New Algorithms, Better Bounds, and a Novel Model for Online Stochastic Matching

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1 New Algorithms, Better Bounds, and a Noel Model for Online Stochastic Matching Brian Brubach, Karthik Abina Sankararaman 2, Araind Sriniasan 3, and Pan Xu 4 Department of Computer Science, Uniersity of Maryland, College Park, USA bbrubach@cs.umd.edu 2 Department of Computer Science, Uniersity of Maryland, College Park, USA kabina@cs.umd.edu 3 Department of Computer Science, Uniersity of Maryland, College Park, USA srin@cs.umd.edu 4 Department of Computer Science, Uniersity of Maryland, College Park, USA panxu@cs.umd.edu Abstract Online matching has receied significant attention oer the last 5 years due 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, much effort has gone into deeloping useful online models that incorporate some stochasticity in the arrial process. One such popular model is the known I.I.D. model where different customer-types arrie online from a known distribution. We deelop algorithms with improed competitie ratios for some basic ariants of this model with integral arrial rates, including: (a) the case of general weighted edges, where we improe the best-known ratio of due to Haeupler, Mirrokni and Zadimoghaddam [] to 0.705; and (b) the ertex-weighted case, where we improe the ratio of Jaillet and Lu [2] to We also consider two extensions, one is known I.I.D. with non-integral arrial rate and stochastic rewards; the other is known I.I.D. b-matching with non-integral arrial rate and stochastic rewards. We present a simple non-adaptie algorithm which works well simultaneously on the two extensions. One of the key ingredients of our 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 solution f; howeer, these gie us less control oer the structure 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 structure of this solution is inspired by Jaillet and Lu (Mathematics of Operations Research, 203) and is a tractable structure for algorithm design and analysis. The appropriate random moe preseres many of the remoed constraints (approximately [exactly] with high probability [in expectation]). This underlies some of our improements, and, we hope, could be of independent interest. 998 ACM Subject Classification F.2 Analysis of Algorithms and Problem Complexity Keywords and phrases Ad-Allocation, Online Matching, Randomized Algorithms Digital Object Identifier /LIPIcs.ESA A full ersion of the paper is aailable at Supported in part by NSF Awards CNS and CCF , and a research award from Adobe, Inc. Brian Brubach, Karthik Abina Sankararaman, Araind Sriniasan, and Pan Xu; licensed under Creatie Commons License CC-BY 24th Annual European Symposium on Algorithms (ESA 206). Editors: Piotr Sankowski and Christos Zaroliagis; Article No. 24; pp. 24: 24:6 Leibniz International Proceedings in Informatics Schloss Dagstuhl Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany

2 24:2 New Algorithms, Better Bounds, and a Noel Model for Online Stochastic Matching Introduction Applications to Internet adertising hae drien the study of online matching problems in recent years [9]. In these problems, we consider a bipartite graph G = (U, V, E) in which the set U is aailable offline while the ertices in V arrie online. Wheneer some ertex arries, it must be matched immediately to at most one ertex in U. Each offline ertex u can be matched to at most one or in the b-matching generalization, at most b ertices in V. In the context of Internet adertising, U is the set of adertisers, V is a set of impressions, and the edges E define the impressions that interest a particular adertiser. When arries, we must choose an aailable adertiser (if any) to match with it. Initially, we consider the case where V can be matched at most once. We later relax this condition to it being matched up to b times. Since adertising forms the key source of reenue for many large Internet companies, finding good matching algorithms and obtaining een small performance gains can hae high impact. Additionally, bipartite matching is a fundamental combinatorial optimization problem. Hence, any improements is interesting from a theoretical standpoint. In the stochastic known I.I.D. model of arrial, we are gien the bipartite graph in adance and each arriing ertex is drawn with replacement from a known distribution on the ertices in V. This captures the fact that we often hae background data about the impressions and can predict the frequency with which each type of impression will arrie. Edge-weighted matching [8] is a general model in the context of adertising: eery adertiser gains a gien reenue for being matched to a particular type of impression. Here, a type of impression refers to a class of users (e.g., a demographic group) who are interested in the same subset 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 reenue generated for matching any of the user types interested in it. In some modern business models, reenue is not generated upon matching adertisements, but only when a user clicks on the adertisement: this is the pay-per-click model. From background data, one can assign the probability of a particular adertisement being clicked by a type of user. Works including [20],[2] capture this notion by assigning a probability to each edge. One unifying theme in most of our approaches is to use an LP benchmark with additional alid constraints that hold for the respectie stochastic-arrial models, combined with some form of dependent rounding.. Related work For readers not familiar with these problems, they are encouraged to first read parts of section 2 for formal definitions before getting into the related work. The study 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 unweighted bipartite matching problem in which ertices arrie in adersarial order. Following that, a series of works hae studied arious related models. The book by Mehta [9] gies a detailed oeriew. The ertex-weighted ersion of this problem was introduced by Aggarwal, Goel and Karande [], where they gie an optimal ( e ) ratio for the adersarial arrial model. The edge-weighted setting has been studied in the adersarial model by Feldman, Korula, Mirrokni and Muthukrishnan [8], where they consider an additional relaxation of free-disposal". Beyond the adersarial model, these problems are studied under the name stochastic matching, where the online ertices either arrie in random order or are drawn I.I.D. from a known distribution. The works [5, 5, 6, 7] among others, study the random arrial order

3 B. Brubach, K. A. Sankararaman, A. Sriniasan, and P. X4:3 model; papers including [4, 9,, 2, 8, 6] study the I.I.D. arrial order model. Another ariant of this problem is when the edges hae stochastic rewards. Models with stochastic rewards hae been preiously studied by [20], [2] among others, but not in the known I.I.D. model. Related Work in the Vertex-Weighted and Unweighted Settings: The ertex-weighted and unweighted settings hae many results starting with Feldman, Mehta, Mirrokni and Muthukrishnan [9] who were the first to beat /e with a competitie ratio of 0.67 for the unweighted problem. This was improed by Manshadi, Gharan, and Saberi [8] to with an adaptie algorithm. In addition, they showed that een in the unweighted ariant with integral arrial rates, no algorithm can achiee a ratio better than e Finally, Jaillet and Lu [2] presented an adaptie algorithm which used a cleer LP to achiee and 2e for the ertex-weighted and unweighted problems, respectiely. Related Work in the Edge-Weighted Setting: For this model, Haeupler, Mirrokni, Zadimoghaddam [] were the first to beat /e by achieing a competitie ratio of They use a discounted 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 constructing two matchings offline to guide their online algorithm. Related Work in Online b-matching: In the model of b-matching, we assume each ertex u has a uniform capacity of b, where b is a parameter which is generally a large integral alue. The model of unweighted b-matching can be iewed as a special case of Adwords or Display Ads. There is extensie literature for Adwords or Display Ads under arious settings (see the book by Mehta [9]). In particular, [3] shows that their algorithm BALANCE is optimal for online b-matching under the adersarial model, which achiees a ratio of. (+/b) b In this paper, we consider edge-weighted b-matching with stochastic rewards under the known I.I.D. model with arbitrary arrial rates. To the best of our knowledge, we are the first to consider this ery general model. Deanur et al [7] 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 budget utilization and when the bid to budget ratios are at most /k. Notice that een the problem of general edge-weighted b-matching with deterministic rewards cannot be captured in the Adwords model. Alaei et al [2] consider the Prophet-Inequality Matching problem, in which arries from a distinct (known) distribution D t, in each round t. They gae a / k + 3 competitie algorithm, where k is the minimum capacity of u. They assume deterministic rewards howeer, and it is non-triial to extend their result to the stochastic reward setting. In this paper, we present a ery simple algorithm which achiees a ratio of b /2+ɛ O(e b2ɛ /3 ) for any gien ɛ > 0. It is worthwhile to see that our algorithm (5) can be triially extended to the case where each ertex u has a distinct capacity b u. The alue of b in the final ratio would be replaced by min u U b u. 2 Preliminaries 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. distribution on V. For each V, we are gien an arrial rate r, which is the expected number of times will arrie. With the exception of Sections 5 and 6, this paper will focus on the integral-arrial-rates setting where E S A 2 0 6

4 24:4 New Algorithms, Better Bounds, and a Noel Model for Online Stochastic Matching all r Z +. As described in [], WLOG we can assume in this setting that V, r =. Let n = V r be the expected number of ertices arriing during the online phase. In the ertex-weighted ariant, eery ertex u U has a weight w u and we seek a maximum weight matching. In the edge-weighted ariant, eery edge e E has a weight w e and we seek a maximum weight matching. In the stochastic rewards ariant, additionally, each edge has a probability p e and we seek to maximize the expected weight of the matching. In the b-matching model, eery ertex in U can be matched upto b times. Throughout, we will use WS to refer to the worst case for arious algorithms. Asymptotic assumption and notation: We will always assume n is large and analyze algorithms as n goes to infinity: e.g., if x ( 2/n) n, we will just write this as x /e 2 instead of the more-accurate x /e 2 + o(). These suppressed o() terms will subtract at most o() from our competitie ratios. Another fact to note is that the competitie ratio is defined slightly different than usual, for this set of problems (Similar to notation used in [9]). In particular, it is defined as E[ALG] E[OP T ]. Algorithms can be adaptie or non-adaptie. When arries, an adaptie algorithm can check which neighbors are still aailable to be matched, but a non-adaptie algorithm cannot. 2. LP Benchmark We will use the following LP to upper bound the optimal offline solution and guide our algorithm. We will first show an LP for the unweighted ariant, then describe changes for the ertex-weighted and edge-weighted settings. As usual, 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 subject to f e (2.) e E e (u) e () f e u U (2.2) f e V (2.3) 0 f e /e e E (2.4) f e + f e /e 2 e, e (u), u U (2.5) Variants: The objectie function is: maximize u U e (u) f ew u in the ertex-weighted ariant and maximize e E f ew e in the edge-weighted ariant. Constraint 2.2 is the matching constraint for ertices in U. Constraint 2.3 is alid because each ertex in V has an arrial rate of. Constraint 2.4 is used in [8] and []. It captures the fact that the expected number of matches for any edge is at most /e. This is alid for large n because the probability that a gien ertex doesn t arrie after n rounds is /e. Constraint 2.5 is similar to the preious one, but for pairs of edges. For any two neighbors of a gien u U, the probability that neither of them arrie is /e 2. Therefore, the sum of ariables for any two distinct edges in (u) cannot exceed /e 2. Notice that constraints 2.4 and 2.5 reduces the gap between the optimal LP solution and the performance The edge realization process is independent from one another. At each step, the algorithm "probes" the edge. With probability p e the edge exists and with remaining probability it doesn t. Once realization of an edge is determined, it doesn t change for the rest of the algorithm

5 B. Brubach, K. A. Sankararaman, A. Sriniasan, and P. X4:5 (C ) u 2 Figure This cycle is the source of the negatie result described by Jaillet and Lu [2]. Thick edges hae f e = while thin edges hae f e = /3. of the optimal online algorithm. In fact, without constraint 2.4, we cannot in general achiee a competitie ratio better than /e. 2.2 Oeriew of ertex-weighted algorithm and contributions A key challenge encountered by [2] was that their special LP could lead to length four cycles of type C shown in Figure. In fact, they used this cycle to show that no algorithm could perform better than 2/e using their LP. They mentioned that tighter LP constraints such as 2.4 and 2.5 in the LP from Section 2 could aoid this bottleneck, but they did not propose a technique to use them. Note that the {0, /3, } solution produced by their LP was an essential component of their Random List algorithm. We show a randomized rounding algorithm to construct a similar, simplified {0, /3, } ector from the solution of a stricter benchmark LP. This allows for the inclusion of additional constraints, most importantly constraint 2.5. Using this rounding algorithm combined with tighter constraints, we will upper bound the probability of a ertex appearing in the cycle C from Figure at 2 3/e (See Lemma 6) Additionally, we show how to deterministically break all other length four cycles which are not of type C without creating any new cycles of type C. Finally, we describe an algorithm which utilizes these techniques to improe preious results in both the ertex-weighted and unweighted settings. For this algorithm, we first sole the LP in Section 2 on the input graph. In Section 4, we show how to use the technique in sub-section 2.6 to obtain a sparse fractional ector. We then present a randomized online algorithm (similar to the one in [2]) which uses the sparse fractional ector as a guide to achiee a competitie ratio of Preiously, there was gap between the best unweighted algorithm with a ratio of 2e 2 due to [2] and the negatie result of e 2 due to [8]. We take a step towards closing that gap by showing that an algorithm can achiee > 2e 2 for both the unweighted and ertex-weighted ariants with integral arrial rates. 2.3 Oeriew of edge-weighted algorithm and contributions A challenge that arises in applying the power of two choices to this setting is when the same edge (u, ) is included in both matchings M and M 2. In this case, the copy of (u, ) in M 2 can offer no benefit and a second arrial of is wasted. To use an example from related work, Haeupler et al. [] choose two matchings in the following way. M is attained by soling an LP with constraints 2.2, 2.3 and 2.4 and rounding to an integral solution. M 2 is constructed by finding a maximum weight matching and remoing any edges which hae already been included 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 construct two or three matchings together in a correlated manner to reduce the probability that some edge is included in all matchings. We will show a E S A 2 0 6

6 24:6 New Algorithms, Better Bounds, and a Noel Model for Online Stochastic Matching general technique to construct an ordered set of k matchings where k is an easily adjustable 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 input graph. We then round the LP solution ector to a sparse integral ector and use this ector to construct a randomly ordered set of matchings which will guide our algorithm during the online phase. We begin Section 3 with a simple warm-up algorithm which uses a set of two matchings as a guide to achiee a competitie ratio, improing the best known result for this problem. We follow it up 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 pseudo-matching algorithm. 2.4 Oeriew of non-integral arrial rates with stochastic rewards contributions This algorithm is presented in Section 5. We beliee the known I.I.D. model with stochastic rewards is an interesting new direction motiated by the work of [20] and [2] in the adersarial model. We introduce a new, more general LP specifically for this setting and show that a simple algorithm using the LP solution directly can achiee a competitie ratio of /e. In [2], it is shown that no randomized algorithm can achiee a ratio better than 0.62 < /e in the adersarial model. Hence, achieing a /e for the i.i.d. model shows that this lower bound does not extend to this model. In Section 6, we extend this simple algorithm 2 to the b-matching generalization of this problem where each offline ertex u can match with up to b arriing ertices. We show that our algorithm achiees a competitie ratio of at least b /2+ɛ O(e b2ɛ /3 ) for any gien ɛ > 0. Note that this result makes progress on Open Question 4 in the online matching and ad allocation surey [9] which asks about stochastic rewards in non-adersarial models. 2.5 Summary of our contributions 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 (4) achiees a competitie ratio of /e. Theorem 4. For edge-weighted online stochastic b-matching with arbitrary arrial rates and stochastic rewards, online algorithm SM b (5) achiees a competitie ratio of at least b /2+ɛ O(e b2ɛ /3 ) for any gien ɛ > 0. 2 Recently, we hae come to know that the result in Section 6 can be obtained as a special case of [3]. Our approach gies an alternatie, and a simpler algorithm for this special case.

7 B. Brubach, K. A. Sankararaman, A. Sriniasan, and P. X4:7 Table Summary of Contributions. Problem Preious Work This Paper Edge-Weighted (Section 3) [] Vertex-Weighted (Section 4) [2] Unweighted [2] Non-integral Stochastic Rewards (Section 5) N/A e b-matching, Stochastic Rewards (Section 6) N/A b /2+ɛ O(e b2ɛ /3 ) 2.6 LP rounding technique For the algorithms presented, we will first sole the benchmark LP in sub-section 2. for the input instance to get a fractional solution ector f. We then round f to an integral solution F using a two step process we call DR[f, k]. The first step is to multiply f by k. The second step is to apply the dependent rounding techniques of Gandhi, Khuller, Parthasarathy and Sriniasan [0] to this new ector. In this paper, we will always choose k to be 2 or 3. This will help us handle the fact that a ertex in V may appear more than once, but probably not more than two or three times. While dependent rounding is typically applied to alues between 0 and, the useful properties extend naturally to our case in which kf e may be greater than for some edge e. To understand this process, it is easiest to imagine splitting each kf e into two edges with the integer alue f e = kf e and fractional alue f e = kf e kf e. The former will remain unchanged by the dependent rounding since it is already an integer while the latter will be rounded to with probability f e and 0 otherwise. Our final alue F e would be the sum of those two rounded alues. The two properties of dependent rounding we will use are:. Marginal distribution: 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 }. 3 Edge-weighted matching with integral arrial rates 3. A simple competitie algorithm As a warm-up, we will describe a simple algorithm which achiees a competitie ratio of and introduces key ideas in our approach. We begin by soling the LP in sub-section 2. to get a fractional solution ector f and applying DR[f, 2] as described in Subsection 2.6 to get an integral ector F. We construct 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 therefore we can decompose it into two matchings using Hall s Theorem. Finally, we randomly permute the two matchings into an ordered pair of matchings, [M, M 2 ]. These matchings sere as a guide for the online phase of the algorithm, similar to []. E S A 2 0 6

8 24:8 New Algorithms, Better Bounds, and a Noel Model for Online Stochastic Matching The entire warm-up algorithm for the edge-weighted model, denoted by EW 0, is summarized in Algorithm. Algorithm : [EW 0 ] Construct and sole the benchmark LP in sub-section 2. for the input instance. 2 Let f be an optimal fraction solution 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 permute 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 u if (u, ) M ; when arries for the second time, try to assign to some if (, ) M 2. 6 When a ertex arries for the third time or more, do nothing in that step. 3.. Analysis of algorithm EW 0 We will show that EW 0 (Algorithm ) achiees a competitie ratio of Let [M, M 2 ] be our randomly ordered pair of matchings. Note that there might exist some edge e which appears in both matchings if f e > /2. Therefore, we consider three types of edges. We say an edge e is of type ψ, denoted by e ψ, iff e appears only in M. Similarly e ψ 2, iff e appears only in M 2 and e ψ b, iff e appears in both M and M 2. Let P, P 2, P b be the probabilities of getting matched for e ψ, e ψ 2, and e ψ b respectiely. According to the result in Haeupler et al. [], the respectie alues are shown as follows. Lemma 5. (Proof details in Section 3 of []) Gien M and M 2, in the worst case () P = ; (2) P 2 = and (3) P b = Proof. (Analysis for EW 0 ) Consider following two cases. Case : 0 f e /2: By the marginal distribution property of dependent rounding, there can be at most one copy of e in G F and the probability of including e in G F is 2f e. Since an edge in G F can appear in either M or M 2 with equal probability /2, we hae Pr[e ψ ] = Pr[e ψ 2 ] = f e. Thus, the ratio is (f e P + f e P 2 )/f e = P + P 2 = Case 2: /2 f e /e: Similarly, by marginal distribution, 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. Thus, the ratio is (( 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 upon the preious warm-up algorithm to get a competitie ratio of 0.7. We start by making an obseration about the performance of the warm-up 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 preious analysis, small edges achieed a much higher competitie ratio of ersus for large edges. This is primarily due to the fact that we may get two copies of a large edge in G F. In this case, the copy in M has a better chance of being matched, since there is no edge which can block it, but the copy that is in M 2 has no chance of being matched.

9 B. Brubach, K. A. Sankararaman, A. Sriniasan, and P. X4:9 To correct this imbalance, we make an additional modification to the f e alues 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 such 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 such that f l > /2. Note that it is possible for e to hae two large neighbors, but we only care about the largest one. We set f s = f s ( (fl +η) f l ). In other words, we increase the alues of large edges while ensuring that for all w U V, f w by reducing the alues of neighboring small edges proportional to their original alues. Note that it is not possible for two large edges to be adjacent since they must both hae f e > /2. For all other small edges which are not adjacent to any large edges, we leae their alues unchanged. We then apply DR[f, 2] to this new ector, multiplying by 2 and applying dependent rounding as before Analysis We can now proe Theorem 2. Proof. As in the warm-up 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-up 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 alue of the largest neighboring edge in the original LP solution. Then s achiees a ratio of ( ) ( ) (fl + η) (fl + η) f s ( )/f s = ( ) f l f l Note that for f l [0, ) this is a decreasing function with respect to f l. So the worst case is f l = /e and we hae a ratio of ( ) ( ) ( /e + η) /e η ( ) = ( ) ( /e) /e 2 < f l 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 since this is a decreasing function with respect to f l. Choosing the optimal alue of η = 0.042, yields an oerall competitie ratio of 0.7 for this new algorithm. 3.3 A competitie algorithm The details of algorithm and the proof of Theorem 2 can be found in the full ersion of this paper. 4 Vertex-weighted stochastic I.I.D. matching with integral arrial rates In this section, we will consider ertex-weighted online stochastic matching on a bipartite graph G under known I.I.D. model with integral arrial rates. We will present an algorithm in which each u has a competitie ratio of at least Recall that after inoking DR[f, 3], E S A 2 0 6

10 24:0 New Algorithms, Better Bounds, and a Noel Model for Online Stochastic Matching /3 u 0. u 0.5 u 0.6 /3 u u () (2) (3) (4) u /3 u /3 /3 u /3 u 0.25 u u u (5) (6) (7) (8) u u u u x x u x 2 x 2 (9) (0) () (2) Figure 2 Illustration for second modification to H. The alue assigned to each edge represents the alue after the second modification. Here, x = and x 2 = we can obtain a (random) integral ector F with F e {0,, 2}. Define H = F/3 and let G H be the graph induced by H and each edge takes the alue H e {0, /3, }. In this section, we focus on the sparse graph G H. The main steps of the algorithm are:. Sole the ertex-weighted benchmark LP in sub-section 2.. Let f be an optimal solution 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 solution H. See sub-section Run the randomized list algorithm (RLA) [2] induced by H on the graph G H. See the details in full ersion of this paper. The WS for ertex-weighted case in [2] is shown in Figure 3, which arries at node u with a competitie ratio of From their analysis, we find node u has a competitie ratio of at least Hence, we boost the performance of u at the cost of u. In other words, we increase the alue of H (u,) and decrease the alue H (u, ). Case (0) and () in Figure 2 illustrates this. After this modification, the new WS for ertex-weighted is now the C cycle shown in Figure. In fact, this is the WS for the unweighted case in [2]. Howeer, Lemma 6 and the cycle breaking algorithm, implies that C cycle can be aoided with probability at least 3/e. This helps us improe the ratio een for the unweighted case in [2]. Lemma 6. For any gien u U, u appears in a C cycle after DR[f, 3] with probability at most 2 3/e.

11 B. Brubach, K. A. Sankararaman, A. Sriniasan, and P. X4: (WS) (C ) (C 2 ) (C 3 ) u u u u u u 3 u u 2 2 Figure 3 Left: The WS for Jaillet and Lu [2] 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 =. Proof. Consider the graph G H obtained after DR[f, 3]. Notice that for some ertex u to appear in a C cycle, it must hae a neighboring edge with H e =. Now we try to bound the probability of this eent. It is easy to see that for some e (u) with f e /3, F e after DR[f, 3], and hence H e = F e /3 /3. Thus only those edges e (u) with f e > /3 will possibly be rounded to H e =. Note that, there can be at most two such edges in (u), since e (u) f e. Hence, we hae the following two cases. Case : (u) 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 function of f e and f e /e from LP constraint 2.4, we hae q 2 3( /e) = 2 3/e. Case 2: (u) 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 mutually exclusie since H u. Using the analysis from case, it follows that q 2 = (3f e ) + (3f e2 ) = 3(f e + f e2 ) 2. From LP constraint 2.5, we know that f e +f e2 /e 2, and hence q 2 3( /e 2 ) 2 < 2 3/e. 4. Two kinds of Modifications to H 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 [2], they gie an efficient way to break C 2 and C 3, as shown in Figure 3. Cycle C cannot be modified further and hence, is the bottleneck for their unweighted case. Notice that, while breaking the cycles of C 2 and C 3, new cycles of C can be created in the graph. Since our randomized construction of solution H gies us control on the probability of cycles C occurring, we would like to break C 2 and C 3 in a controlled way, so as to not create any new C cycles. This procedure is summarized in Algorithm 2. The proof of Lemma 7 can be found in the full ersion of this paper. Lemma 7. After applying Algorithm 2 to G H, we hae () the alue 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. E S A 2 0 6

12 24:2 New Algorithms, Better Bounds, and a Noel Model for Online Stochastic Matching Algorithm 2: [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 return to the first step. The second modification is to decrease the rates of lists associated with those nodes u with H u = /3 or H u = and increase the rates of lists associated with nodes u with H u =. All details can be found in the full ersion. Let H be the solution ector obtained by applying two kinds of modifications to H. The algorithm for the ertex-weighted case, denoted by VW, is summarized below. The detailed analysis can be found in the full ersion of this paper. Algorithm 3: VW [Vertex Weighted] Construct and sole the LP in sub-section 2. for the input instance. 2 Inoke DR[f, 3] to output F and H. Apply the two kinds of modifications to morph H to H. 3 Run RLA[H ] on the graph G H. 5 Non-integral arrial rates with stochastic rewards The setting here is strictly generalized oer the preious sections in the following ways. Firstly, it allows an arbitrary arrial rate (say r ) which can be fractional for each stochastic ertex. Notice that, r = n where n is the total number of rounds. Secondly, each e = (, u) E is associated with a alue p e, which indicates the probability that edge e = (u, ) is present when we assign to u. We assume this process is independent of the stochastic arrial of each. We will show that the simple non-adaptie algorithm introduced in [] can be extended to this general case. This achiees a competitie ratio of ( e ). Note that Manshadi et al. [8] show that no non-adaptie algorithm can possibly achiee a ratio better than ( /e) for the non-integral arrial rates, een for the case of all p e =. Thus, our algorithm is an optimal non-adaptie algorithm for this model. max w e f e p e : (5.) s.t. e E e (u) e () f e p e, u U (5.2) f e r, V (5.3) We use a similar LP as [2] for the case of non-integral arrial rates. For each e E, let f e be the probability that e gets matched in the offline optimal algorithm. Our algorithm is summarized in Algorithm 4. Notice that the last constraint ensures that step 2 in the algorithm is alid. Let us now proe theorem 3. Proof. Let B(u, t) be the eent that u is safe at beginning of round t and A(u, t) to be the eent that ertex u is matched during the round t conditioned on B(u, t). From

13 B. Brubach, K. A. Sankararaman, A. Sriniasan, and P. X4:3 Algorithm 4: SM Construct and sole LP (5.). WLOG assume {f e e E} is an optimal solution. 2 When a ertex arries, assign to each of its neighbor u with a probability f (u,) r. the algorithm, we know Pr[A(u, t)] r n f u, r p e n, which follows by P r[b(u, t)] = [ u t ] Pr i= ( A(u, i)) ( t. n) Consider an edge e = (u, ) in the graph. Notice that the probability that e gets matched in SM should be Pr[e is matched] = n Pr[ arries at t and B(u, t) ] fep e t= n t= ( ) t ( r f e p e ) f e p e. n n r e r 6 Extension to b-matching with stochastic rewards In this section, we further generalize the model in Section 5 to the case where each u in the offline set U has a uniform integral capacity b (i.e., each ertex u can be matched at most b times). Otherwise, we retain the same setting as Section 5; we allow non-integral arrial rates and stochastic rewards. We will generalize the simple algorithm used in the preious setting (i.e., Section 5) to this new setting. Consider the following updated LP: max w e f e p e : (6.) s.t. e E e (u) e () f e p e b, u U (6.2) f e r, V (6.3) We modify Algorithm 4 for the b-matching problem as shown in Algorithm 5. Let us now proe Theorem 4. Algorithm 5: SM b Construct and sole LP (6.). WLOG assume {f e e E} is an optimal solution. 2 When a ertex arries, assign to each of its neighbor u with a probability f (u,) r. Proof. The proof is similar to that of Theorem 3. Let A t be the number of times u has been matched at the beginning of round t. Let B(u, t) be the eent that u is safe at the beginning of round t, which is defined as A t b. For any gien edge e, let X e be the number of times that e gets matched oer the n rounds. Thus we hae E[X e ] = n t= Pr[B(u, t)] r f e p e = f ep e n r n n Pr[A t b ]. t= E S A 2 0 6

14 24:4 New Algorithms, Better Bounds, and a Noel Model for Online Stochastic Matching Now we upper bound the alue of Pr[A t b]. For each i t, let Z i be the indicator random ariable for u to be matched during round i. Thus A t+ = t i= Z i. Notice that for each i, we hae E[Z i ] u r n f (u,) r p (u,) b n. It follows that for any t n( τ) with 0 < τ <, we hae E[A t+ ] ( τ)b. By applying Chernoff-Hoeffding bounds, we get Pr[A t+ b] e bτ 2 /3. Therefore E[X e ] = f ep e n f ep e n n Pr[A t b ] t= n( τ) t= ( e bτ 2 /3 ) = f e p e ( τ)( e bτ 2 /3 ) For any gien ɛ > 0, choose τ = b /2+ɛ to get a competitie ratio of b /2+ɛ O(e b2ɛ /3 ). 7 Conclusion and Future Directions In this paper, we gae improed algorithms for the Edge-Weighted and Vertex-Weighted models. Preiously, there was a gap between the best unweighted algorithm with a ratio of 2e 2 due to [2] and the negatie result of e 2 due to [8]. We took a step towards closing that gap by showing that an algorithm can achiee > 2e 2 for both the unweighted and ertex-weighted ariants with integral arrial rates. In doing so, we made progess on Open Questions 3 and 4 in the online matching and ad allocation surey [9]. This was possible because our approach of rounding to a simpler fractional solution allowed us to employ a stricter LP. For the edge-weighted ariant, we showed that one can significantly improe the power of two choices approach by generating two matchings from the same LP solution. For the ariant with edge weights, non-integral arrial rates, and stochastic rewards, we presented a ( /e)-competitie algorithm. This showed that the 0.62 < /e bound gien in [2] for the adersarial model with stochastic rewards does not extend to the known I.I.D. model. Furthermore, we considered the online edge-weighted b-matching problem with stochastic rewards under the known IID setting. We gae a ery simple non-adaptie algorithm which achiees a ratio of b /2+ɛ O(e b2ɛ /3 ) for any gien ɛ > 0. A natural next step in the edge-weighted setting is to use an adaptie strategy. For the ertex-weighted problem, one can easily see that the stricter LP we use still has a gap. In addition, we only utilize fractional solutions {0, /3, }. Howeer, dependent rounding gies solutions in {0, /k, 2/k,..., k( /e) /k}; allowing for random lists of length greater than three. Stricter LPs and longer lists could both yield improed results. In the stochastic rewards model with non-integral arrial rates, an open question is to either improe upon the ( e ) ratio or consider a simpler model with integral arrial rates and improe the ratio for this restricted model. Lastly, there is a gap between our result for b-matching with stochastic rewards and the results of [7] and [2] for similar problems with deterministic rewards. It would be nice to see a result for this problem that is O(k /2 ). Acknowledgements. The authors would like to thank Aranyak Mehta and the anonymous reiewers for their aluable comments, which hae significantly helped improe the presentation of this paper.

15 B. Brubach, K. A. Sankararaman, A. Sriniasan, and P. X4:5 References Gagan Aggarwal, Gagan Goel, Chinmay Karande, and Aranyak Mehta. Online ertexweighted bipartite matching and single-bid budgeted allocations. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages SIAM, Saeed Alaei, MohammadTaghi Hajiaghayi, and Vahid Liaghat. Online prophet-inequality matching with applications to ad allocation. In Proceedings of the 3th ACM Conference on Electronic Commerce, pages ACM, Saeed Alaei, MohammadTaghi Hajiaghayi, and Vahid Liaghat. The online stochastic generalized assignment problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques: 6th International Workshop, APPROX, and 7th International Workshop, RANDOM, pages 25. Springer Berlin Heidelberg, Bahman Bahmani and Michael Kapralo. Improed bounds for online stochastic matching. In European Symposium on Algorithms (ESA), pages Springer, Nikhil R Deanur and Thomas P Hayes. The adwords problem: online keyword matching with budgeted bidders under random permutations. In Proceedings of the 0th ACM conference on Electronic commerce, pages ACM, Nikhil R. Deanur, Kamal Jain, Balasubramanian Sian, and Christopher A. Wilkens. Near optimal online algorithms and fast approximation algorithms for resource allocation problems. In Proceedings of the 2th ACM Conference on Electronic Commerce, pages ACM, Nikhil R Deanur, Balasubramanian Sian, and Yossi Azar. Asymptotically optimal algorithm for stochastic adwords. In Proceedings of the 3th ACM Conference on Electronic Commerce, pages ACM, Jon Feldman, Nitish Korula, Vahab Mirrokni, S Muthukrishnan, and Martin Pál. Online ad assignment with free disposal. In Internet and network economics, pages Springer, Jon Feldman, Aranyak Mehta, Vahab Mirrokni, and S Muthukrishnan. Online stochastic matching: Beating -/e. In Foundations of Computer Science (FOCS), pages IEEE, Raji Gandhi, Samir Khuller, Sriniasan Parthasarathy, and Araind Sriniasan. Dependent rounding and its applications to approximation algorithms. Journal of the ACM (JACM), 53(3): , Bernhard Haeupler, Vahab S. Mirrokni, and Morteza Zadimoghaddam. Online stochastic weighted matching: Improed approximation algorithms. In Internet and Network Economics, olume 7090 of Lecture Notes in Computer Science, pages Springer Berlin Heidelberg, Patrick Jaillet and Xin Lu. Online stochastic matching: New algorithms with better bounds. Mathematics of Operations Research, 39(3): , Bala Kalyanasundaram and Kirk R Pruhs. An optimal deterministic algorithm for online b-matching. Theoretical Computer Science, 233():39 325, Richard M Karp, Umesh V Vazirani, and Vijay V Vazirani. An optimal algorithm for on-line bipartite matching. In Proceedings of the twenty-second annual ACM symposium on Theory of computing, pages ACM, Thomas Kesselheim, Klaus Radke, Andreas Tönnis, and Berthold Vöcking. An optimal online algorithm for weighted bipartite matching and extensions to combinatorial auctions. In European Symposium on Algorithms (ESA), pages Springer, Nitish Korula and Martin Pál. Algorithms for secretary problems on graphs and hypergraphs. In Automata, Languages and Programming, pages Springer, E S A 2 0 6

16 24:6 New Algorithms, Better Bounds, and a Noel Model for Online Stochastic Matching 7 Mohammad Mahdian and Qiqi Yan. Online bipartite matching with random arrials: an approach based on strongly factor-reealing LPs. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages ACM, Vahideh H Manshadi, Shayan Oeis Gharan, and Amin Saberi. Online stochastic matching: Online actions based on offline statistics. Mathematics of Operations Research, 37(4): , Aranyak Mehta. Online matching and ad allocation. Foundations and Trends in Theoretical Computer Science, 8(4): , Aranyak Mehta and Debmalya Panigrahi. Online matching with stochastic rewards. In Foundations of Computer Science (FOCS), pages IEEE, Aranyak Mehta, Bo Waggoner, and Morteza Zadimoghaddam. Online stochastic matching with unequal probabilities. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, 205.