We present a 0.659-competitive Quadratic Ranking algorithm for the Oblivious Bipartite Matching problem, a distribution-free version of Query-Commit Matching. This result breaks the $1-\frac{1}{e}$ barrier, addressing an open question raised by Tang, Wu, and Zhang (JACM 2023). Moreover, the competitive ratio of this distribution-free algorithm improves the best existing 0.641 ratio for Query-Commit Matching achieved by the distribution-dependent algorithm of Chen, Huang, Li, and Tang (SODA 2025). Quadratic Ranking is a novel variant of the classic Ranking algorithm. We parameterize the algorithm with two functions, and let two key expressions in the definition and analysis of the algorithm be quadratic forms of the two functions. We show that the quadratic forms are the unique choices that satisfy a set of natural properties. Further, they allow us to optimize the choice of the two functions using powerful quadratic programming solvers.

The prophet secretary problem is a combination of the prophet inequality and the secretary problem, where elements are drawn from known independent distributions and arrive in uniformly random order. In this work, we design 1) a 0.688-competitive algorithm, that breaks the 0.675 barrier of blind strategies (Correa, Saona, Ziliotto, 2021), and 2) a 0.641-competitive algorithm for the prophet secretary matching problem, that breaks the 1 — 1/e ≈ 0.632 barrier for the first time. Our second result also applies to the query-commit model of weighted stochastic matching and improves the state-of-the-art ratio (Derakhshan and Farhadi, 2023).

We initiate the study of Stochastic Online Correlated Selection (SOCS), a family of online rounding algorithms for the general Non-IID model of Stochastic Online Submodular Welfare Maximization and its special cases such as Online Stochastic Matching, Stochastic Ad-Words, and Stochastic Display Ads. At each time step, the algorithm sees the type of an online item and a fractional allocation of the item, then immediately allocates the item to an agent. We propose a metric called the convergence rate that measures the quality of SOCS algorithms in the above special cases. This is cleaner than most metrics in the related Online Correlated Selection (OCS) literature and may be of independent interest. We propose a Type Decomposition framework that reduces the design of SOCS algorithms to the easier special case of two-way SOCS. First, we sample a surrogate type whose fractional allocation is half-integer. The rounding is trivial for a one-way surrogate type fully allocated to one agent. For a two-way surrogate type split equally between two agents, we round it using a two-way SOCS. We design the distribution of surrogate types to get two-way types as often as possible, while respecting the original fractional allocation in expectation. Following this framework, we make progress on nu-merous problems including two open questions related to AdWords.

This paper considers the hidden-action model of the principal-agent problem, in which a principal incentivizes an agent to work on a project using a contract. We investigate whether contracts with bounded payments are learnable and approximately optimal. Our main results are two learning algorithms that can find a nearly optimal bounded contract using a polynomial number of queries, under two standard assumptions in the literature: a costlier action for the agent leads to a better outcome distribution for the principal, and the agent's cost/effort has diminishing returns. Our polynomial query complexity upper bound shows that standard assumptions are sufficient for achieving an exponential improvement upon the known lower bound for general instances. Unlike the existing algorithms which relied on discretizing the contract space, our algorithms directly learn the underlying outcome distributions. As for the approximate optimality of bounded contracts, we find that they could be far from optimal in terms of multiplicative or additive approximation, but satisfy a notion of mixed approximation.

Abstract. Mehta and Panigrahi ( FOCS 2012, IEEE, Piscataway, NJ, 2012, pp. 728–737) introduce the problem of online matching with stochastic rewards, where edges are associated with success probabilities and a match succeeds with the probability of the corresponding edge. It is one of the few online matching problems that have defied the randomized online primal dual framework by Devanur, Jain, and Kleinberg ( SODA 2013, SIAM, Philadelphia, 2013, pp. 101–107) thus far. This paper unlocks the power of randomized online primal dual in online matching with stochastic rewards by employing the configuration linear program rather than the standard matching linear program used in previous works. Our main result is a 0.572 competitive algorithm for the case of vanishing and unequal probabilities, improving the best previous bound of 0.534 by Mehta, Waggoner, and Zadimoghaddam ( SODA 2015, SIAM, Philadelphia, 2015, pp. 1388–1404) and, in fact, is even better than the best previous bound of 0.567 by Mehta and Panigrahi ( FOCS 2012, IEEE, Piscataway, NJ, 2012, pp. 728–737) for the more restricted case of vanishing and equal probabilities. For vanishing and equal probabilities, we get a better competitive ratio of 0.576. Our results further generalize to the vertex-weighted case due to the intrinsic robustness of the randomized online primal dual analysis.

We initiate the study of fairness among classes of agents in online bipartite matching where there is a given set of offline vertices (aka agents) and another set of vertices (aka items) that arrive online and must be matched irrevocably upon arrival. In this setting, agents are partitioned into a set of classes and the matching is required to be fair with respect to the classes. We adopt popular fairness notions (e.g. envy-freeness, proportionality, and maximin share) and their relaxations to this setting and study deterministic and randomized algorithms for matching indivisible items (leading to integral matchings) and for matching divisible items (leading to fractional matchings). For matching indivisible items, we propose an adaptive-priority-based algorithm, MATCH-AND-SHIFT, prove that it achieves (1/2)-approximation of both class envy-freeness up to one item and class maximin share fairness, and show that each guarantee is tight. For matching divisible items, we design a water-filling-based algorithm, EQUAL-FILLING, that achieves (1-1/e)-approximation of class envy-freeness and class proportionality; we prove (1-1/e) to be tight for class proportionality and establish a 3/4 upper bound on class envy-freeness.

Consider Myerson's optimal auction with respect to an inaccurate prior, e.g., estimated from data, which is an underestimation of the true value distribution. Can the auctioneer expect getting at least the optimal revenue w.r.t. the inaccurate prior since the true value distribution is larger? This so-called strong revenue monotonicity is known to be true for single-parameter auctions when the feasible allocations form a matroid. We find that strong revenue monotonicity fails to generalize beyond the matroid setting, and further show that auctions in the matroid setting are the only downward-closed auctions that satisfy strong revenue monotonicity. On the flip side, we recover an approximate version of strong revenue monotonicity that holds for all single-parameter auctions, even without downward-closedness. As applications, we get sample complexity upper bounds for single-parameter auctions under matroid constraints, downward-closed constraints, and general constraints. They improve the state-of-the-art upper bounds and are tight up to logarithmic factors.

We study the power of multiple choices in online stochastic matching. Despite a long line of research, existing algorithms still only consider two choices of offline neighbors for each online vertex because of the technical challenge in analyzing multiple choices. This paper introduces two approaches for designing and analyzing algorithms that use multiple choices. For unweighted and vertex-weighted matching, we adopt the online correlated selection (OCS) technique into the stochastic setting, and improve the competitive ratios to 0.716, from 0.711 and 0.7 respectively. For edge-weighted matching with free disposal, we propose the Top Half Sampling algorithm. We directly characterize the progress of the whole matching instead of individual vertices, through a differential inequality. This improves the competitive ratio to 0.706, breaking the 1−1/e barrier in this setting for the first time in the literature. Finally, for the harder edge-weighted problem without free disposal, we prove that no algorithms can be 0.703 competitive, separating this setting from the aforementioned three.

Online bipartite matching is one of the most fundamental problems in the online algorithms literature. Karp, Vazirani, and Vazirani (STOC 1990) gave an elegant algorithm for unweighted bipartite matching that achieves an optimal competitive ratio of 1-1/e . Aggarwal et al. (SODA 2011) later generalized their algorithm and analysis to the vertex-weighted case. Little is known, however, about the most general edge-weighted problem aside from the trivial 1/2-competitive greedy algorithm. In this article, we present the first online algorithm that breaks the long-standing 1/2 barrier and achieves a competitive ratio of at least 0.5086. In light of the hardness result of Kapralov, Post, and Vondrák (SODA 2013), which restricts beating a 1/2 competitive ratio for the more general monotone submodular welfare maximization problem, our result can be seen as strong evidence that edge-weighted bipartite matching is strictly easier than submodular welfare maximization in an online setting. The main ingredient in our online matching algorithm is a novel subroutine called online correlated selection (OCS), which takes a sequence of pairs of vertices as input and selects one vertex from each pair. Instead of using a fresh random bit to choose a vertex from each pair, the OCS negatively correlates decisions across different pairs and provides a quantitative measure on the level of correlation. We believe our OCS technique is of independent interest and will find further applications in other online optimization problems.

This paper explores a theory of generalization for learning problems on product distributions, complementing the existing learning theories in the sense that it does not rely on any complexity measures of the hypothesis classes. The main contributions are two general sample complexity bounds: (1) $\tilde{O} \big( \frac{nk}{\epsilon^2} \big)$ samples are sufficient and necessary for learning an $\epsilon$-optimal hypothesis in \emph{any problem} on an $n$-dimensional product distribution, whose marginals have finite supports of sizes at most $k$; (2) $\tilde{O} \big( \frac{n}{\epsilon^2} \big)$ samples are sufficient and necessary for any problem on $n$-dimensional product distributions if it satisfies a notion of strong monotonicity from the algorithmic game theory literature. As applications of these theories, we match the optimal sample complexity for single-parameter revenue maximization (Guo et al., STOC 2019), improve the state-of-the-art for multi-parameter revenue maximization (Gonczarowski and Weinberg, FOCS 2018) and prophet inequality (Correa et al., EC 2019; Rubinstein et al., ITCS 2020), and provide the first and tight sample complexity bound for Pandora’s problem.

This paper studies online correlated selection (OCS). Suppose that we receive a pair of elements in each round and select one of them. Can we select with negative correlation to be more effective than independent random selections? Our contributions are threefold. For semi-OCS, which considers the probability that an element remains unselected after appearing in$k$rounds, we give an optimal algorithm that minimizes this probability for all k. It leads to 0.536-competitive unweighted and vertex-weighted on-line bipartite matching algorithms that randomize over only two options in each round, improving the previous 0.508-competitive ratio by Fahrbach et al. (2020). Further, we develop the first multi-way semi-OCS that allows an arbitrary number of elements with arbitrary masses in each round. As an application, it rounds the Balance algorithm in unweighted and vertex-weighted online bi-partite matching to get a 0.593-competitive ratio. Finally, we study OCS, which further considers the probability that an element is unselected in any subset of rounds. We prove that the optimal “level of negative correlation” is between 0.167 and 0.25, improving the previous bounds of 0.109 and 1 by Fahrbach et al. (2020). Our OCS gives a 0.519-competitive edge-weighted online bipartite matching algorithm, improving the previous 0.508-competitive ratio by Fahrbach et al. (2020).

This paper introduces the targeted sampling model in optimal auction design. In this model, the seller may specify a quantile interval and sample from a buyer's prior restricted to the interval. This can be interpreted as allowing the seller to, for example, examine the top 40% bids from previous buyers with the same characteristics. The targeting power is quantified with a parameter Δ ∈ [0, 1] which lower bounds how small the quantile intervals could be. When Δ = 1, it degenerates to Cole and Roughgarden's model of i.i.d. samples; when it is the idealized case of Δ = 0, it degenerates to the model studied by [7]. For instance, for n buyers with bounded values in [0, 1], ~O(ε-1) targeted samples suffice while it is known that at least ~Ømega(n ε-2) i.i.d. samples are needed. In other words, targeted sampling with sufficient targeting power allows us to remove the linear dependence in n, and to improve the quadratic dependence in ε-1 to linear. In this work, we introduce new technical ingredients and show that the number of targeted samples sufficient for learning an ε-optimal auction is substantially smaller than the sample complexity of i.i.d. samples for the full spectrum of Δ ∈ [0, 1). Even with only mild targeting power, i.e., whenever Δ = o(1), our targeted sample complexity upper bounds are strictly smaller than the optimal sample complexity of i.i.d. samples.

We study the online stochastic matching problem. Consider a bipartite graph with offline vertices on one side, and with i.i.d.online vertices on the other side. The offline vertices and the distribution of online vertices are known to the algorithm beforehand. The realization of the online vertices, however, is revealed one at a time, upon which the algorithm immediately decides how to match it. For maximizing the cardinality of the matching, we give a 0.711-competitive online algorithm, which improves the best previous ratio of 0.706. When the offline vertices are weighted, we introduce a 0.7009-competitive online algorithm for maximizing the total weight of the matched offline vertices, which improves the best previous ratio of 0.662.

Today's IaaS clouds allow dynamic scaling of VMs allocated to a user, according to real-time demand of the user. There are two types of scaling: horizontal scaling (scale-out) by allocating more VM instances to the user, and vertical scaling (scale-up) by boosting resources of VMs owned by the user. It has been a daunting issue how to efficiently allocate the resources on physical servers to meet the scaling demand of users on the go, which achieves the best server utilization and user utility. An accompanying critical challenge is how to effectively charge the incremental resources, such that the economic benefits of both the cloud provider and cloud users are guaranteed. There has been online auction design dealing with dynamic VM provisioning, where the resource bids are not related to each other, failing to handle VM scaling where later bids may rely on earlier bids of the same user. As the first in the literature, this paper designs an efficient, truthful online auction for resource provisioning and pricing in the practical cases of dynamic VM scaling, where: (i) users bid for customized VMs to use in future durations, and can bid again in the following time to increase resources, indicating both scale-up and scale-out options; (ii) the cloud provider packs the demanded VMs on heterogeneous servers for energy cost minimization on the go. We carefully design resource prices maintained for each type of resource on each server to achieve threshold-based online allocation and charging, as well as a novel competitive analysis technique based on submodularity of the offline objective, to show a good competitive ratio is achieved. The efficacy of the online auction is validated through solid theoretical analysis and trace-driven simulations.

Karp, Vazirani, and Vazirani (STOC 1990) initiated the study of online bipartite matching, which has held a central role in online algorithms ever since. Of particular importance are the Ranking algorithm for integral matching and the Water-filling algorithm for fractional matching. Most algorithms in the literature can be viewed as adaptations of these two in the corresponding models. Recently, Huang et al. (SODA 2019, JACM 2020) introduced a more general model called fully online matching, which considers general graphs and allows all vertices to arrive online. They also generalized Ranking and Water-filling to fully online matching and gave some tight analysis: Ranking is Ω ≈ 0.567-competitive on bipartite graphs where the Ω-constant satisfies ΩeΩ=1, and Water-filling is 2-√2 ≈ 0.585-competitive on general graphs. We propose fully online matching algorithms strictly better than Ranking and Water-filling. For integral matching on bipartite graphs, we build on the online primal dual analysis of Ranking and Water-filling to design a 0.569-competitive hybrid algorithm called Balanced Ranking. To our knowledge, it is the first integral algorithm in the online matching literature that successfully integrates ideas from Water-filling. For fractional matching on general graphs, we give a 0.592-competitive algorithm called Eager Water-filling, which may match a vertex on its arrival. By contrast, the original Water-filling algorithm always matches vertices at their deadlines. Our result for fractional matching further shows a separation between fully online matching and the general vertex arrival model by Wang and Wong (ICALP 2015), due to an upper bound of 0.5914 in the latter model by Buchbinder, Segev, and Tkach (ESA 2017).

Abstract. Three decades ago, Karp, Vazirani, and Vazirani [ Proceedings of the 22 nd Annual ACM Symposium on Theory of Computing, 1990, pp. 352–358] defined the online matching problem and gave an optimal [Formula: see text]-competitive algorithm. Fifteen years later, Mehta et al. [ J. ACM, 54 (2007), pp. 22:1–22:19] introduced the first generalization called AdWords driven by online advertising and obtained the optimal [Formula: see text] competitive ratio in the special case of small bids. It has been open ever since whether there is an algorithm for general bids better than the 0.5-competitive greedy algorithm. This paper presents a 0.5016-competitive algorithm for AdWords, answering this open question on the positive end. The algorithm builds on several ingredients, including a combination of the online primal dual framework and the configuration linear program of matching problems recently explored by Huang and Zhang [ Proceedings of the 52 nd ACM Symposium on Theory of Computing, 2020], a novel formulation of AdWords which we call the panorama view, and a generalization of the online correlated selection by Fahrbach et al. [ Proceedings of the 61 st Annual IEEE Symposium on Foundations of Computer Science, 2020], which we call the panoramic online correlated selection.

Online bipartite matching is one of the most fundamental problems in the online algorithms literature. Karp, Vazirani, and Vazirani (STOC 1990) introduced an elegant algorithm for the unweighted bipartite matching that achieves an optimal competitive ratio of 1-1/e. Aggarwal et al. (SODA 2011) later generalized their algorithm and analysis to the vertex-weighted case. Little is known, however, about the most general edge-weighted problem aside from the trivial 1/2-competitive greedy algorithm. In this paper, we present the first online algorithm that breaks the long-standing 1/2barrier and achieves a competitive ratio of at least 0.5086. In light of the hardness result of Kapralov, Post, and Vondrák (SODA 2013) that restricts beating a 1/2competitive ratio for the more general problem of monotone submodular welfare maximization, our result can be seen as strong evidence that edge-weighted bipartite matching is strictly easier than submodular welfare maximization in the online setting. The main ingredient in our online matching algorithm is a novel subroutine called online correlated selection (OCS), which takes a sequence of pairs of vertices as input and selects one vertex from each pair. Instead of using a fresh random bit to choose a vertex from each pair, the OCS negatively correlates decisions across different pairs and provides a quantitative measure on the level of correlation. We believe our OCS technique is of independent interest and will find further applications in other online optimization problems.

We consider a generalization of the third degree price discrimination problem studied in [4](Bergemann et al., 2015), where an intermediary between the buyer and the seller can design market segments to maximize any linear combination of consumer surplus and seller revenue. Unlike in [4], we assume that the intermediary only has partial information about the buyer's value. We consider three different models of information, with increasing order of difficulty. In the first model, we assume that the intermediary's information allows him to construct a probability distribution of the buyer's value. Next we consider the sample complexity model, where we assume that the intermediary only sees samples from this distribution. Finally, we consider a bandit online learning model, where the intermediary can only observe past purchasing decisions of the buyer, rather than her exact value. For each of these models, we present algorithms to compute optimal or near optimal market segmentation.

Mehta and Panigrahi (FOCS 2012) introduce the problem of online matching with stochastic rewards, where edges are associated with success probabilities and a match succeeds with the probability of the corresponding edge. It is one of the few online matching problems that have defied the randomized online primal dual framework by Devanur, Jain, and Kleinberg (SODA 2013) thus far. This paper unlocks the power of randomized online primal dual in online matching with stochastic rewards by employing the configuration linear program rather than the standard matching linear program used in previous works. Our main result is a 0.572 competitive algorithm for the case of vanishing and unequal probabilities, improving the best previous bound of 0.534 by Mehta, Waggoner, and Zadimoghaddam (SODA 2015) and, in fact, is even better than the best previous bound of 0.567 by Mehta and Panigrahi (FOCS 2012) for the more restricted case of vanishing and equal probabilities. For vanishing and equal probabilities, we get a better competitive ratio of 0.576. Our results further generalize to the vertex-weighted case due to the intrinsic robustness of the randomized online primal dual analysis.

We introduce a fully online model of maximum cardinality matching in which all vertices arrive online. On the arrival of a vertex, its incident edges to previously arrived vertices are revealed. Each vertex has a deadline that is after all its neighbors’ arrivals. If a vertex remains unmatched until its deadline, then the algorithm must irrevocably either match it to an unmatched neighbor or leave it unmatched. The model generalizes the existing one-sided online model and is motivated by applications including ride-sharing platforms, real-estate agency, and so on. We show that the Ranking algorithm by Karp et al. (STOC 1990) is 0.5211-competitive in our fully online model for general graphs. Our analysis brings a novel charging mechanic into the randomized primal dual technique by Devanur et al. (SODA 2013), allowing a vertex other than the two endpoints of a matched edge to share the gain. To our knowledge, this is the first analysis of Ranking that beats 0.5 on general graphs in an online matching problem, a first step toward solving the open problem by Karp et al. (STOC 1990) about the optimality of Ranking on general graphs. If the graph is bipartite, then we show a tight competitive ratio ≈0.5671 of Ranking. Finally, we prove that the fully online model is strictly harder than the previous model as no online algorithm can be 0.6317 < 1- 1/e-competitive in our model, even for bipartite graphs.

Cloud computing has been widely adopted to support various computation services. A fundamental problem faced by cloud providers is how to efficiently allocate resources upon user requests and price the resource usage, in order to maximize resource efficiency and hence provider profit. Existing studies establish detailed performance models of cloud resource usage, and propose offline or online algorithms to decide allocation and pricing. Differently, we adopt a blackbox approach, and leverage model-free Deep Reinforcement Learning (DRL) to capture dynamics of cloud users and better characterize inherent connections between an optimal allocation/pricing policy and the states of the dynamic cloud system. The goal is to learn a policy that maximizes net profit of the cloud provider through trial and error, which is better than decisions made on explicit performance models. We combine long short-term memory (LSTM) units with fully-connected neural networks in our DRL to deal with online user arrivals, and adjust the output and update methods of basic DRL algorithms to address both resource allocation and pricing. Evaluation based on real-world datasets shows that our DRL approach outperforms basic DRL algorithms and state-of-theart white-box online cloud resource allocation/pricing algorithms significantly, in terms of both profit and the number of accepted users.

We introduce an $(ε, δ)$-jointly differentially private algorithm for packing problems. Our algorithm not only achieves the optimal trade-off between the privacy parameter $ε$ and the minimum supply requirement (up to logarithmic factors), but is also scalable in the sense that the running time is linear in the number of agents $n$. Previous algorithms either run in cubic time in $n$, or require a minimum supply per resource that is $\sqrt{n}$ times larger than the best possible.

Huang et al. (STOC 2018) introduced the fully online matching problem, a generalization of the classic online bipartite matching problem in that it allows all vertices to arrive online and considers general graphs. They showed that the ranking algorithm by Karp et al. (STOC 1990) is strictly better than 0.5-competitive and the problem is strictly harder than the online bipartite matching problem in that no algorithms can be (1 – 1/e)-competitive. This paper pins down two tight competitive ratios of classic algorithms for the fully online matching problem. For the fractional version of the problem, we show that a natural instantiation of the water-filling algorithm is 2 – ≈ 0.585-competitive, together with a matching hardness result. Interestingly, our hardness result applies to arbitrary algorithms in the edge-arrival models of the online matching problem, improving the state-of-art upper bound. For integral algorithms, we show a tight competitive ratio of ≈ 0.567 for the ranking algorithm on bipartite graphs, matching a hardness result by Huang et al. (STOC 2018).

This paper settles the sample complexity of single-parameter revenue maximization by showing matching upper and lower bounds, up to a poly-logarithmic factor, for all families of value distributions that have been considered in the literature. The upper bounds are unified under a novel framework, which builds on the strong revenue monotonicity by Devanur, Huang, and Psomas (STOC 2016), and an information theoretic argument. This is fundamentally different from the previous approaches that rely on either constructing an є-net of the mechanism space, explicitly or implicitly via statistical learning theory, or learning an approximately accurate version of the virtual values. To our knowledge, it is the first time information theoretical arguments are used to show sample complexity upper bounds, instead of lower bounds. Our lower bounds are also unified under a meta construction of hard instances.

We consider revenue maximization in online auction/pricing problems. A seller sells an identical item in each period to a new buyer, or a new set of buyers. For the online pricing problem, both when the arriving buyer bids or only responds to the posted price, we design algorithms whose regret bounds scale with the best fixed price in-hindsight, rather than the range of the values. Under the bidding model, we further show our algorithms achieve a revenue convergence rate that matches the offline sample complexity of the single-item single-buyer auction. We also show regret bounds that are scale free, and match the offline sample complexity, when comparing to a benchmark that requires a lower bound on the market share. We further expand our results beyond pricing to multi-buyer auctions, and obtain online learning algorithms for auctions, with convergence rates matching the known sample complexity upper bound of online single-item multi-buyer auctions. These results are obtained by generalizing the classical learning from experts and multi-armed bandit problems to their multi-scale versions. In this version, the reward of each action is in a different range, and the regret with respect to a given action scales with its own range, rather than the maximum range. We obtain almost optimal multi-scale regret bounds by introducing a new Online Mirror Descent (OMD) algorithm whose mirror map is the multi-scale version of the negative entropy function. We further generalize to the bandit setting by introducing the stochastic variant of this OMD algorithm.

We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and we prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time increases, and each online vertex matches the neighbor with the highest offer at its arrival.

We consider the online makespan minimization problem on identical machines. Chen and Vestjens (ORL 1997) show that the largest processing time first (LPT) algorithm is 1.5-competitive. For the special case of two machines, Noga and Seiden (TCS 2001) introduce the SLEEPY algorithm that achieves a competitive ratio of $(5 - \sqrt{5})/2 \approx 1.382$, matching the lower bound by Chen and Vestjens (ORL 1997). Furthermore, Noga and Seiden note that in many applications one can kill a job and restart it later, and they leave an open problem whether algorithms with restart can obtain better competitive ratios. We resolve this long-standing open problem on the positive end. Our algorithm has a natural rule for killing a processing job: a newly-arrived job replaces the smallest processing job if 1) the new job is larger than other pending jobs, 2) the new job is much larger than the processing one, and 3) the processed portion is small relative to the size of the new job. With appropriate choice of parameters, we show that our algorithm improves the 1.5 competitive ratio for the general case, and the 1.382 competitive ratio for the two-machine case.

We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time goes by, and each online vertex matches the neighbor with the highest offer at its arrival.

State-of-the-art cloud platforms adopt pay-as-you-go pricing, where users pay for the resources on demand according to occupation time. Simple and intuitive as it is, such a pricing scheme is a mismatch for new workloads today such as large-scale machine learning, whose completion time is hard to estimate beforehand. To supplement existing cloud pricing schemes, we propose an occupation-oblivious online pricing mechanism for cloud jobs without pre-specified time duration and for users who prefer a pre-determined cost for job execution. Our strategy posts unit resource prices upon user arrival and decides a fixed charge for completing the user's job, without the need to know how long the job is to occupy the requested resources. At the core of our design is a novel multi-armed bandit based online learning algorithm for estimating unknown input by exploration and exploitation of past resource sales, and deciding resource prices to maximize profit of the cloud provider in an online setting. Our online learning algorithm achieves a low regret sublinear with the time horizon, in terms of overall provider profit, compared with an omniscient benchmark. We also conduct trace-driven simulations to verify efficacy of the algorithm in real-world settings.

We consider the problem of learning optimal reserve price in repeated auctions against non-myopic bidders, who may bid strategically in order to gain in future rounds even if the single-round auctions are truthful. Previous algorithms, e.g., empirical pricing, do not provide non-trivial regret rounds in this setting in general. We introduce algorithms that obtain small regret against non-myopic bidders either when the market is large, i.e., no bidder appears in a constant fraction of the rounds, or when the bidders are impatient, i.e., they discount future utility by some factor mildly bounded away from one. Our approach carefully controls what information is revealed to each bidder, and builds on techniques from differentially private online learning as well as the recent line of works on jointly differentially private algorithms.

We consider privacy-preserving k-means clustering. For the objective of minimizing the Wasserstein distance between the output and the optimal solution, we show that there is a polynomial-time (ε,δ)-differentially private algorithm which, for any sufficiently large Φ2 well-separated datasets, outputs k centers that are within Wasserstein distance Ø(Φ2) from the optimal. This result improves the previous bounds by removing the dependence on ε, number of centers k, and dimension d. Further, we prove a matching lower bound that no (ε, δ)-differentially private algorithm can guarantee Wasserstein distance less than Ømega (Φ2) and, thus, our positive result is optimal up to a constant factor. For minimizing the k-means objective when the dimension d is bounded, we propose a polynomial-time private local search algorithm that outputs an αn-additive approximation when the size of the dataset is at least ~Ø (k3/2 · d · ε-1 · poly(α-1)).

We present an improved (ε, δ-jointly differentially private algorithm for packing problems. Our algorithm gives a feasible output that is approximately optimal up to an αn additive factor as long as the supply of each resource is at least , where m is the number of resources. This improves the previous result by Hsu et al. (SODA ’16), which requires the total supply to be at least Õ(m2/αε), and only guarantees approximate feasibility in terms of total violation. Further, we complement our algorithm with an almost matching hardness result, showing that supply is necessary for any (ε, δ)-jointly differentially private algorithm to compute an approximately optimal packing solution. Finally, we introduce an alternative approach that runs in linear time, is exactly truthful, can be implemented online, and can be ε-jointly differentially private, but requires a larger supply of each resource.

We introduce a fully online model of maximum cardinality matching in which all vertices arrive online. On the arrival of a vertex, its incident edges to previously-arrived vertices are revealed. Each vertex has a deadline that is after all its neighbors’ arrivals. If a vertex remains unmatched until its deadline, the algorithm must then irrevocably either match it to an unmatched neighbor, or leave it unmatched. The model generalizes the existing one-sided online model and is motivated by applications including ride-sharing platforms, real-estate agency, etc. We show that the Ranking algorithm by Karp et al. (STOC 1990) is 0.5211-competitive in our fully online model for general graphs. Our analysis brings a novel charging mechanic into the randomized primal dual technique by Devanur et al. (SODA 2013), allowing a vertex other than the two endpoints of a matched edge to share the gain. To our knowledge, this is the first analysis of Ranking that beats 0.5 on general graphs in an online matching problem, a first step towards solving the open problem by Karp et al. (STOC 1990) about the optimality of Ranking on general graphs. If the graph is bipartite, we show that the competitive ratio of Ranking is between 0.5541 and 0.5671. Finally, we prove that the fully online model is strictly harder than the previous model as no online algorithm can be 0.6317 < 1−1/e-competitive in our model even for bipartite graphs.

We study the problem of setting a price for a potential buyer with a valuation drawn from an unknown distribution $D$. The seller has “data” about $D$ in the form of $m \ge 1$ independent and identically distributed samples, and the algorithmic challenge is to use these samples to obtain expected revenue as close as possible to what could be achieved with advance knowledge of $D$. Our first set of results quantifies the number of samples $m$ that are necessary and sufficient to obtain a $(1-\epsilon)$-approximation. For example, for an unknown distribution that satisfies the monotone hazard rate (MHR) condition, we prove that $\tilde{\Theta}(\epsilon^{-3/2})$ samples are necessary and sufficient. Remarkably, this uses fewer samples than is necessary to accurately estimate the expected revenue obtained for such a distribution by even a single reserve price. We also prove essentially tight sample complexity bounds for regular distributions, bounded-support distributions, and a wide class of irregular distributions. Our lower bound approach, which applies to all randomized pricing strategies, borrows tools from differential privacy and information theory, and we believe it could find further applications in auction theory. Our second set of results considers the single-sample case. While no deterministic pricing strategy is better than $\tfrac{1}{2}$-approximate for regular distributions, for MHR distributions we show how to do better: there is a simple deterministic pricing strategy that guarantees expected revenue at least 0.589 times the maximum possible. We also prove that no deterministic pricing strategy achieves an approximation guarantee better than $\frac{e}{4} \approx .68$.

We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying matroid. In each round when a new element arrives, the algorithm may accept the new element into its feasible set and possibly remove elements from it, provided that the resulting set is still independent. The goal is to maximize the value of the final feasible set under some monotone submodular function, to which the algorithm has oracle access. For k -uniform matroids, we give a deterministic algorithm with competitive ratio at least 0.2959, and the ratio approaches 1/α ∞ ≈ 0.3178 as k approaches infinity, improving the previous best ratio of 0.25 by Chakrabarti and Kale (IPCO 2014), Buchbinder et al. (SODA 2015), and Chekuri et al. (ICALP 2015). We also show that our algorithm is optimal among a class of deterministic monotone algorithms that accept a new arriving element only if the objective is strictly increased. Further, we prove that no deterministic monotone algorithm can be strictly better than 0.25-competitive even for partition matroids, the most modest generalization of k -uniform matroids, matching the competitive ratio by Chakrabarti and Kale (IPCO 2014) and Chekuri et al. (ICALP 2015). Interestingly, we show that randomized algorithms are strictly more powerful by giving a (non-monotone) randomized algorithm for partition matroids with ratio 1/α ∞ ≈ 0.3178.

We consider the problem of online scheduling of jobs on unrelated machines with dynamic speed scaling to minimize the sum of energy and weighted flow-time. We give an algorithm with an almost optimal competitive ratio for arbitrary power functions. (No earlier results handled arbitrary power functions for unrelated machines.) For power functions of the form f ( s ) = s α for some constant α > 1, we get a competitive ratio of O (α / log α), improving upon a previous competitive ratio of O (α 2 ) by Anand et al. (2012), along with a matching lower bound of Ω(α / log α). Further, in the resource augmentation model, with a 1+ ϵ speed up, we give a 2(1/ϵ + 1) competitive algorithm, with essentially the same techniques, improving the bound of 1 + O (1/ϵ 2 ) by Gupta et al. (2010) and matching the bound of Anand et al. (2012) for the special case of fixed speed unrelated machines. Unlike the previous results most of which used an amortized local competitiveness argument or dual fitting methods, we use a primal-dual method, which is useful not only to analyze the algorithms but also to design the algorithm itself.

We consider revenue maximization in online auctions and pricing. A seller sells an identical item in each period to a new buyer, or a new set of buyers. For the online posted pricing problem, we show regret bounds that scale with the best fixed price, rather than the range of the values. We also show regret bounds that are almost scale free, and match the offline sample complexity, when comparing to a benchmark that requires a lower bound on the market share. These results are obtained by generalizing the classical learning from experts and multi-armed bandit problems to their multi-scale versions. In this version, the reward of each action is in a different range, and the regret w.r.t. a given action scales with its own range, rather than the maximum range.

We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying matroid. In each round when a new element arrives, the algorithm may accept the new element into its feasible set and possibly remove elements from it, provided that the resulting set is still independent. The goal is to maximize the value of the final feasible set under some monotone submodular function, to which the algorithm has oracle access. For k-uniform matroids, we give a deterministic algorithm with competitive ratio at least 0.2959, and the ratio approaches as k approaches infinity, improving the previous best ratio of 0.25 by Chakrabarti and Kale (IPCO 2014), Buchbinder et al. (SODA 2015) and Chekuri et al. (ICALP 2015). We also show that our algorithm is optimal among a class of deterministic monotone algorithms that accept a new arriving element only if the objective is strictly increased. Further, we prove that no deterministic monotone algorithm can be strictly better than 0.25-competitive even for partition matroids, the most modest generalization of k-uniform matroids, matching the competitive ratio by Chakrabarti and Kale (IPCO 2014) and Chekuri et al. (ICALP 2015). Interestingly, we show that randomized algorithms are strictly more powerful by giving a (non-monotone) randomized algorithm for partition matroids with ratio Finally, our techniques can be extended to a more general problem that generalizes both the online sub- modular maximization problem and the online bipartite matching problem with free disposal. Using the techniques developed in this paper, we give constant- competitive algorithms for the submodular online bipartite matching problem.

With the recent advent of network functions virtualization (NFV), enterprises and businesses are looking into network service provisioning through the service chains of virtual network functions (VNFs), instead of relying on dedicated hardware middleboxes. Accompanying this trend, an NFV market is emerging, where NFV service providers create VNF instances, assemble VNF service chains, and sell them for the use of customers, using resources (computing, bandwidth) that they own or rent from other resource suppliers. Efficient service chain provisioning and pricing mechanisms are still missing, to charge assembled service chains according to demand and the supply of resources at any time. We propose an online stochastic auction mechanism for on-demand service chain provisioning and pricing at an NFV provider. Our auction takes in buy bids for service chains from multiple customers and sell bids from various resource suppliers to supplement the NFV provider's geo-distributed resource pool, with resource occupation/contribution durations. We extend online primal-dual optimization framework for handling both buyers and sellers, with a new competitive analysis. The online mechanism maximizes the expected social welfare of the NFV ecosystem (the NFV provider, customers and resource suppliers) with a good competitive ratio as compared with the expected offline optimal social welfare, while guaranteeing truthfulness in bidding, individual rationality for both buyers and sellers, and polynomial time for computation. We evaluate our mechanism through trace-driven simulation studies, and demonstrate a close-to-offline-optimal performance in expected social welfare under realistic settings.

Auction design has recently been studied for dynamic resource bundling and virtual machine (VM) provisioning in IaaS clouds, but is mostly restricted to one-shot or offline setting. This paper targets a more realistic case of online VM auction design, where: 1) cloud users bid for resources into the future to assemble customized VMs with desired occupation durations, possibly located in different data centers; 2) the cloud provider dynamically packs multiple types of resources on heterogeneous physical machines (servers) into the requested VMs; 3) the operational costs of servers are considered in resource allocation; and 4) both social welfare and the cloud provider's net profit are to be maximized over the system running span. We design truthful, polynomial time auctions to achieve social welfare maximization and/or the provider's profit maximization with good competitive ratios. Our mechanisms consist of two main modules: 1) an online primal-dual optimization framework for VM allocation to maximize the social welfare with server costs, and for revealing the payments through the dual variables to guarantee truthfulness and 2) a randomized reduction algorithm to convert the social welfare maximizing auctions to ones that provide a maximal expected profit for the provider, with competitive ratios comparable to those for social welfare. We adopt a new application of Fenchel duality in our primal-dual framework, which provides richer structures for convex programs than the commonly used Lagrangian duality, and our optimization framework is general and expressive enough to handle various convex server cost functions. The efficacy of the online auctions is validated through careful theoretical analysis and trace-driven simulation studies.

This paper studies the cloud market for computing jobs with completion deadlines, and designs efficient online auctions for cloud resource provisioning. A cloud user bids for future cloud resources to execute its job. Each bid includes: 1) a utility, reflecting the amount that the user is willing to pay for executing its job and 2) a soft deadline, specifying the preferred finish time of the job, as well as a penalty function that characterizes the cost of violating the deadline. We target cloud job auctions that executes in an online fashion, runs in polynomial time, provides truthfulness guarantee, and achieves optimal social welfare for the cloud ecosystem. Towards these goals, we leverage the following classic and new auction design techniques. First, we adapt the posted pricing auction framework for eliciting truthful online bids. Second, we address the challenge posed by soft deadline constraints through a new technique of compact exponential-size LPs coupled with dual separation oracles. Third, we develop efficient social welfare approximation algorithms using the classic primal-dual framework based on both LP duals and Fenchel duals. Empirical studies driven by real-world traces verify the efficacy of our online auction design.

We present online algorithms for covering and packing problems with (non-linear) convex objectives. The convex covering problem is defined as: minxϵR+nf(x) s.t. Ax ≥ 1, where f:R+n→ R+is a monotone convex function, and A is an m×n matrix with non-negative entries. In the online version, a new row of the constraint matrix, representing a new covering constraint, is revealed in each step and the algorithm is required to maintain a feasible and monotonically non-decreasing assignment x over time. We also consider a convex packing problem defined as: maxyϵR+mΣj=1myj - g(ATy), where g:R+n→R+is a monotone convex function. In the online version, each variable yj arrives online and the algorithm must decide the value of yj on its arrival. This represents the Fenchel dual of the convex covering program, when g is the convex conjugate of f. We use a primal-dual approach to give online algorithms for these generic problems, and use them to simplify, unify, and improve upon previous results for several applications.

We present a general method for approximately solving convex programs defined by private information from agents, when the solution can be naturally partitioned among the agents. This class of problems includes multi-commodity flow problems, general allocation problems, and multi-dimensional knapsack problems, among other examples. The accuracy of our algorithm depends on the number of coupling constraints, which bind multiple agents. On the other hand, our accuracy is nearly independent of the number of variables, and in many cases, actually improves as the number of agents increases. A special case of our result (solving general allocation problems beyond “Gross Substitute” preferences) resolves the main open problem from [Hsu et al. STOC 2014]. We also consider strategic agents who have preferences over their part of the solution. For any convex program in our class that maximizes social welfare, we show how to create an approximately dominant strategy truthful mechanism, approximately maximizing welfare. The central idea is to charge agents prices based on the approximately optimal dual variables, which are themselves computed under differential privacy. Our results substantially broaden the class of problems that are known to be solvable under privacy and/or incentive constraints.

Traditionally, the Bayesian optimal auction design problem has been considered either when the bidder values are i.i.d, or when each bidder is individually identifiable via her value distribution. The latter is a reasonable approach when the bidders can be classified into a few categories, but there are many instances where the classification of bidders is a continuum. For example, the classification of the bidders may be based on their annual income, their propensity to buy an item based on past behavior, or in the case of ad auctions, the click through rate of their ads. We introduce an alternate model that captures this aspect, where bidders are a priori identical, but can be distinguished based (only) on some side information the auctioneer obtains at the time of the auction. We extend the sample complexity approach of Dhangwatnotai et al. and Cole and Roughgarden to this model and obtain almost matching upper and lower bounds. As an aside, we obtain a revenue monotonicity lemma which may be of independent interest. We also show how to use Empirical Risk Minimization techniques to improve the sample complexity bound of Cole and Roughgarden for the non-identical but independent value distribution case.

We consider a private variant of the classical allocation problem: given $k$ goods and $n$ agents with private valuation functions over bundles of goods, how can we allocate goods to agents to maximize social welfare? An important special case is when agents desire at most one good, and specify their (private) value for each good: in this case, the problem is exactly the maximum-weight matching problem in a bipartite graph. Private matching and allocation problems have not been considered in the differential privacy literature for a good reason: they are plainly impossible to solve under differential privacy. Informally, the allocation must match agents to their preferred goods in order to maximize social welfare, but this preference is exactly what agents wish to hide! Therefore, we consider the problem under the relaxed constraint of joint differential privacy: for any agent $i$, no coalition of agents excluding $i$ should be able to learn about the valuation function of agent $i$. In this setting, the full allocation is no longer published---instead, each agent is told what good to receive. We first show that if there are several identical copies of each good, it is possible to efficiently and accurately solve the matching problem while guaranteeing joint differential privacy. We then consider the more general allocation problem where bidder valuations satisfy the gross substitutes condition. Finally, we prove that the allocation problem cannot be solved to nontrivial accuracy under joint differential privacy without requiring multiple copies of each type of good.

We study the problem of setting a price for a potential buyer with a valuation drawn from an unknown distribution D. The seller has "data" about D in the form of m ≥ 1 i.i.d. samples, and the algorithmic challenge is to use these samples to obtain expected revenue as close as possible to what could be achieved with advance knowledge of D.

Auction design has recently been studied for dynamic resource bundling and VM provisioning in IaaS clouds, but is mostly restricted to the one-shot or offline setting. This work targets a more realistic case of online VM auction design, where: (i) cloud users bid for resources into the future to assemble customized VMs with desired occupation durations; (ii) the cloud provider dynamically packs multiple types of resources on heterogeneous physical machines (servers) into the requested VMs; (iii) the operational costs of servers are considered in resource allocation; (iv) both social welfare and the cloud provider's net profit are to be maximized over the system running span. We design truthful, polynomial time auctions to achieve social welfare maximization and/or the provider's profit maximization with good competitive ratios. Our mechanisms consist of two main modules: (1) an online primal-dual optimization framework for VM allocation to maximize the social welfare with server costs, and for revealing the payments through the dual variables to guarantee truthfulness; and (2) a randomized reduction algorithm to convert the social welfare maximizing auctions to ones that provide a maximal expected profit for the provider, with competitive ratios comparable to those for social welfare. We adopt a new application of Fenchel duality in our primal-dual framework, which provides richer structures for convex programs than the commonly used Lagrangian duality, and our optimization framework is general and expressive enough to handle various convex server cost functions. The efficacy of the online auctions is validated through careful theoretical analysis and trace-driven simulation studies.

We study online combinatorial auctions with production costs proposed by Blum et al. [4] using the online primal dual framework. In this model, buyers arrive online, and the seller can produce multiple copies of each item subject to a non-decreasing marginal cost per copy. The goal is to allocate items to maximize social welfare less total production cost. For arbitrary (strictly convex and differentiable) production cost functions, we characterize the optimal competitive ratio achievable by online mechanisms/algorithms. We show that online posted pricing mechanisms, which are incentive compatible, can achieve competitive ratios arbitrarily close to the optimal, and construct lower bound instances on which no online algorithms, not necessarily incentive compatible, can do better. Our positive results improve or match the results in several previous work, e.g., Bartal et al. [a], Blum et al. [4], and Buchbinder and Conen [6]. Our lower bounds apply to randomized algorithms and resolve an open problem by Buchbinder and Gonen [6].

In recent years, there has been a growing interest in speed scaling algorithms, where a set of jobs need to be scheduled on a machine with variable speed so as to optimize the flow-times of the jobs and the energy consumed by the machine. A series of results have culminated in constant-competitive algorithms for this problem in the clairvoyant model, i.e., when job parameters are revealed on releasing a job (Bansal, Pruhs, and Stein, SODA 2007; Bansal, Chan, and Pruhs, SODA 2009). Our main contribution in this paper is the first constant-competitive speed scaling algorithm in the non-clairvoyant model, which is typically used in the scheduling literature to model practical settings where job volume is revealed only after the job has been completely processed. Unlike in the clairvoyant model, the speed scaling problem in the non-clairvoyant model is non-trivial even for a single job. Our non-clairvoyant algorithm is defined by using the existing clairvoyant algorithm in a novel inductive way, which then leads to an inductive analytical tool that may be of independent interest for other online optimization problems. We also give additional algorithmic results and lower bounds for speed scaling on multiple identical parallel machines.

A coverage function $f$ over a ground set $[m]$ is associated with a universe $U$ of weighted elements and $m$ sets $A_1,\ldots,A_m \subseteq U$, and for any $T\subseteq [m]$, $f(T)$ is defined as the total weight of the elements in the union $\cup_{j\in T} A_j$. Coverage functions are an important special case of submodular functions, and arise in many applications, for instance, as a class of utility functions of agents in combinatorial auctions. Naïve representations of coverage functions have size exponential in $m$, and in algorithmic applications, an access to a value oracle is assumed. In this paper, we ask whether one can recognize if a given oracle is that of a coverage function or not. We demonstrate an algorithm which makes $O(m|U|)$ queries to an oracle of a coverage function and completely reconstructs it. This is polynomial time whenever $|U|$ is polynomially bounded implying the function has a succinct description. To complement the above result, we show a negative result. We prove that “noncoverageness” needs large certificates---there exists a function which is not coverage and yet any algorithm making fewer than $2^{m-1}$ queries cannot distinguish this function from some coverage function. Our positive result shows that the property of coverageness has $O(m|U|)$-query proximity oblivious testers, while our negative result shows an exponential lower bound. We believe our lower bound also goes through for general property testers, and provide some evidence of the same.

We consider the problem of online scheduling of jobs on unrelated machines with dynamic speed scaling to minimize the sum of energy and weighted flow time. We give an algorithm with an almost optimal competitive ratio for arbitrary power functions. (No earlier results handled arbitrary power functions for minimizing flow time plus energy with unrelated machines.) For power functions of the form f(s) = s^alpha for some constant alpha > 1, we get a competitive ratio of O(alpha/log(alpha)), improving upon a previous competitive ratio of O(alpha^2) by Anand et al., along with a matching lower bound of . Further, in the resource augmentation model, with a 1+epsilon speed up, we give a O(1/epsilon) competitive algorithm, with essentially the same techniques, improving the bound of O(1/epsilon^2) by Gupta et al. and matching the bound of Anand et al. [3] for the special case of fixed speed unrelated machines. Unlike the previous results most of which used an amortized local competitiveness argument or dual fitting methods, we use a primal-dual method, which is useful not only to analyze the algorithms but also to design the algorithm itself. Copyright © 2014 by the Society for Industrial and Applied Mathematics.

We consider the problem of privately answering queries defined on databases which are collections of points belonging to some metric space. We give simple, computationally efficient algorithms for answering distance queries defined over an arbitrary metric. Distance queries are specified by points in the metric space, and ask for the average distance from the query point to the points contained in the database, according to the specified metric. Our algorithms run efficiently in the database size and the dimension of the space, and operate in both the online query release setting, and the offline setting in which they must in polynomial time generate a fixed data structure which can answer all queries of interest. This represents one of the first subclasses of linear queries for which efficient algorithms are known for the private query release problem, circumventing known hardness results for generic linear queries.

We consider a private variant of the classical allocation problem: given k goods and n agents with individual, private valuation functions over bundles of goods, how can we partition the goods amongst the agents to maximize social welfare? An important special case is when each agent desires at most one good, and specifies her (private) value for each good: in this case, the problem is exactly the maximum-weight matching problem in a bipartite graph.

In the context of online ad serving, display ads may appear on different types of web-pages, where each page includes several ad slots and therefore multiple ads can be shown on each page. The set of ads that can be assigned to ad slots of the same page needs to satisfy various pre-specified constraints including exclusion constraints, diversity constraints, and the like. Upon arrival of a user, the ad serving system needs to allocate a set of ads to the current web-page respecting these per-page allocation constraints. Previous slot-based settings ignore the important concept of a page, and may lead to highly suboptimal results in general. In this paper, motivated by these applications in display advertising and inspired by the submodular welfare maximization problem with online bidders, we study a general class of page-based ad allocation problems, present the first (tight) constant-factor approximation algorithms for these problems, and confirm the performance of our algorithms experimentally on real-world data sets.

We provide a Polynomial Time Approximation Scheme (PTAS) for the Bayesian optimal multi-item multi-bidder auction problem under two conditions. First, bidders are independent, have additive valuations and are from the same population. Second, every bidder's value distributions of items are independent but not necessarily identical monotone hazard rate (MHR) distributions. For non-i.i.d. bidders, we also provide a PTAS when the number of bidders is small. Prior to our work, even for a single bidder, only constant factor approximations are known. Another appealing feature of our mechanism is the simple allocation rule. Indeed, the mechanism we use is either the second-price auction with reserve price on every item individually, or VCG allocation with a few outlying items that requires additional treatments. It is surprising that such simple allocation rules suffice to obtain nearly optimal revenue.

We consider the item pricing problem for revenue maximization, where a single seller with multiple distinct items caters to multiple buyers with unknown subadditive valuation functions who arrive in a sequence. The seller sets the prices on individual items, and we design randomized pricing strategies to maximize expected revenue. We consider dynamic uniform strategies, which can change the price upon the arrival of each buyer but the price on all unsold items is the same at all times, and static nonuniform strategies, which can assign different prices to different items but can never change it after setting it initially. We design pricing strategies that guarantee poly-logarithmic (in number of items) approximation to maximum possible social welfare, which is an upper bound on revenue. We also show that any static uniform pricing strategy cannot yield such approximation, thus highlighting a large gap between the powers of dynamic and static pricing. Finally, our pricing strategies imply poly-logarithmic approximation for revenue-optimal incentive compatible mechanisms, in multiparameter combinatorial auctions with subaddititve buyer valuations, which is the best known guarantee given by efficient mechanisms for both prior-free and Bayesian settings.

In this paper we show that for any mechanism design problem with the objective of maximizing social welfare, the exponential mechanism can be implemented as a truthful mechanism while still preserving differential privacy. Our instantiation of the exponential mechanism can be interpreted as a generalization of the VCG mechanism in the sense that the VCG mechanism is the extreme case when the privacy parameter goes to infinity. To our knowledge, this is the first general tool for designing mechanisms that are both truthful and differentially private.

We study the generalized sorting problem where we are given a set of n elements to be sorted but only a subset of all possible pairwise element comparisons is allowed. The goal is to determine the sorted order using the smallest possible number of allowed comparisons. The generalized sorting problem may be equivalently viewed as follows. Given an undirected graph G(V, E) where V is the set of elements to be sorted and E defines the set of allowed comparisons, adaptively find the smallest subset E' ⊆ E of edges to probe such that the directed graph induced by E' contains a Hamiltonian path. When G is a complete graph, we get the standard sorting problem, and it is well-known that Θ(n log n) comparisons are necessary and sufficient. An extensively studied special case of the generalized sorting problem is the nuts and bolts problem where the allowed comparison graph is a complete bipartite graph between two equal-size sets. It is known that for this special case also, there is a deterministic algorithm that sorts using Θ(n log n) comparisons. However, when the allowed comparison graph is arbitrary, to our knowledge, no bound better than the trivial Õ(n2) bound is known. Our main result is a randomized algorithm that sorts any allowed comparison graph using O(n3/2) comparisons with high probability (provided the input is sortable). We also study the sorting problem in randomly generated allowed comparison graphs, and show that when the edge probability is p, Õ(min{p2/n, n3/2√p}) comparisons suffice on average to sort.

Very recently, Hartline and Lucier [14] studied single-parameter mechanism design problems in the Bayesian setting. They proposed a black-box reduction that converted Bayesian approximation algorithms into Bayesian-Incentive-Compatible (BIC) mechanisms while preserving social welfare. It remains a major open question if one can find similar reduction in the more important multi-parameter setting. In this paper, we give positive answer to this question when the prior distribution has finite and small support. We propose a black-box reduction for designing BIC multi-parameter mechanisms. The reduction converts any algorithm into an ε-BIC mechanism with only marginal loss in social welfare. As a result, for combinatorial auctions with sub-additive agents we get an ε-BIC mechanism that achieves constant approximation.

We study the ITEM PRICING problem for revenue maximization in the limited supply setting, where a single seller with n distinct items caters to m buyers with unknown subadditive valuation functions who arrive in a sequence. The seller sets the prices on individual items. Each buyer buys a subset of yet unsold items that maximizes her utility. Our goal is to design pricing strategies that guarantee an expected revenue that is within a small multiplicative factor of the optimal social welfare an upper bound on the maximum revenue that can be generated by any pricing mechanism. Most earlier work has focused on the unlimited supply setting, where selling an item to a buyer does not affect the availability of the item to the future buyers. Recently, Balcan et. al. studied the limited supply setting, giving a randomized pricing strategy that achieves a 2O(?(log n log log n)-approximation; their strategy assigns a single price to all items (uniform pricing), and never changes it (static pricing). They also showed that no pricing strategy that is both static and uniform can give better than 2??(log1/4 n)-approximation. Our first result is a strengthening of the lower bound on approximation achievable by static uniform pricing to 2??(log n). We then design dynamic uniform pricing strategies (all items are identically priced but item prices can change over time), that achieves O(log2n)-approximation, and also show a lower bound of ? ((log n/ log log n)2) for this class of strategies. Our strategies are simple to implement, and in particular, one strategy is to smoothly decrease the price over time. We also design a static nonuniform pricing strategy (different items can have different prices but prices do not change over time), that give poly-logarithmic approximation in a more restricted setting with few buyers. Thus in the limited supply setting, our results highlight a strong separation between the power of dynamic and non-uniform pricing strategies versus static uniform pricing strategy. To our knowledge, this is the first non-trivial analysis of dynamic and non-uniform pricing schemes for revenue maximization in a setting with multiple distinct items.