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Matching, capturing allocation of items to unit-demand buyers, or tasks to workers, or pairs of collaborators, is a central problem in economics. Indeed, the growing prevalence of matching-based markets, many of which online in nature, has motivated much research in economics, operations research, computer science, and their intersection. This brief survey is meant as an introduction to the area of online matching, with an emphasis on recent trends, both technical and conceptual., Comment: Also in SIGECOM Exchanges

We study multi-buyer multi-item sequential item pricing mechanisms for revenue maximization with the goal of approximating a natural fractional relaxation -- the ex ante optimal revenue. We assume that buyers' values are subadditive but make no assumptions on the value distributions. While the optimal revenue, and therefore also the ex ante benchmark, is inapproximable by any simple mechanism in this context, previous work has shown that a weaker benchmark that optimizes over so-called ``buy-many" mechanisms can be approximable. Approximations are known, in particular, for settings with either a single buyer or many unit-demand buyers. We extend these results to the much broader setting of many subadditive buyers. We show that the ex ante buy-many revenue can be approximated via sequential item pricings to within an $O(\log^2 m)$ factor, where $m$ is the number of items. We also show that a logarithmic dependence on $m$ is necessary. Our approximation is achieved through the construction of a new multi-dimensional Online Contention Resolution Scheme (OCRS), that provides an online rounding of the optimal ex ante solution. Chawla et al. arXiv:2204.01962 previously constructed an OCRS for revenue for unit-demand buyers, but their construction relied heavily on the ``almost single dimensional" nature of unit-demand values. Prior to that work, OCRSes have only been studied in the context of social welfare maximization for single-parameter buyers. For the welfare objective, constant-factor approximations have been demonstrated for a wide range of combinatorial constraints on item allocations and classes of buyer valuation functions. Our work opens up the possibility of a similar success story for revenue maximization., Comment: 39 pages

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.

Causal discovery with latent variables is a crucial but challenging task. Despite the emergence of numerous methods aimed at addressing this challenge, they are not fully identified to the structure that two observed variables are influenced by one latent variable and there might be a directed edge in between. Interestingly, we notice that this structure can be identified through the utilization of higher-order cumulants. By leveraging the higher-order cumulants of non-Gaussian data, we provide an analytical solution for estimating the causal coefficients or their ratios. With the estimated (ratios of) causal coefficients, we propose a novel approach to identify the existence of a causal edge between two observed variables subject to latent variable influence. In case when such a causal edge exits, we introduce an asymmetry criterion to determine the causal direction. The experimental results demonstrate the effectiveness of our proposed method., Comment: Accepted by AAAI 2024

Mehta and Panigrahi (2012) proposed Online Matching with Stochastic Rewards, which generalizes the Online Bipartite Matching problem of Karp, Vazirani, and Vazirani (1990) by associating the edges with success probabilities. This new feature captures the pay-per-click model in online advertising. Recently, Huang and Zhang (2020) studied this problem under the online primal dual framework using the Configuration Linear Program (LP), and got the best known competitive ratios of the Stochastic Balance algorithm. Their work suggests that the more expressive Configuration LP is more suitable for this problem than the Matching LP. This paper advances the theory of Configuration LP in two directions. Our technical contribution includes a characterization of the joint matching outcome of an offline vertex and \emph{all its neighbors}. This characterization may be of independent interest, and is aligned with the spirit of Configuration LP. By contrast, previous analyses of Ranking generally focus on only one neighbor. Second, we designed a Stochastic Configuration LP that captures a stochastic benchmark proposed by Goyal and Udwani (2020), who used a Path-based LP. The Stochastic Configuration LP is smaller and simpler than the Path-based LP. Moreover, using the new LP we improved the competitive ratio of Stochastic Balance from $0.596$ to $0.611$ when the success probabilities are infinitesimal, and to $0.613$ when the success probabilities are further equal.

We show that a simple greedy algorithm is $4.75$ probability-competitive for the Laminar Matroid Secretary Problem, improving the $3\sqrt{3} \approx 5.196$-competitive algorithm based on the forbidden sets technique (Soto, Turkieltaub, and Verdugo, 2018).

We analyze union-find using potential functions motivated by continuous algorithms, and give alternate proofs of the $O(\log\log{n})$, $O(\log^{*}n)$, $O(\log^{**}n)$, and $O(\alpha(n))$ amortized cost upper bounds. The proof of the $O(\log\log{n})$ amortized bound goes as follows. Let each node's potential be the square root of its size, i.e., the size of the subtree rooted from it. The overall potential increase is $O(n)$ because the node sizes increase geometrically along any tree path. When compressing a path, each node on the path satisfies that either its potential decreases by $\Omega(1)$, or its child's size along the path is less than the square root of its size: this can happen at most $O(\log\log{n})$ times along any tree path., Comment: 13 pages, 1 figure

Causal discovery with latent confounders is an important but challenging task in many scientific areas. Despite the success of some overcomplete independent component analysis (OICA) based methods in certain domains, they are computationally expensive and can easily get stuck into local optima. We notice that interestingly, by making use of higher-order cumulants, there exists a closed-form solution to OICA in specific cases, e.g., when the mixing procedure follows the One-Latent-Component structure. In light of the power of the closed-form solution to OICA corresponding to the One-Latent-Component structure, we formulate a way to estimate the mixing matrix using the higher-order cumulants, and further propose the testable One-Latent-Component condition to identify the latent variables and determine causal orders. By iteratively removing the share identified latent components, we successfully extend the results on the One-Latent-Component structure to the Multi-Latent-Component structure and finally provide a practical and asymptotically correct algorithm to learn the causal structure with latent variables. Experimental results illustrate the asymptotic correctness and effectiveness of the proposed method., Comment: Accepted by ICML 2023

This paper presents a novel solution to address the challenges in achieving energy efficiency and cooperation for collision avoidance in UAV swarms. The proposed method combines Artificial Potential Field (APF) and Particle Swarm Optimization (PSO) techniques. APF provides environmental awareness and implicit coordination to UAVs, while PSO searches for collision-free and energy-efficient trajectories for each UAV in a decentralized manner under the implicit coordination. This decentralized approach is achieved by minimizing a novel cost function that leverages the advantages of the active contour model from image processing. Additionally, future trajectories are predicted by approximating the minima of the novel cost function using calculus of variation, which enables proactive actions and defines the initial conditions for PSO. We propose a two-branch trajectory planning framework that ensures UAVs only change altitudes when necessary for energy considerations. Extensive experiments are conducted to evaluate the effectiveness and efficiency of our method in various situations.

We study the fine-grained complexity of graph connectivity problems in unweighted undirected graphs. Recent development shows that all variants of edge connectivity problems, including single-source-single-sink, global, Steiner, single-source, and all-pairs connectivity, are solvable in $m^{1+o(1)}$ time, collapsing the complexity of these problems into the almost-linear-time regime. While, historically, vertex connectivity has been much harder, the recent results showed that both single-source-single-sink and global vertex connectivity can be solved in $m^{1+o(1)}$ time, raising the hope of putting all variants of vertex connectivity problems into the almost-linear-time regime too. We show that this hope is impossible, assuming conjectures on finding 4-cliques. Moreover, we essentially settle the complexity landscape by giving tight bounds for combinatorial algorithms in dense graphs. There are three separate regimes: (1) all-pairs and Steiner vertex connectivity have complexity $\hat{\Theta}(n^{4})$, (2) single-source vertex connectivity has complexity $\hat{\Theta}(n^{3})$, and (3) single-source-single-sink and global vertex connectivity have complexity $\hat{\Theta}(n^{2})$. For graphs with general density, we obtain tight bounds of $\hat{\Theta}(m^{2})$, $\hat{\Theta}(m^{1.5})$, $\hat{\Theta}(m)$, respectively, assuming Gomory-Hu trees for element connectivity can be computed in almost-linear time., Comment: 23 pages, 4 figures

All-gather collective communication is one of the most important communication primitives in parallel and distributed computation, which plays an essential role in many HPC applications such as distributed Deep Learning (DL) with model and hybrid parallelism. To solve the communication bottleneck of All-gather, optical interconnection network can provide unprecedented high bandwidth and reliability for data transfer among the distributed nodes. However, most traditional All-gather algorithms are designed for electrical interconnection, which cannot fit well for optical interconnect systems, resulting in poor performance. This paper proposes an efficient scheme, called OpTree, for All-gather operation on optical interconnect systems. OpTree derives an optimal $m$-ary tree corresponding to the optimal number of communication stages, achieving minimum communication time. We further analyze and compare the communication steps of OpTree with existing All-gather algorithms. Theoretical results exhibit that OpTree requires much less number of communication steps than existing All-gather algorithms on optical interconnect systems. Simulation results show that OpTree can reduce communication time by 72.21%, 94.30%, and 88.58%, respectively, compared with three existing All-gather schemes, WRHT, Ring, and NE., Comment: This paper is under review at a conference

The maximization of Nash welfare, which equals the geometric mean of agents' utilities, is widely studied because it balances efficiency and fairness in resource allocation problems. Banerjee, Gkatzelis, Gorokh, and Jin (2022) recently introduced the model of online Nash welfare maximization for $T$ divisible items and $N$ agents with additive utilities with predictions of each agent's utility for receiving all items. They gave online algorithms whose competitive ratios are logarithmic. We initiate the study of online Nash welfare maximization \emph{without predictions}, assuming either that the agents' utilities for receiving all items differ by a bounded ratio, or that their utilities for the Nash welfare maximizing allocation differ by a bounded ratio. We design online algorithms whose competitive ratios only depend on the logarithms of the aforementioned ratios of agents' utilities and the number of agents.

Given a spectrally sparse signal $\mathbf{y} = \sum_{i=1}^s x_i\mathbf{f}(\tau_i) \in \mathbb{C}^{2n+1}$ consisting of $s$ complex sinusoids, we consider the super-resolution problem, which is about estimating frequency components $\{\tau_i\}_{i=1}^s$ of $\mathbf y$. We consider the OMP-type algorithms for super-resolution, which is more efficient than other approaches based on Semi-Definite Programming. Our analysis shows that a two-stage algorithm with OMP initialization can recover frequency components under the separation condition $n\Delta \gtrsim \text{dyn}(\mathbf{x})$ and the dependency on $\text{dyn}(\mathbf{x})$ is inevitable for the vanilla OMP algorithm. We further show that the Sliding-OMP algorithm, a variant of the OMP algorithm with an additional refinement step at each iteration, is provable to recover $\{\tau_i\}_{i=1}^s$ under the separation condition $n\Delta \geq c$. Moreover, our result can be extended to an incomplete measurement model with $O( s^2\log n)$ measurements., Comment: Add numerical experiments

Communication efficiency plays an important role in accelerating the distributed training of Deep Neural Networks (DNN). All-reduce is the crucial communication primitive to reduce model parameters in distributed DNN training. Most existing all-reduce algorithms are designed for traditional electrical interconnect systems, which cannot meet the communication requirements for distributed training of large DNNs due to the low data bandwidth of the electrical interconnect systems. One of the promising alternatives for electrical interconnect is optical interconnect, which can provide high bandwidth, low transmission delay, and low power cost. We propose an efficient scheme called WRHT (Wavelength Reused Hierarchical Tree) for implementing all-reduce operation in optical interconnect systems. WRHT can take advantage of WDM (Wavelength Division Multiplexing) to reduce the communication time of distributed data-parallel DNN training. We further derive the required number of wavelengths, the minimum number of communication steps, and the communication time for the all-reduce operation on optical interconnect. The constraint of insertion loss is also considered in our analysis. Simulation results show that the communication time of all-reduce by WRHT is reduced by 80.81%, 64.36%, and 82.12%, respectively, compared with three traditional all-reduce algorithms according to our simulation results of an optical interconnect system. Our results also show that WRHT can reduce the communication time of all-reduce operation by 92.42% and 91.31% compared to two existing all-reduce algorithms running in the electrical interconnect system., Comment: This paper is accepted by PPoPP as a poster

The deterministic $k$-server conjecture states that there is a $k$-competitive deterministic algorithm for the $k$-server problem for any metric space. We show that the work function algorithm is $3$-competitive for the $3$-server problem on circle metrics, a case left open by Coester and Koutsoupias (2021). Our analysis follows the existing framework but introduces a new potential function which may be viewed as a relaxation of the counterpart by Coester and Koutsoupias (2021). We further notice that the new potential function and many existing ones can be rewritten in a canonical form. Through a computer-aided verification, however, we find that no such canonical potential function can resolve the deterministic $3$-server conjecture for general metric spaces under the current analysis framework.

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.

Multi-UAV collision avoidance is a challenging task for UAV swarm applications due to the need of tight cooperation among swarm members for collision-free path planning. Centralized Training with Decentralized Execution (CTDE) in Multi-Agent Reinforcement Learning is a promising method for multi-UAV collision avoidance, in which the key challenge is to effectively learn decentralized policies that can maximize a global reward cooperatively. We propose a new multi-agent critic-actor learning scheme called MACA for UAV swarm collision avoidance. MACA uses a centralized critic to maximize the discounted global reward that considers both safety and energy efficiency, and an actor per UAV to find decentralized policies to avoid collisions. To solve the credit assignment problem in CTDE, we design a counterfactual baseline that marginalizes both an agent's state and action, enabling to evaluate the importance of an agent in the joint observation-action space. To train and evaluate MACA, we design our own simulation environment MACAEnv to closely mimic the realistic behaviors of a UAV swarm. Simulation results show that MACA achieves more than 16% higher average reward than two state-of-the-art MARL algorithms and reduces failure rate by 90% and response time by over 99% compared to a conventional UAV swarm collision avoidance algorithm in all test scenarios.

In the classical version of online bipartite matching, there is a given set of offline vertices (aka agents) and another set of vertices (aka items) that arrive online. When each item arrives, its incident edges -- the agents who like the item -- are revealed and the algorithm must irrevocably match the item to such agents. We initiate the study of class fairness in this setting, where 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 from the fair division literature such as envy-freeness (up to one item), proportionality, and maximin share fairness to our setting. Our class versions of these notions demand that all classes, regardless of their sizes, receive a fair treatment. We study deterministic and randomized algorithms for matching indivisible items (leading to integral matchings) and for matching divisible items (leading to fractional matchings). We design and analyze three novel algorithms. 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. Finally, we build upon EQUAL-FILLING to design a randomized algorithm for matching indivisible items, EQAUL-FILLING-OCS, which achieves 0.593-approximation of class proportionality. The algorithm and its analysis crucially leverage the recently introduced technique of online correlated selection (OCS) [Fahrbach et al., 2020]., Comment: 31 pages

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 $1-\frac{1}{e}$ 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., Comment: STOC 2022

Online algorithm is an important branch in algorithm design. Designing online algorithms with a bounded competitive ratio (in terms of worst-case performance) can be hard and usually relies on problem-specific assumptions. Inspired by adversarial training from Generative Adversarial Net (GAN) and the fact that competitive ratio of an online algorithm is based on worst-case input, we adopt deep neural networks to learn an online algorithm for a resource allocation and pricing problem from scratch, with the goal that the performance gap between offline optimum and the learned online algorithm can be minimized for worst-case input. Specifically, we leverage two neural networks as algorithm and adversary respectively and let them play a zero sum game, with the adversary being responsible for generating worst-case input while the algorithm learns the best strategy based on the input provided by the adversary. To ensure better convergence of the algorithm network (to the desired online algorithm), we propose a novel per-round update method to handle sequential decision making to break complex dependency among different rounds so that update can be done for every possible action, instead of only sampled actions. To the best of our knowledge, our work is the first using deep neural networks to design an online algorithm from the perspective of worst-case performance guarantee. Empirical studies show that our updating methods ensure convergence to Nash equilibrium and the learned algorithm outperforms state-of-the-art online algorithms under various settings.

Fully Connected Neural Network (FCNN) is a class of Artificial Neural Networks widely used in computer science and engineering, whereas the training process can take a long time with large datasets in existing many-core systems. Optical Network-on-Chip (ONoC), an emerging chip-scale optical interconnection technology, has great potential to accelerate the training of FCNN with low transmission delay, low power consumption, and high throughput. However, existing methods based on Electrical Network-on-Chip (ENoC) cannot fit in ONoC because of the unique properties of ONoC. In this paper, we propose a fine-grained parallel computing model for accelerating FCNN training on ONoC and derive the optimal number of cores for each execution stage with the objective of minimizing the total amount of time to complete one epoch of FCNN training. To allocate the optimal number of cores for each execution stage, we present three mapping strategies and compare their advantages and disadvantages in terms of hotspot level, memory requirement, and state transitions. Simulation results show that the average prediction error for the optimal number of cores in NN benchmarks is within 2.3%. We further carry out extensive simulations which demonstrate that FCNN training time can be reduced by 22.28% and 4.91% on average using our proposed scheme, compared with traditional parallel computing methods that either allocate a fixed number of cores or allocate as many cores as possible, respectively. Compared with ENoC, simulation results show that under batch sizes of 64 and 128, on average ONoC can achieve 21.02% and 12.95% on reducing training time with 47.85% and 39.27% on saving energy, respectively., Comment: 14 pages, 10 figures. This paper is under the second review of IEEE Transactions of Computers

This paper studies the online correlated selection (OCS) problem. It was introduced by Fahrbach, Huang, Tao, and Zadimoghaddam (2020) to obtain the first edge-weighted online bipartite matching algorithm that breaks the $0.5$ barrier. Suppose that we receive a pair of elements in each round and immediately 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 online bipartite matching algorithms that randomize over only two options in each round, improving the $0.508$-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 bipartite matching and is $0.593$-competitive. Finally, we study OCS, which further considers the probability that an element is unselected in an arbitrary 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)., Comment: Compared to the first version, this version adds a discussion on two concurrent works on the same topic, gives a more accurate description of previous results, and improves the presentation based on the feedbacks by anonymous reviewers. The conference version appears in FOCS 2021

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 $40$ percents bids from previous buyers with the same characteristics. The targeting power is quantified with a parameter $\Delta \in [0, 1]$ which lower bounds how small the quantile intervals could be. When $\Delta = 1$, it degenerates to Cole and Roughgarden's model of i.i.d. samples; when it is the idealized case of $\Delta = 0$, it degenerates to the model studied by Chen et al. (2018). For instance, for $n$ buyers with bounded values in $[0, 1]$, $\tilde{O}(\epsilon^{-1})$ targeted samples suffice while it is known that at least $\tilde{\Omega}(n \epsilon^{-2})$ 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 $\epsilon^{-1}$ to linear. In this work, we introduce new technical ingredients and show that the number of targeted samples sufficient for learning an $\epsilon$-optimal auction is substantially smaller than the sample complexity of i.i.d. samples for the full spectrum of $\Delta \in [0, 1)$. Even with only mild targeting power, i.e., whenever $\Delta = o(1)$, our targeted sample complexity upper bounds are strictly smaller than the optimal sample complexity of i.i.d. samples., Comment: To appear in The Twenty-Second ACM Conference on Economics and Computation (EC 21)

Energy efficiency is of critical importance to trajectory planning for UAV swarms in obstacle avoidance. In this paper, we present $E^2Coop$, a new scheme designed to avoid collisions for UAV swarms by tightly coupling Artificial Potential Field (APF) with Particle Swarm Planning (PSO) based trajectory planning. In $E^2Coop$, swarm members perform trajectory planning cooperatively to avoid collisions in an energy-efficient manner. $E^2Coop$ exploits the advantages of the active contour model in image processing for trajectory planning. Each swarm member plans its trajectories on the contours of the environment field to save energy and avoid collisions to obstacles. Swarm members that fall within the safeguard distance of each other plan their trajectories on different contours to avoid collisions with each other. Simulation results demonstrate that $E^2Coop$ can save energy up to 51\% compared with two state-of-the-art schemes., Comment: The 31st International Conference on Automated Planning and Scheduling 2021

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 $0.662$. Conceptually, we find that the analysis of online algorithms simplifies if the online vertices follow a Poisson process, and establish an approximate equivalence between this Poisson arrival model and online stochstic matching. Technically, we propose a natural linear program for the Poisson arrival model, and demonstrate how to exploit its structure by introducing a converse of Jensen's inequality. Moreover, we design an algorithmic amortization to replace the analytic one in previous work, and as a result get the first vertex-weighted online stochastic matching algorithm that improves the results in the weaker random arrival model., Comment: 24 pages. Accepted by STOC 2021

Three decades ago, Karp, Vazirani, and Vazirani (STOC 1990) defined the online matching problem and gave an optimal $1-\frac{1}{e} \approx 0.632$-competitive algorithm. Fifteen years later, Mehta, Saberi, Vazirani, and Vazirani (FOCS 2005) introduced the first generalization called AdWords driven by online advertising and obtained the optimal $1-\frac{1}{e}$ 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 (STOC 2020), a novel formulation of AdWords which we call the panorama view, and a generalization of the online correlated selection by Fahrbach, Huang, Tao, and Zadimorghaddam (FOCS 2020) which we call the panoramic online correlated selection.

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.~(STOC 2018, SODA 2019) introduced a more general model called \emph{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 $\Omega \approx 0.567$-competitive on bipartite graphs where the $\Omega$-constant satisfies $\Omega e^\Omega = 1$, and Water-filling is $2-\sqrt{2} \approx 0.585$-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).

While prior researches focus on CPU-based microservices, they are not applicable for GPU-based microservices due to the different contention patterns. It is challenging to optimize the resource utilization while guaranteeing the QoS for GPU microservices. We find that the overhead is caused by inter microservice communication, GPU resource contention and imbalanced throughput within microservice pipeline. We propose Camelot, a runtime system that manages GPU micorservices considering the above factors. In Camelot, a global memory-based communication mechanism enables onsite data sharing that significantly reduces the end-to-end latencies of user queries. We also propose two contention aware resource allocation policies that either maximize the peak supported service load or minimize the resource usage at low load while ensuring the required QoS. The two policies consider the microservice pipeline effect and the runtime GPU resource contention when allocating resources for the microservices. Compared with state-of-the-art work, Camelot increases the supported peak load by up to 64.5% with limited GPUs, and reduces 35% resource usage at low load while achieving the desired 99%-ile latency target., Comment: 13 pages

Online bipartite matching and its variants are among the most fundamental problems in the online algorithms literature. Karp, Vazirani, and Vazirani (STOC 1990) introduced an elegant algorithm for the unweighted problem that achieves an optimal competitive ratio of $1-1/e$. Later, Aggarwal et al. (SODA 2011) 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 $1/2$ barrier and achieves a competitive ratio of at least $0.5086$. In light of the hardness result of Kapralov, Post, and Vondr\'ak (SODA 2013) that restricts beating a $1/2$ competitive 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., Comment: 36 pages, 5 figures. This work merges and refines the results in arXiv:1704.05384, arXiv:1910.02569, and arXiv:1910.03287. In particular, we fix a bug in arXiv:1910.03287 and have a smaller competitive ratio as a result

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 consider a generalization of the third degree price discrimination problem studied in 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 Bergemann et al. (2015), 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.

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 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), and provide the first and tight sample complexity bound for Pandora's problem., Comment: 38pages, submitted to SODA21