{"title": "Bi-Objective Online Matching and Submodular  Allocations", "book": "Advances in Neural Information Processing Systems", "page_first": 2739, "page_last": 2747, "abstract": "Online allocation problems have been widely studied due to their numerous practical applications (particularly to Internet advertising), as well as considerable theoretical interest. The main challenge in such problems is making assignment decisions in the face of uncertainty about future input; effective algorithms need to predict which constraints are most likely to bind, and learn the balance between short-term gain and the value of long-term resource availability.  In many important applications, the algorithm designer is faced with multiple objectives to optimize. In particular, in online advertising it is fairly common to optimize multiple metrics, such as clicks, conversions, and impressions, as well as other metrics which may be largely uncorrelated such as \u2018share of voice\u2019, and \u2018buyer surplus\u2019. While there has been considerable work on multi-objective offline optimization (when the entire input is known in advance), very little is known about the online case, particularly in the case of adversarial input. In this paper, we give the first results for bi-objective online submodular optimization, providing almost matching upper and lower bounds for allocating items to agents with two submodular value functions. We also study practically relevant special cases of this problem related to Internet advertising, and obtain improved results. All our algorithms are nearly best possible, as well as being efficient and easy to implement in practice.", "full_text": "Bi-Objective Online Matching and Submodular\n\nAllocations\n\nHossein Esfandiari\nUniversity of Maryland\nCollege Park, MD 20740\nhossein@cs.umd.edu\n\nNitish Korula\nGoogle Research\n\nNew York, NY 10011\nnitish@google.com\n\nAbstract\n\nVahab Mirrokni\nGoogle Research\n\nNew York, NY 10011\n\nmirrokni@google.com\n\nOnline allocation problems have been widely studied due to their numerous prac-\ntical applications (particularly to Internet advertising), as well as considerable\ntheoretical interest. The main challenge in such problems is making assignment\ndecisions in the face of uncertainty about future input; effective algorithms need to\npredict which constraints are most likely to bind, and learn the balance between\nshort-term gain and the value of long-term resource availability.\nIn many important applications, the algorithm designer is faced with multiple\nobjectives to optimize. In particular, in online advertising it is fairly common to\noptimize multiple metrics, such as clicks, conversions, and impressions, as well\nas other metrics which may be largely uncorrelated such as \u2018share of voice\u2019, and\n\u2018buyer surplus\u2019. While there has been considerable work on multi-objective of\ufb02ine\noptimization (when the entire input is known in advance), very little is known\nabout the online case, particularly in the case of adversarial input. In this paper,\nwe give the \ufb01rst results for bi-objective online submodular optimization, providing\nalmost matching upper and lower bounds for allocating items to agents with two\nsubmodular value functions. We also study practically relevant special cases of\nthis problem related to Internet advertising, and obtain improved results. All our\nalgorithms are nearly best possible, as well as being ef\ufb01cient and easy to implement\nin practice.\n\n1\n\nIntroduction\n\nAs a central optimization problem with a wide variety of applications, online resource allocation\nproblems have attracted a large body of research in networking, distributed computing, and electronic\ncommerce. Here, items arrive one at a time (i.e. online), and when each item arrives, the algorithm\nmust irrevocably assign it to an agent; each agent has a limited resource budget / capacity for items\nassigned to him. A big challenge in developing good algorithms for these problems is to predict future\nbinding constraints or learn future capacity availability, and allocate items one by one to agents who\nare unlikely to hit their capacity in the future. Various stochastic and adversarial models have been\nproposed to study such online allocation problems, and many techniques have been developed for\nthese problems. For stochastic input, a natural approach is to build a predicted instance (for instance,\nvia sampling, or using historical data), and some of these techniques solve a dual linear program\nto learn dual variables that are used by the online algorithm moving forward [6, 10, 2, 23, 16, 18].\nHowever, stochastic approaches may provide poor results on some input (for example, when there\nare unexpected spikes in supply / demand), and hence such problems have been extensively studied\nin adversarial models as well. Here, the algorithm typically maintains a careful balance between\ngreedily exploiting the current item by assigning it to agents with high value for it, and assigning the\nitem to a lower-value agent for whom the value is further from the distribution of \u2018typical\u2019 items they\n\n30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.\n\n\fa\u2208A fa(Sa) and(cid:80)\n\ntwo social welfare functions based on fa\u2019s and ga, i.e.,(cid:80)\n\nhave received. Again, primal-dual techniques have been applied to learn the dual variables used by\nthe algorithm in an online manner [17, 3, 9].\nA central practical application of such online algorithms is the online allocation of impressions or\npage-views to ads on the Internet [9, 2, 23, 5, 7]. Such problems are present both in the context of\nsponsored search advertising where advertisers have global budget constraints [17, 6, 3], or in display\nadvertising where each ad campaign has a desired goal or a delivery constraint [9, 10, 2, 23, 5, 7].\nMany of these online optimization techniques apply to general optimization problems including the\nonline submodular welfare maximization problem (SWM) [20, 13].\nFor many real-world optimization problems, the goal is to optimize multiple objective functions [14,\n1]. For instance, in Internet advertising, such objectives might include revenue, clicks, or conversions.\nA variety of techniques have been developed for multi-objective optimization problems; however, in\nmost cases, these techniques are only applicable for of\ufb02ine multi-objective optimization problems [21,\n26], and they do not apply to online settings, especially for online competitive algorithms that work\nagainst an adversarial input [17, 9] or in the presence of traf\ufb01c spikes [18, 8] or hard-to-predict traf\ufb01c\npatterns [5, 4, 22].\nOur contributions. Motivated by the above applications and the increasing need to satisfy multiple\nobjectives, we study a wide class of multi-objective online optimization problems, and present both\nhardness results and (almost tight) bi-objective approximation algorithms for them. In particular, we\nstudy resource allocation problems in which a sequence of items (also referred to as impressions)\ni from an unknown set I arrive one by one, and we have to allocate each item to one agent (for\nexample, one advertiser) a in a given set of agents A. Each agent a has two monotone submodular\nset functions fa, ga : 2I \u2192 R associated with it. Let Sa be the set of items assigned to bin a as\na result of online allocation decisions. The goal of the online allocation algorithm is to maximize\na\u2208A ga(Sa). We \ufb01rst\npresent almost tight online approximation algorithms for the general online bi-objective submodular\nwelfare maximization problem (see Theorems 2.3 and 2.5, and Fig. 1). We show that a simple random\nselection rule along with the greedy algorithm (when each item arrives, randomly pick one objective\nto greedily optimize) results in almost optimal algorithms. Our allocation rule is thus both very\nfast to run and trivially easy to implement. The main technical result of this part is the hardness\nresult showing that the achieved approximation factor is almost tight unless P=NP. Furthermore, we\nconsider special cases of this problem motivated by online ad allocation. In particular, for the special\ncases of online budgeted allocation and online weighted matching, motivated by sponsored search\nand display advertising (respectively), we present improved primal-dual-based algorithms along with\nimproved hardness results for these problems (see, for example, the tight Theorem 3.1).\nRelated Work. It is known that the greedy algorithm leads to a 1/2-approximation for the submodular\nsocial welfare maximization problem (SWM) [11], and this problem admits a 1 \u2212 1/e-approximation\nin the of\ufb02ine setting [24], which is tight [19]. However, for the online setting, the problem does\nnot admit a better than 1/2-approximation algorithm unless P= NP [12]. Bi-objective online\nallocation problems have been studied in two previous papers [14, 1]. The \ufb01rst paper presents [14]\nan online bi-objective algorithm for the problem of maximizing a general weight function and the\ncardinality function, and the second paper [1] presents results for the combined budgeted allocation\nand cardinality constraints. Our results in this paper improve and generalize those results for\nmore general settings. Submodular partitioning problems have also been studied based on mixed\nrobust/average-case objectives [25].\nOur work is related to online ad allocation problems, including the Display Ads Allocation (DA)\nproblem [9, 10, 2, 23], and the Budgeted Allocation (AdWords) problem [17, 6]. In both of these\nproblems, the publisher must assign online impressions to an inventory of ads, optimizing ef\ufb01ciency\nor revenue of the allocation while respecting pre-speci\ufb01ed contracts. The Display Ad (DA) problem is\nthe online matching problem described above with a single weight objective [9, 7]. In the Budgeted\nAllocation problem, the publisher allocates impressions resulting from search queries. Advertiser\na has a budget B(a) on the total spend, instead of a bound n(a) on the number of impressions.\nAssigning impression i to advertiser a consumes wia units of a\u2019s budget instead of 1 of the n(a)\nslots, as in the DA problem. For both of these problems, 1 \u2212 1\ne -approximation algorithms have been\ndesigned under the assumption of large capacities [17, 3, 9]. None of the above papers for adversarial\nmodels studies multiple objectives at the same time.\n\n2\n\n\f2 Bi-Objective Online Submodular Welfare Maximization\n\nthe set of items I. The welfare of allocation S is de\ufb01ned as(cid:80)\n\n2.1 Model and Overview\nFor any allocation S, let Sa denote the set of items assigned to agent a \u2208 A by this allocation. In the\nclassic Submodular Welfare Maximization problem (SWM) for which there is a single monotone\nsubmodular objective, each agent a \u2208 A is associated with a submodular function fa de\ufb01ned on\na fa(Sa), and the goal of SWM is to\nmaximize this welfare. In the classic SWM, the natural greedy algorithm is to assign each item (when\nit arrives) to the agent whose gain increases the most. This greedy algorithm (note that it is an online\nalgorithm) is (1/2 + 1/n)-competitive, and this is the best possible [15].\nIn this section, we consider the extension of online SWM to two monotone submodular functions.\nFormally, each agent a \u2208 A is associated with two submodular functions - fa and ga - de\ufb01ned on I.\na ga(Sa). We\nmeasure the performance of the algorithm by comparison to the of\ufb02ine optimum for each objective:\na ga(Sa). An algorithm\na fa(Sa) \u2265\n\nThe goal is to \ufb01nd an allocation S that does well on both objectives(cid:80)\na fa(Sa) and(cid:80)\nLet S\u2217f = arg maxallocations S(cid:80)\nA is (\u03b1, \u03b2)-competitive if, for every input, it produces an allocation S such that(cid:80)\n\u03b1(cid:80)\n\na fa(Sa) and S\u2217g = arg maxallocations S(cid:80)\n\na ga(Sa) \u2265 \u03b2(cid:80)\n\na ) and(cid:80)\n\na ga(S\u2217g\na ).\n\na fa(S\u2217f\n\nA (1, 1)-competitive algorithm would be one that \ufb01nds an allocation which is simultaneously optimal\nin both objectives, but since the objectives are distinct, no single allocation may maximize both,\neven ignoring computational dif\ufb01culties or lack of knowledge of the future. One could attempt to\nmaximize a linear combination of the two submodular objectives, but since the linear combination is\nitself submodular, this is no harder than the classic online SWM. Instead, we provide algorithms with\nthe stronger guarantee that they are simultaneously competitive with the optimal solution for each\nobjective separately. Further, our algorithms are parametrized, so the user can balance the importance\nof the two objectives.\nSimilar to previous approaches for bi-objective online allocation problems [14], we run two simulta-\nneous greedy algorithms, each based on one of the objective functions. Upon arrival of each online\nitem, with probability p we pass the item to the greedy algorithm based on the objective function f,\nand with probability 1 \u2212 p we pass the item to the greedy algorithm based on g.\nFirst, as a warmup, we provide a charging argument to show that the greedy algorithm for (single-\nobjective) SWM is 1/2-competitive. This charging argument is similar to the usual primal-dual\nanalysis for allocation problems. However, since the objective functions are not linear, it may not be\npossible to interpret the proof using a primal-dual technique. Later, we modify our charging argument\nand show that if we run the greedy algorithm for SWM but only consider items for allocation with\n1+p. (Note that a naive analysis would yield a competitive ratio\nprobability p, the competitive ratio is\nof p/2, since we lose a factor of p in the sampling and a factor of 1/2 due to the greedy algorithm.)\nSince our algorithm for bi-objective online SWM passes items to the \u2018\ufb01rst\u2019 greedy algorithm with\nprobability p and passes items to the second greedy algorithm with probability 1 \u2212 p, the modi\ufb01ed\ncharging argument immediately implies that our algorithm is ( p\n2\u2212p ) competitive, as we state in\nTheorem 2.3 below. Also, using a factor-revealing framework, assuming N P (cid:54)= RP , we provide an\nalmost tight hardness result, which holds even if the objective functions have the simpler \u2018coverage\u2019\nstructure. Both our competitive ratio and the associated hardness result are presented in Figure 1.\n\n1+p , 1\u2212p\n\np\n\n2.2 Algorithm for Bi-Objective online SWM\n\nWe de\ufb01ne some notation and ideas that we use to bound the competitive ratio of our algorithm. Let Gr\nbe the greedy algorithm and let Opt be a \ufb01xed optimum allocation. For an agent j, and an algorithm\nAlg, let Algj be the set of online items allocated to the agent j by Alg; Optj denotes the set of online\nitems allocated to j in Opt. Trivially, for any two agents j and k, we have Algj \u2229 Algk = \u2205.\nFor each online item i we de\ufb01ne a variable \u03b1i, and for each agent j we de\ufb01ne a variable \u03b2j. In order\nto bound the competitive ratio of the algorithm Alg by c, it suf\ufb01ces to set the values of \u03b1is and \u03b2js\nand 2) the value of Opt is at most\nsuch that 1) the value of Alg is at least c\n\n(cid:16)(cid:80)n\ni=1 \u03b1i +(cid:80)m\n\nj=1 \u03b2j\n\n(cid:17)\n\n(cid:80)n\ni=1 \u03b1i +(cid:80)m\n\nj=1 \u03b2j.\n\n3\n\n\fFigure 1: The lower (blue) curve is the competitive ratio of our algorithm, and the red curve is the\nupper bound on the competitive ratio of any algorithm.\n\nto the value of Gr. Thus, the value of Gr is clearly 0.5\n\nTheorem 2.1. (Warmup) The greedy algorithm is 0.5-competitive for online SWM.\nProof. For each online item i, let \u03b1i be the marginal gain by Gr from allocating item i upon its\ni=1 \u03b1i is equal to the value of Gr. For each agent j, let \u03b2j be the total\nj=1 \u03b2j is equal\n\narrival. It is easy to see that(cid:80)n\nvalue of the allocation to j at the end of the algorithm. By de\ufb01nition, we know that(cid:80)m\n(cid:16)(cid:80)n\ni=1 \u03b1i +(cid:80)m\nbounded by \u03b2j +(cid:80)\nby summing over all agents, we can upper-bound the value of Opt by(cid:80)n\nNow, we just need to show that for any agent j we have fj(Optj) \u2264 \u03b2j +(cid:80)\n\nRecall that fj(.) denotes the valuation function of agent j. Below, we show that fj(Optj) is upper-\n\u03b1i. Note that for distinct agents j and k, Optj and Optk are disjoint. Thus,\nj=1 \u03b2j. This\n\n\u03b1i. Note that for\nany item i \u2208 Optj, the value of \u03b1i is at least the marginal gain that would have been obtained from\nassigning i to j when it arrives. Applying submodularity of fj, we have \u03b1i \u2265 fj(Grj \u222a i) \u2212 fj(Grj).\nMoreover, by de\ufb01nition we have \u03b2j = fj(Grj). Thus, we have:\n\n(cid:17)\ni=1 \u03b1i +(cid:80)m\n\nmeans that the competitive ratio of Gr is 0.5.\n\nj=1 \u03b2j\n\ni\u2208Optj\n\ni\u2208Optj\n\n.\n\n(cid:88)\n\ni\u2208Optj\n\n\u03b2j +\n\n(cid:88)\n\n\u03b1i \u2265 fj(Grj) +\n\n(fj(Grj \u222a i) \u2212 fj(Grj))\n\n\u2265 fj(Grj) +(cid:0)fj(Grj \u222a Optj) \u2212 fj(Grj)(cid:1)\n\ni\u2208Optj\n\n= fj(Grj \u222a Optj) \u2265 fj(Optj),\n\nwhere the second inequality follows by submodularity, and the last inequality by monotonicity. This\ncompletes the proof.\n\np\n\n1+p -competitive for online SWM.\n\nLemma 2.2. Let Grp be an algorithm that with probability p passes each online item to Gr for\nallocation, and leaves it unmatched otherwise. Grp is\nProof. The proof here is fairly similar to Theorem 2.1. For each online item i, set \u03b1i to be the\nmarginal gain that would have been achieved from allocating item i upon its arrival (assuming i is\npassed to Gr), given the current allocation of items. Note that \u03b1i is a random variable (depending on\nthe outcome of previous decisions to pass items to Gr or not), but it is independent of the coin toss\nthat determines whether it is passed to Gr, and so the expected marginal gain of allocating item i,\n(given all previous allocations) is pE[\u03b1i]. Thus, by linearity of expectation, the expected value of Grp\ni=1 \u03b1i]. On the other hand, for each agent j, set \u03b2j to be the value of the actual allocations\nj=1 \u03b2j equal to the value of Grp. Combining these\n.\n\nis pE[(cid:80)n\nto j at the end of the algorithm. Again, we have(cid:80)m\n\n(cid:16)(cid:80)n\ni=1 E[\u03b1i] +(cid:80)m\n\ntwo, we conclude that the expected value of Grp is equal to\n\nj=1 E[\u03b2j]\n\n(cid:17)\n\n1\n\n1+1/p\n\n4\n\n\fAs before, we show that fj(Optj) is upper-bounded by \u03b2j +(cid:80)\nIt remains only to show that for any agent j, we have fj(Optj) \u2264 \u03b2j +(cid:80)\n\nthat the competitive ratio of Grp is\n\n1+1/p = p\n\ni\u2208Optj\n\n1+p.\n\n1\n\n\u03b1i. This is exactly\nthe same as our proof for Theorem 2.1: By submodularity of fj we have, \u03b1i \u2265 fj(Grp(j) \u222a i) \u2212\nfj(Grp(j)), and by de\ufb01nition we have \u03b2j = fj(Grp(j)). We provide the complete proof in the full\nversion.\nThe main theorem of this section follows immediately.\nTheorem 2.3. For any 0 < p < 1, there is a ( p\nonline SWM.\n\n2\u2212p )-competitive algorithm for bi-objective\n\n1+p , 1\u2212p\n\ni\u2208Optj\n\n\u03b1i. Therefore, we can conclude\n\n2.3 Hardness of Bi-Objective online SWM\n\nWe now prove that Theorem 2.3 is almost tight, by describing a hard instance for bi-objective online\nSWM. To describe this instance, we de\ufb01ne notions of super nodes and super edges, which capture\nthe hardness of maximizing a submodular function even in the of\ufb02ine setting. Using the properties of\nsuper nodes and edges, we construct and analyze a hard example for bi-objective online SWM.\nOur construction generalizes that of Kapralov et al. [12], who prove the upper bound corresponding to\nthe two points (0.5, 0) and (0, 0.5) in the curve shown in Figure 1. They use the following result: For\nany \ufb01xed c0 and \u0001(cid:48) it is NP-hard to distinguish between the following two cases for of\ufb02ine SWM with\nn agents and m = kn items. This holds even for submodular functions with \u2018coverage\u2019 valuations.\n\n\u2022 There is an allocation with value n.\n\u2022 For any l \u2264 c0, no allocation allocates kl items and gets a value more than 1 \u2212 e\u2212l + \u0001(cid:48).\nIntuitively, in the former case, we can assign k items to each agent and obtain value 1 per agent. In the\nlatter case, even if we assign 2k items (however they are split across agents), we can obtain total value\nat most 0.865. It also follows that there exist \u2018hard\u2019 instances such that there is an optimal solution\nof value n, but for any l < 1, any assigment of ml edges obtains value at most (1 \u2212 e\u2212l + \u0001(cid:48))n.\nWe now de\ufb01ne a super edge to be a hard instance of of\ufb02ine SWM as de\ufb01ned above. We refer to the\nset of agents in a super edge as the agent super node, and the set of items in the super edge as the\nitem super node. If two super edges share a super node, it means that they share the agents / items\ncorresponding to that super node in the same order. If (in expectation) we allocate ml items of a\nsuper edge, we say the load of that super edge is l. Similarly, if (in expectation) we allocate ml items\nto an agent super node, we say the load of that super node is l. Using the de\ufb01nition of super edge and\nsuper node, the hardness result of Kapralov et al. [12] gives us the following lemma:\nLemma 2.4. Assume RP (cid:54)= N P and let \u0001 be an arbitrary small constant. If the (expected) load of a\nrandomized polynomial algorithm on an agent super node is l, the expected welfare of all agents is at\nmost (1 \u2212 e\u2212l + \u0001)n.\nNow with Lemma 2.4 in hand, we are ready to present an upper bound for bi-objective online SWM.\nTheorem 2.5. Assume RP (cid:54)= N P . The competitive ratio (\u03b1, \u03b2) of any algorithm for bi-objective\nonline SWM is upper bounded by the red curve in Figure 1. More precisely (assuming w.l.o.g. that\n\u03b1 \u2265 \u03b2), for any \u03b3 \u2208 [0, 1], there is no algorithm with \u03b1 > 0.5+\u03b32/6\n\nand \u03b2 > \u03b3\u03b1.\n\n1+\u03b32\n\n3 Bi-Objective Online Weighted Matching\n\nij and wg\n\nIn this section, we consider two special cases of bi-objective online SWM, each of which generalizes\nthe (single objective) online weighted matching problem (with free disposal). Here, each item i has\ntwo weights wf\nij for agent j, and each agent j has (large) capacity Cj. The weights of item i\nare revealed when it arrives, and the algorithm must allocate it to some agent immediately.\nIn the \ufb01rst model, after the algorithm terminates, and each agent j has received items Sj, it chooses a\nsubset S(cid:48)\nij, and\nij. Intuitively, each agent must pick a subset of its items, and it\n\nj \u2286 Sj of at most Cj items. The total value in the \ufb01rst objective is then(cid:80)\n\nin the second objective(cid:80)\n\n(cid:80)\n\n(cid:80)\n\ni\u2208S(cid:48)\n\nwg\n\nwf\n\nj\n\nj\n\nj\n\ni\u2208S(cid:48)\n\nj\n\n5\n\n\fExponential Weight Algorithm.\nSet \u03b2j to 0 for each agent j.\nUpon arrival of each item i:\n\n1. If there is agent j with wij \u2212 \u03b2j > 0\n\n(a) Let j be the agent that maximizes wij \u2212 \u03b2j\n(b) Assign i to j, and set \u03b1i to wij \u2212 \u03b2j.\n(c) Let w1, w2, . . . , wCj be the weights of the Cj highest weight items, matched to j in\n\na non-increasing order.\n1+ 1\nCj\n\n(cid:17)j\u22121\n(cid:80)Cj\nj=1 wj\nCj((1+1/Cj )Cj \u22121)\n\n(cid:16)\n\n(d) Set \u03b2j to\n\n.\n\n2. Else: Leave i unassigned.\n\nFigure 2: Exponential weight algorithm for online matching with free disposal.\n\ngets paid its (additive) value for these items. In the (single-objective) case where each agent can only\nbe allocated Cj items, this is the online weighted b-matching problem, where vertices are arriving\nonline, and we have edge weights in the bipartite (item, agent) graph. This problem is completely\nintractable in the online setting, while the free disposal variant [9] in which additional items can be\nassigned, but at most Cj items count towards the objective, is of theoretical and practical interest.\nIn the second model, after the algorithm terminates and agent j has received items Sj, it chooses two\n(not necessarily disjoint) subsets S\nj are counted towards the \ufb01rst objective,\nand those in S(cid:48)g\nTheorem 3.1. For any (\u03b1, \u03b2) such that \u03b1 + \u03b2 \u2264 1 \u2212 1\ne , there is an (\u03b1, \u03b2)-competitive algorithm for\nthe \ufb01rst model of the bi-objective online weighted matching. For any constant \u0001 > 0, there is no such\nalgorithm when \u03b1 + \u03b2 > 1 \u2212 1\n\nj ; items in S(cid:48)f\nj are counted towards the second objective.\n\n(cid:48)f\nj and S(cid:48)g\n\ne + \u0001.\n\ne ), (1 \u2212 p)(1 \u2212 1\n\nTo obtain the positive result, with probability p, run the exponential weight algorithm (see Figure 2)\nfor the \ufb01rst objective (for all items), and with probability 1 \u2212 p run the exponential weight algorithm\nfor the second objective for all items; this combination is (p(1 \u2212 1\ne ))-competitive.\nWe deffer the proof of this and the matching hardness results to the full version.\nHaving given matching upper and lower bounds for the \ufb01rst model, we now consider the second model,\nwhere if we assign a set Sj of items / edges to an agent j we can select two subsets S(cid:48)f\nj \u2286 Sj\nand use them for the \ufb01rst and second objective functions respectively.\nTheorem 3.2. There is a (p(1 \u2212 1\nobjective online weighted matching problem in the second model as minj{Cj} tends to in\ufb01nity.\nTheorem 3.3. The competitive ratio of any algorithm for bi-objective online weighted matching in\nthe second model is upper bounded by the curve in \ufb01gure 3.\n\ne1/(1\u2212p) ))-competitive algorithm for the bi-\n\ne1/p ), (1 \u2212 p)(1 \u2212\n\nj ,S(cid:48)g\n\n1\n\n4 Bi-Objective Online Budgeted Allocation\n\nat most Cj items; its score is(cid:80)\n\nIn this section, we consider the bi-objective online allocation problem where one of the objectives is\na budgeted allocation problem and the other objective function is weighted matching. Here, each\nitem i has a weight wij and a bid bij for agent j. Each agent j has a capacity Cj and a budget Bj. If\nan agent is allocated items Sj, for the \ufb01rst objective (weighted matching), it chooses a subset S(cid:48)\nj of\nbij, Bj}.\nNote that in the second objective, the agent does not need to choose a subset; it obtains the sum of the\nbids of all items assigned to it, capped at its budget Bj.\nClearly, if we set all bids bij to 1, the goal of the budgeted allocation part will be maximizing the\ncardinality. Thus, this is a clear generalization of the bi-objective online allocation to maximize\nweight and cardinality, and the same hardness results hold here.\n\nwij. For the second objective, its score is min{(cid:80)\n\ni\u2208Sj\n\ni\u2208S(cid:48)\n\nj\n\n6\n\n\fFigure 3: The blue curve is the competitive ratio of our algorithm in the second model, while the red\nline and the green curves are the upper bounds on the competitive ratio of any algorithm.\n\n1\n\ne1/p ), (1 \u2212 p)(1 \u2212\n\n1\n\ne1/p ), (1 \u2212 p)(1 \u2212\n\nAs is standard, throughout this section we assume that the bid of each agent for each item is\nvanishingly small compared to the budget of each bidder. Interestingly, again here, we provide a\n(p(1 \u2212 1\ne1/(1\u2212p) ))-competitive algorithm, which is almost tight. At the end, as a\ncorollary of our results, we provide a a (p(1 \u2212 1\ne1/(1\u2212p) ))-competitive algorithm,\nfor the case that both objectives are budgeted allocation problems (with separate budgets).\nOur algorithm here is roughly the same as for two weight objectives. For each item, with probability\n1\u2212 p, we pass it to the Exponential Weight algorithm for matching, and allocate it based on its weight.\nWith the remaining probability p, we assign the algorithm based on its bids and count it towards the\nBudgeted Allocation objective. However, the algorithm we use for Budgeted Allocation is slightly\ndifferent: We virtually run the Balance algorithm of Mehta et al. [17] for Budgeted Allocation (Fig. 4),\nas though we were assigning all items (not just those passed to this algorithm), but with each item\u2019s\nbids scaled down by a factor of p. For those p fraction of items to be assigned by the Budgeted\nAllocation algorithm, assign them according to the recommendation of the virtual Balance algorithm.\nTheorem 3.2 from the previous section shows that our algorithm is (1 \u2212 p)(1 \u2212\ne1/(1\u2212p) )-competitive\nagainst the optimum weighted matching objective. Thus, in the rest of this section, we only need\nto show that this algorithm is p(1 \u2212 1\ne1/p )-competitive against the optimum Budgeted Allocation\nsolution. First, using a primal dual approach, we show that the outcome of the virtual Balance\nalgorithm (that runs on p fraction of the value of each item) is p(1 \u2212 1\ne1/p ) against the optimum with\nthe actual weights. Then, using the Hoeffding inequality, we show that the expected value of our\nallocation for the budgeted allocation objective is fairly close to the virtual algorithm\u2019s value, i.e. the\ndifference between the competitive ratio of our allocation and the virtual allocation is o(1).\nLemma 4.1. When maxi,j bij\np fraction of the value of each bid is at least p(1 \u2212 1\nvalues.\n\n\u2192 0, the total allocation of the virtual balance algorithm that runs on\ne1/p ) times that of the optimum with the actual\n\nBj\n\n1\n\nThe proof of this lemma is similar to the analysis of Buchbinder et al. [3] for the basic Budgeted\nAllocation problem. We provide this proof in the full version.\nLemma 4.2. For any constant p, assuming maxi,j bij\nalgorithm tends to the value of Balance with p fraction of each bid, with high probability.\n\n\u2192 0, the budgeted allocation value of our\n\nBj\n\nIn the virtual Balance algorithm, we allocate p fraction of each item, while in our real algorithm, we\nallocate every item according to the virtual Balance algorithm with probability p. Since each item\u2019s\nbids are small compared to the budgets, the lemma follows from a straightforward concentration\nargument. We present the complete proof in the full version.\nThe following lemma is an immediate result of combining Lemma 4.1 and Lemma 4.2.\n\n7\n\n\fVirtual Balance algorithm on p fraction of values.\nSet \u03b2j and yj to 0 for each agent j.\nUpon arrival of each item i:\n\n1. If i has a neighbor with bij(1 \u2212 \u03b2j) > 0\n\n(a) Let j be the agent that maximizes bij(1 \u2212 \u03b2j)\n(b) Assign i to j i.e. set xij to 1.\n(c) Set \u03b1i to bij(1 \u2212 \u03b2j).\n(d) Increase yj by bij\nBj\n(e) Increase \u03b2j by eyj\u22121/p\n1\u2212e\u22121/p\n2. Else: Leave i unassigned.\n\nbij\nBj\n\nFigure 4: Maintaining solution to primal and dual LPs.\n\nLemma 4.3. For any constant p, assuming maxi,j bij\nBj\nagainst the optimum budgeted allocation solution.\n\n\u2192 0, our algorithm is p(1 \u2212 1\n\ne1/p )-competitive\n\nLemma 4.3 immediately gives us the following theorem.\ne1/p ), (1 \u2212 p)(1 \u2212\nTheorem 4.4. For any constant p, assuming maxi,j bij\ne1/(1\u2212p) ))-competitive algorithm for the bi-objective online allocation with two budgeted allocation\nobjectives.\n\n\u2192 0, there is a (p(1 \u2212 1\n\nBj\n\n1\n\nMoreover, if we pass each item to the exponential weight algorithm with probability p, the expected\nsize of the output matching is at least p(1 \u2212 1\ne1/p ) that of the optimum [14]. Together with Lemma\n4.3, this gives us the following theorem.\ne1/p ), (1 \u2212 p)(1 \u2212\nTheorem 4.5. For any constant p, assuming maxi,j bij\ne1/(1\u2212p) ))-competitive algorithm for the bi-objective online allocation with a budgeted allocation\nobjective and a weighted matching objective.\n\n\u2192 0, there is a (p(1 \u2212 1\n\nBj\n\n1\n\n5 Conclusions\n\nIn this paper, we gave the \ufb01rst algorithms for several bi-objective online allocation problems. Though\nthese are nearly tight, it would be interesting to consider other models for bi-objective online\nallocation, special cases where one may be able to go beyond our hardness results, and other\nobjectives such as fairness to agents.\n\nReferences\n[1] Gagan Aggarwal, Yang Cai, Aranyak Mehta, and George Pierrakos. Biobjective online bipartite matching.\n\nIn WINE, pages 218\u2013231, 2014.\n\n[2] Shipra Agrawal, Zizhuo Wang, and Yinyu Ye. A dynamic near-optimal algorithm for online linear\n\nprogramming. Computing Research Repository, 2009.\n\n[3] N. Buchbinder, Kamal Jain, and J. Naor. Online primal-dual algorithms for maximizing ad-auctions\n\nrevenue. In ESA, pages 253\u2013264. Springer, 2007.\n\n[4] Dragos Florin Ciocan and Vivek F. Farias. Model predictive control for dynamic resource allocation. Math.\n\nOper. Res., 37(3):501\u2013525, 2012.\n\n[5] N. R. Devanur, K. Jain, B. Sivan, and C. A. Wilkens. Near optimal online algorithms and fast approximation\n\nalgorithms for resource allocation problems. In EC, pages 29\u201338. ACM, 2011.\n\n[6] Nikhil Devanur and Thomas Hayes. The adwords problem: Online keyword matching with budgeted\n\nbidders under random permutations. In EC, pages 71\u201378, 2009.\n\n8\n\n\f[7] Nikhil R. Devanur, Zhiyi Huang, Nitish Korula, Vahab S. Mirrokni, and Qiqi Yan. Whole-page optimization\n\nand submodular welfare maximization with online bidders. In EC, pages 305\u2013322, 2013.\n\n[8] Hossein Esfandiari, Nitish Korula, and Vahab Mirrokni. Online allocation with traf\ufb01c spikes: Mixing\n\nstochastic and adversarial inputs. In EC. ACM, 2015.\n\n[9] J. Feldman, N. Korula, V. Mirrokni, S. Muthukrishnan, and M. Pal. Online ad assignment with free disposal.\n\nIn WINE, 2009.\n\n[10] Jon Feldman, Monika Henzinger, Nitish Korula, Vahab S. Mirrokni, and Cliff Stein. Online stochastic\n\npacking applied to display ad allocation. In ESA, pages 182\u2013194. Springer, 2010.\n\n[11] M. L. Fisher, G. L. Nemhauser, and L. A. Wolsey. An analysis of approximations for maximizing\n\nsubmodular set functions. II. Math. Programming Stud., 8:73\u201387, 1978. Polyhedral combinatorics.\n\n[12] Michael Kapralov, Ian Post, and Jan Vondr\u00e1k. Online submodular welfare maximization: Greedy is optimal.\n\nIn SODA, pages 1216\u20131225, 2013.\n\n[13] Nitish Korula, Vahab Mirrokni, and Morteza Zadimoghaddam. Online submodular welfare maximization:\n\nGreedy beats 1/2 in random order. In STOC, pages 889\u2013898. ACM, 2015.\n\n[14] Nitish Korula, Vahab S. Mirrokni, and Morteza Zadimoghaddam. Bicriteria online matching: Maximizing\n\nweight and cardinality. In WINE, pages 305\u2013318, 2013.\n\n[15] Lehman, Lehman, and N. Nisan. Combinatorial auctions with decreasing marginal utilities. Games and\n\nEconomic Behaviour, pages 270\u2013296, 2006.\n\n[16] Mohammad Mahdian and Qiqi Yan. Online bipartite matching with random arrivals: A strongly factor\n\nrevealing LP approach. In STOC, pages 597\u2013606, 2011.\n\n[17] Aranyak Mehta, Amin Saberi, Umesh Vazirani, and Vijay Vazirani. Adwords and generalized online\n\nmatching. J. ACM, 54(5):22, 2007.\n\n[18] Vahab S. Mirrokni, Shayan Oveis Gharan, and Morteza ZadiMoghaddam. Simultaneous approximations\n\nof stochastic and adversarial budgeted allocation problems. In SODA, pages 1690\u20131701, 2012.\n\n[19] Vahab S. Mirrokni, Michael Schapira, and Jan Vondr\u00e1k. Tight information-theoretic lower bounds for\n\nwelfare maximization in combinatorial auctions. In EC, pages 70\u201377, 2008.\n\n[20] G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher. An analysis of approximations for maximizing\n\nsubmodular set functions. I. Math. Programming, 14(3):265\u2013294, 1978.\n\n[21] Christos H Papadimitriou and Mihalis Yannakakis. On the approximability of trade-offs and optimal access\n\nof web sources. In FOCS, pages 86\u201392. IEEE, 2000.\n\n[22] Bo Tan and R Srikant. Online advertisement, optimization and stochastic networks. In CDC-ECC, pages\n\n4504\u20134509. IEEE, 2011.\n\n[23] Erik Vee, Sergei Vassilvitskii, and Jayavel Shanmugasundaram. Optimal online assignment with forecasts.\n\nIn EC, pages 109\u2013118, 2010.\n\n[24] Jan Vondr\u00e1k. Optimal approximation for the Submodular Welfare Problem in the value oracle model. In\n\nSTOC, pages 67\u201374, 2008.\n\n[25] Kai Wei, Rishabh K Iyer, Shengjie Wang, Wenruo Bai, and Jeff A Bilmes. Mixed robust/average\nsubmodular partitioning: Fast algorithms, guarantees, and applications. In Advances in Neural Information\nProcessing Systems, pages 2233\u20132241, 2015.\n\n[26] Mihalis Yannakakis. Approximation of multiobjective optimization problems. In WADS, page 1, 2001.\n\n9\n\n\f", "award": [], "sourceid": 1396, "authors": [{"given_name": "Hossein", "family_name": "Esfandiari", "institution": "University of Maryland"}, {"given_name": "Nitish", "family_name": "Korula", "institution": "Google Research"}, {"given_name": "Vahab", "family_name": "Mirrokni", "institution": "Google"}]}