 3 ) Figre 2: Left: The WS for Jaillet and L [3] for their ertex-weighted case. Right: The three possible types of cycles of length 4 after applying DR[f, 3]. Thin edges hae H e = /3 and thick edges hae H e =. The first modification is to break the cycles deterministically. There are three possible cycles of length 4 in the graph G H, denoted C, C 2, and C 3. In [3], they gie an efficient way to break C 2 and C 3, as shown in Figre 2. Cycle C cannot be modified frther and hence, is the bottleneck for their nweighted case. Notice that, while breaking the cycles of C 2 and C 3, new cycles of C can be created in the graph. Since or randomized constrction of soltion H gies s control on the probability of cycles C occrring, we wold like to break C 2 and C 3 in a controlled way, so as not to create any new C cycles. This procedre is smmarized in Algorithm 7 and its correctness is proed in Lemma 6. Lemma 5. After applying Algorithm 7 to G H, we hae () the ale H w is presered for each w U V ; (2) no cycle of type C 2 or C 3 exists; (3) no new cycle of type C is added. 6

17 Proof of Lemma 5. Lemma 5 follows from the following three Claims: Claim. Breaking cycles will not change the ale H w for any w U V. Claim 2. After breaking a cycle of type C 2, the ertices, 2,, and 2 can neer be part of any length for cycle. Claim 3. When all length for cycles are of type C or C 3, breaking exactly one cycle of type C 3 cannot create a new cycle of type C. Proof of Claim. As shown in Figre 2, we increase and decrease edge ales f e in sch a way that their sms H w at any ertex w will be presered. Notice that C 2 cycles can be freely broken withot creating new C cycles. After remoing all cycles of type C 2, remoing a single cycle of type C 3 cannot create any cycles of type C. Hence, Algorithm 7 remoes all C 2 and C 3 cycles withot creating any new C cycles. Proof of Claim 2. Consider the strctre after breaking a cycle of type C 2. Note that the edge ( 2, 2 ) has been permanently remoed and hence, these for ertices together can neer be part of a cycle of length for. The ertices and hae H = and H = respectiely. So they cannot hae any other edges and therefore cannot appear in any length for cycle. The ertices 2 and 2 can each hae one additional edge, bt since the edge ( 2, 2 ) has been remoed, they can neer be part of any cycle with length less than six. Proof of Claim 3. First, we note that since no edges will be added dring this process, we cannot create a new cycle of length for or join with a cycle of type C. Therefore, the only cycles which cold be affected are of type C 3. Howeer, eery cycle c of type C 3 falls into one of two cases: Case : c is the cycle we are breaking. In this case, c cannot become a cycle of type C since we remoe two of its edges and break the cycle. Case 2: c is not the cycle we are breaking. In this case, c can hae at most one of its edges conerted to a edge. Let c be the length for cycle we are breaking. Note that c and c will differ by at least one ertex. When we break c, the two edges which are conerted to will coer all for ertices of c. Therefore, at most one of these edges can be in c. Note that breaking one cycle of type C 3 cold create cycles of type C 2, bt these cycles are always broken in the next iteration, before breaking another cycle of type C The Second modification to H. Informally, this second modification decreases the rates of lists associated with those nodes with H = /3 or H = and increases the rates of lists associated with nodes with H =. We will illstrate or intition on the following example. Consider the graph G in Figre 3. Let thin and thick edges represent H e = /3 and H e = respectiely. We will now calclate the competitie ratio after applying RLA on G. Let P denote the probability that gets matched after the algorithm. Let B denote the eent that among the n random lists, there exists a list starting with and G denote the eent that among the n lists, there exists sccessie lists sch that () Each of those lists starts with a and δ() and (2) The lists arrie in an order which ensres will be matched by the algorithm. From lemma 4 and Corollary in [3], the following lemma follows: 7

18 /3 / Figre 3: An example of the need for the second modification. For the left: competitie analysis shows that in this case, and 2 can achiee a high competitie ratio at the expense of. For the right: an example of balancing strategy by making slightly more likely to pick when it comes. Lemma 6. Sppose is not a part of any cycle of length 4. We hae P = ( Pr[B ]) ( Pr[G ]) + o(/n) For the node, we hae Pr[B ] = e. From definition, G is the eent that among the n lists, the random list R = (, ) comes at least twice. Notice that the list R = (, ) comes with probability 3n. Ths we hae Pr[G ] = Pr[X 2] = e /3 ( + /3), where X Pois(/3). Similarly, we can get Pr[G 2 ] = e ( + ) and the resltant P = Obsere 9e 2 that P Pr[B ] = e /3 and P 2 Pr[B 2 ] = e. Let R[RLA, ], R[RLA, /3] and R[RLA, ] be the competitie ratio achieed by RLA for, and 2 respectiely. Hence, we hae R[RLA, ] while R[RLA, /3] 3( e /3 ) and R[RLA, ] Intitiely, one can improe the worst case ratio by increasing the arrial rate for R = (, ) while redcing that for R = (, ). Sppose one modifies H (, ) and H (, ) to H (, ) = 0. and H (, ) = 0.9, the arrial rate for R = (, ) and R = (, ) gets modified to 0./n and 0.9/n respectiely. The reslting changes are Pr[B ] = e 0.9 /3, Pr[G ] = e 0. ( + 0.), R[RLA, ] = 0.75, Pr[B ] = e /3, Pr[G ] and R[RLA, /3] 0.8. Hence, the performance on WS instance improes. Notice that after the modifications, H = H (, ) + H (, 2 ) = /3. Figre 4 describes the arios modifications applied to H ector. The ales on top of the edge, denote the new ales. Cases () and (2) help improe pon the WS described in Figre Vertex-Weighted Algorithm VW 4.3. Analysis of algorithm VW. The fll details of or ertex-weighted algorithm are stated as follows. Algorithm 8: VW Constrct and sole the LP in sb-section 2. for the inpt instance. 2 Inoke DR[f, 3] to otpt F and H. Apply the two kinds of modifications to morph H to H. 3 Rn RLA[H ] on the graph G H. The algorithm VW consists of two different random processes: sb-rotine DR[f, 3] in the offline phase and RLA in the online phase. Conseqently, the analysis consists of two parts. First, for a 8