In online combinatorial allocation, agents arrive sequentially and items are allocated in an online manner. The algorithm designer only knows the distribution of each agent’s valuation, while the actual realization of the valuation is revealed only upon her arrival. Against the offline benchmark, Feldman, Gravin, and Lucier (SODA 2015) designed an optimal 0.5-competitive algorithm for XOS agents. An emerging line of work focuses on designing approximation algorithms against the (computationally unbounded) optimal online algorithm. The primary goal is to design algorithms with approximation ratios strictly greater than 0.5, surpassing the impossibility result against the offline optimum. Positive results are established for unit-demand agents (Papadimitriou, Pollner, Saberi, Wajc, MOR 2024), and for \(k\)-demand agents (Braun, Kesselheim, Pollner, Saberi, EC 2024).

Abstract. We introduce a new measure for the performance of online algorithms in Bayesian settings, where the input is drawn from a known prior, but the realizations are revealed one-by-one in an online fashion. Our new measure is called an order-competitive ratio. It is defined as the worst case (over all distribution sequences) ratio between the performance of the best order-unaware and order-aware algorithms, and quantifies the loss that is incurred due to lack of knowledge of the arrival order. Despite the growing interest in the role of the arrival order on the performance of online algorithms, this loss has been overlooked thus far. We study the order-competitive ratio in the paradigmatic prophet inequality problem, for the two common objective functions of (i) maximizing the expected value, and (ii) maximizing the probability of obtaining the largest value; and with respect to two families of algorithms, namely, (i) adaptive algorithms, and (ii) single-threshold algorithms. We provide tight bounds for all four combinations, with respect to deterministic algorithms, and preliminary results for randomized algorithms. Our analysis requires new ideas and departs from standard techniques. In particular, our adaptive algorithms inevitably go beyond single-threshold algorithms. In contrast to the classic competitive ratio measure, where the optimal performance is obtained by deterministic single-threshold algorithms, our results for order-competitive ratio capture the intuition that adaptive algorithms may be more powerful than single-threshold ones, and randomized algorithms outperform deterministic ones.

We study cooperative games where players join sequentially, and the value generated by those who have joined at any point must be irrevocably divided among these players. We introduce two desiderata for the value division mechanism: that the players should have incentives to join as early as possible, and that the division should be considered fair. For the latter, we require that each player's expected share in the mechanism should equal her Shapley value if the players' arrival order is uniformly at random. When the value generation function is submodular, allocating the marginal value to the player satisfies these properties. This is no longer true for more general functions. Our main technical contribution is a complete characterization of 0-1 value games for which desired mechanisms exist. We show that a natural mechanism, Rewarding First Critical Player (RFC), is complete, in that a 0-1 value function admits a mechanism with the properties above if and only if RFC satisfies them; we analytically characterize all such value functions. Moreover, we give an algorithm that decomposes, in an online fashion, any value function into 0-1 value functions, on each of which RFC can be run. In this way, we design an extension of RFC for general monotone games, and the properties are proved to be maintained.

We revisit the celebrated Ranking algorithm by Karp, Vazirani, and Vazirani (STOC 1990) for online bipartite matching under the random arrival model, that is shown to be 0.696-competitive for unweighted graphs by Mahdian and Yan (STOC 2011) and 0.662-competitive for vertex-weighted graphs by Jin and Williamson (WINE 2021).

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).

The classical prophet inequalities problem introduced by Krengel and Sucheston [1977, 1978] assumed complete knowledge of distributions. However, such an assumption may be unrealistic both in practice and for some applications.

There is a rising interest for studying the online benchmark as an alternative of the classical offline benchmark in online stochastic settings. Ezra, Feldman, Gravin, and Tang (SODA 2023) introduced the notion of order-competitive ratio, defined as the worst-case ratio between the performance of the best order-unaware algorithm and the best order-aware algorithm, to quantify the loss incurred by the lack of knowledge of the arrival order. They showed in the online single selection setting (a.k.a. the prophet problem), the optimal order-competitive ratio achieved by deterministic algorithms is 1/ϕ ≈ 0.618, and left with an open question whether randomized algorithms can do better.

In Bayesian online settings, every element is associated with a value drawn from a known underlying distribution. This distribution, representing the population from which the element is drawn, is referred to as the element's identity. The elements arrive sequentially, with their values being revealed in an online manner. Most previous work has assumed that, upon the arrival of a new element, the online algorithm observes its value and its identity. However, practical scenarios frequently require algorithms to make decisions based solely on the element's value, disregarding its identity. This necessity emerges either from the algorithm's lack of knowledge about the element's identity or in the pursuit of fairness, aiming for bias-free decisions across varying identities. We call such algorithms identity-blind algorithms, and propose the identity-blindness gap as a metric to evaluate the performance loss in online algorithms caused by identity-blindness. This gap is defined as the maximum ratio between the expected performance of an identity-blind online algorithm and an optimal online algorithm that knows the arrival order, thus also the identities.

We consider the robust contract design problem when the principal only has limited information about the actions the agent can take. The principal evaluates a contract according to its worst-case performance caused by the uncertain action space. Carroll (AER 2015) showed that a linear contract is optimal among deterministic contracts. Recently, Kambhampati (JET 2023) showed that the principal's payoff can be strictly increased via randomization over linear contracts. In this paper, we characterize the optimal randomized contract, which remains linear and admits a closed form of its cumulative density function. The advantage of randomized contracts over deterministic contracts can be arbitrarily large even when the principal knows only one non-trivial action of the agent. Furthermore, our result generalizes to the model of contracting with teams, by Dai and Toikka (Econometrica 2022).

Online bipartite matching with one-sided arrival and its variants have been extensively studied since the seminal work of Karp, Vazirani, and Vazirani (STOC 1990). Motivated by real-life applications with dynamic market structures, e.g., ride-sharing, two generalizations of the classical one-sided arrival model are proposed to allow non-bipartite graphs and to allow all vertices to arrive online. Namely, online matching with general vertex arrival is introduced by Wang and Wong (ICALP 2015), and fully online matching is introduced by Huang et al. (JACM 2020).

Ranking and Balance are arguably the two most important algorithms in the online matching literature. They achieve the same optimal competitive ratio of 1-1/e for the integral version and fractional version of online bipartite matching by Karp, Vazirani, and Vazirani (STOC 1990) respectively. The two algorithms have been generalized to weighted online bipartite matching problems, including vertex-weighted online bipartite matching and AdWords, by utilizing a perturbation function. The canonical choice of the perturbation function is f(x) = 1-e^{x-1} as it leads to the optimal competitive ratio of 1-1/e in both settings. We advance the understanding of the weighted generalizations of Ranking and Balance in this paper, with a focus on studying the effect of different perturbation functions. First, we prove that the canonical perturbation function is the unique optimal perturbation function for vertex-weighted online bipartite matching. In stark contrast, all perturbation functions achieve the optimal competitive ratio of 1-1/e in the unweighted setting. Second, we prove that the generalization of Ranking to AdWords with unknown budgets using the canonical perturbation function is at most 0.624 competitive, refuting a conjecture of Vazirani (2021). More generally, as an application of the first result, we prove that no perturbation function leads to the prominent competitive ratio of 1-1/e by establishing an upper bound of 1-1/e-0.0003. Finally, we propose the online budget-additive welfare maximization problem that is intermediate between AdWords and AdWords with unknown budgets, and we design an optimal 1-1/e competitive algorithm by generalizing Balance.

We consider ordinal online problems, i.e., tasks that only require pairwise comparisons between elements of the input. A classic example is the secretary problem and the game of googol, as well as its multiple combinatorial extensions such as $(J, K)$-secretary, 2-sided game of googol, ordinal-competitive matroid secretary. A natural approach to these tasks is to use ordinal online algorithms that at each step only consider relative ranking among the arrived elements, without looking at the numerical values of the input. We formally study the question of how cardinal algorithms (that can use numerical values of the input) can improve upon ordinal algorithms. We give first a universal construction of the input distribution for any ordinal online problem, such that the advantage of any cardinal algorithm over the ordinal algorithms is at most $1+\varepsilon$ for arbitrary small $\varepsilon\gt 0$. This implies that lower bounds from [Buchbinder, Jain, Singh, MOR 2014], [Nuti and Vondrák, SODA 2023] hold not only against any ordinal algorithm, but also against any online algorithm. Another immediate corollary is that cardinal algorithms are no better than ordinal algorithms in the matroid secretary problem with ordinal-competitive objective of [Soto, Turkieltaub, Verdugo, MOR 2021]. However, the value range of the input elements in our construction is huge: $N=$ $O\left(\frac{n^{3} \cdot n ! \cdot n !}{\varepsilon}\right) \uparrow \uparrow(n-1)$ (tower of exponents) for an input sequence of length n. As a second result, we identify a class of natural ordinal problems and find cardinal algorithm with a matching advantage of $1+\Omega\left(\frac{1}{\log (c) N}\right)$, where $\log^{(c)} N=\log \log \ldots \log N$ with c iterative logs and c is an arbitrary constant $c \leq n-2$. This suggests that for relatively small input numerical values N the cardinal algorithms may be significantly better than the ordinal algorithms on the ordinal tasks, which are typically assumed to be almost indistinguishable prior to our work. This observation leads to a natural complexity measure (we dub it cardinal complexity) for any given ordinal online task: the minimum size $N(\varepsilon)$ of different numerical values in the input such the advantage of cardinal over ordinal algorithms is at most $1+\varepsilon$ for any given $\varepsilon\gt 0$. As a third result, we show that the game of googol has much lower cardinal complexity of $N=O\left(\left(\frac{n}{\varepsilon}\right)^{n}\right)$.

Motivated by practical concerns in the online advertising industry, we study a bidder subset selection problem in single-item auctions. In this problem, a large pool of candidate bidders have independent values sampled from known prior distributions. The seller needs to pick a subset of bidders and run a given auction format on the selected subset to maximize her expected revenue. We propose two frameworks for the subset restrictions: (i) capacity constraint on the set of selected bidders; and (ii) incurred costs for the bidders invited to the auction. For the second-price auction with anonymous reserve (SPA-AR), we give constant approximation polynomial time algorithms in both frameworks (in the latter framework under mild assumptions about the market). Our results are in stark contrast to the previous work of Mehta, Nadav, Psomas, Rubinstein [NeurIPS 2020], who showed hardness of approximation for the SPA without a reserve price. We also give complimentary approximation results for other well-studied auction formats such as anonymous posted pricing and sequential posted pricing. On a technical level, we find that the revenue of SPA-AR as a set function f(S) of its bidders S is fractionally-subadditive but not submodular. Our bidder selection problem with invitation costs is a natural question about (approximately) answering a demand oracle for f(·) under a given vector of costs, a common computational assumption in the literature on combinatorial auctions. * This work is supported by Science and Technology Innovation 2030 –“New Generation of Artificial Intelligence” Major Project No.(2018AAA0100903), Innovation Program of Shanghai Municipal Education Commission, Program for Innovative Research Team of Shanghai University of Finance and Economics (IRTSHUFE) and the Fundamental Research Funds for the Central Universities. Zhihao Gavin Tang is supported by NSFC grant 61902233. Nick Gravin is supported by NSFC grant 62150610500.

We introduce a new measure for the performance of online algorithms in Bayesian settings, where the input is drawn from a known prior, but the realizations are revealed one-by-one in an online fashion. Our new measure is called order-competitive ratio. It is defined as the worst case (over all distribution sequences) ratio between the performance of the best order-unaware and order-aware algorithms, and quantifies the loss that is incurred due to lack of knowledge of the arrival order. Despite the growing interest in the role of the arrival order on the performance of online algorithms, this loss has been overlooked thus far. We study the order-competitive ratio in the paradigmatic prophet inequality problem, for the two common objective functions of (i) maximizing the expected value, and (ii) maximizing the probability of obtaining the largest value; and with respect to two families of algorithms, namely (i) adaptive algorithms, and (ii) single-threshold algorithms. We provide tight bounds for all four combinations, with respect to deterministic algorithms. Our analysis requires new ideas and departs from standard techniques. In particular, our adaptive algorithms inevitably go beyond single-threshold algorithms. The results with respect to the order-competitive ratio measure capture the intuition that adaptive algorithms are stronger than single-threshold ones, and may lead to a better algorithmic advice than the classical competitive ratio measure. * This work is supported by Science and Technology Innovation 2030 –“New Generation of Artificial Intelligence” Major Project No.(2018AAA0100903), Innovation Program of Shanghai Municipal Education Commission, Program for Innovative Research Team of Shanghai University of Finance and Economics (IRTSHUFE) and the Fundamental Research Funds for the Central Universities. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. 866132), by the Israel Science Foundation (grant number 317/17), by an Amazon Research Award, and by the NSF-BSF (grant number 2020788). Tomer Ezra was partially supported by the ERC Advanced Grant 788893 AMDROMA “Algorithmic and Mechanism Design Research in Online Markets” and MIUR PRIN project ALGADIMAR “Algorithms, Games, and Digital Markets”. Zhihao Gavin Tang is supported by NSFC grant 61902233. Nick Gravin is supported by NSFC grant 62150610500.

A large body of work in Computer Science and Operations Research study online algorithms for stochastic resource allocation problems. The most common assumption is that the online requests have randomly generated i.i.d. types. This assumption is well justified for static markets and/or relatively short time periods. We consider dynamic markets, whose states evolve as a random walk in a market-specific Markov Chain. This is a new model that generalizes previous i.i.d. settings. We identify important parameters of the Markov chain that is crucial for obtaining good approximation guarantees to the expected value of the optimal offline algorithm which knows realizations of all requests in advance. We focus on a stylized single-resource setting and: (i) generalize the well-known Prophet Inequality from the optimal stopping theory (single-unit setting) to Markov Chain setting; (ii) in multi-unit setting, design a simple algorithm that is asymptotically optimal under mild assumptions on the underlying Markov chain.

There has been a long history of studying randomized greedy matching algorithms since the work by Dyer and Frieze [ 9 ]. We follow this trend and consider the problem formulated in the oblivious setting, in which the vertex set of a graph is known to the algorithm but not the edge set. The algorithm can make queries for the existence of the edge between any pair of vertices but must include the edge into the matching if it exists, i.e., as in the query-commit model by Gamlath et al. [ 12 ]. We revisit theModified Randomized Greedy (MRG)algorithm by Aronson et al. [ 1 ] that is proved to achieve a (0.5+ε)-approximation. In each step of the algorithm, an unmatched vertex is chosen uniformly at random and matched to a randomly chosen neighbor (if exists). We study a weaker version of the algorithm namedRandom Decision Order (RDO)that, in each step, randomly picks an unmatched vertex and matches it to an arbitrary neighbor (if exists). We prove that theRDOalgorithm provides a 0.639-approximation for bipartite graphs and 0.531-approximation for general graphs. As a corollary, we substantially improve the approximation ratio ofMRG. Furthermore, we generalize theRDOalgorithm to the edge-weighted case and prove that it achieves a 0.501-approximation ratio. This result solves the open question by Chan et al. [ 4 ] and Gamlath et al. [ 12 ] about the existence of an algorithm that beats greedy in edge-weighted general graphs, where the greedy algorithm probes the edges in descending order of edge-weights. We also present a variant of the algorithm that achieves a (1-1/e)-approximation for edge-weighted bipartite graphs, which generalizes the (1-1/e)-approximation ratio of Gamlath et al. [ 12 ] for the stochastic setting to the case when the realizations of edges are arbitrarily correlated, where in the stochastic setting, there is a known probability associated with each pair of vertices that indicates the probability that an edge exists between the two vertices, when the pair is probed.

In the classical prophet inequality, a gambler faces a sequence of items, whose values are drawn independently from known distributions. Upon the arrival of each item, its value is realized and the gambler either accepts it and the game ends, or irrevocably rejects it and continues to the next item. The goal is to maximize the value of the selected item and compete against the expected maximum value of all items. A tight competitive ratio of $\frac{1}{2}$ is established in the classical setting and various relaxations have been proposed to surpass the barrier, including the i.i.d. model, the order selection model, and the random order model. In this paper, we advance the study of the order selection prophet inequality, in which the gambler is given the extra power for selecting the arrival order of the items. Our main result is a 0.725-competitive algorithm, that substantially improves the state-of-the-art 0.669 ratio by Correa, Saona and Ziliotto (Math. Program. 2021), achieved in the harder random order model. Recently, Agrawal, Sethuraman and Zhang (EC 2020) proved that the task of selecting the optimal order is NP-hard. Despite this fact, we introduce a novel algorithm design framework that translates the discrete order selection problem into a continuous arrival time design problem. From this perspective, we can focus on the arrival time design without worrying about the threshold optimization afterwards. As a side result, we achieve the optimal 0.745 competitive ratio by applying our algorithm to the i.i.d. model.

Online algorithms for secretary matching in bipartite weighted graphs have been studied extensively in recent years. We generalize this study to secretary matching in general weighted graphs, for both vertex and edge arrival models.

Motivated by display advertising on the internet, the online stochastic matching problem is proposed by Feldman, Mehta, Mirrokni, and Muthukrishnan (FOCS 2009). Consider a stochastic bipartite graph with offline vertices on one side and with i.i.d. online vertices on the other side. The algorithm knows the offline vertices and the distribution of the online vertices in advance. Upon the arrival of each online vertex, its type is realized and the algorithm immediately and irrevocably decides how to match it. In the vertex-weighted version of the problem, each offline vertex is associated with a weight and the goal is to maximize the total weight of the matching.

We introduce the Online Food Delivery Problem (OFDP) to model the delivery problem commonly encountered in online food-ordering-and-delivery platforms. In the OFDP the requests (orders) are submitted online, and the depot (restaurant) needs to decide when to send out a server to serve the submitted requests. In addition, the server has to return to the depot (to pickup foods) before serving new requests. The objective is to minimize maximum flow time, i.e., the maximum time between the submission and completion of a request. This problem can also be viewed as a variant of the Online Dial-a-Ride problem, for which however the max flow time objective is inapproximable in general. We study the OFDP on star graphs, and give both algorithmic and hardness results. We analyze a natural greedy strategy and show that it achieves the optimal competitive ratio 3 among all myopic algorithms, which are algorithms that immediately send out the server whenever there are unserved requests. Then we prove that a far-sighted (i.e., non-myopic) algorithm with proper waiting strategy can achieve 8/3-competitive ratio. On the negative side, we give a simple lower bound example that excludes the possibility of any ( 2 − ϵ ) -competitive algorithms. • This paper introduces the Online Food Delivery Problem encountered in online food-ordering-and-delivery platforms. • We show that a natural greedy strategy achieves the optimal competitive ratio 3 among all myopic algorithms. • A far-sighted algorithm with proper waiting strategy can achieve 8/3-competitive ratio.

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.

With a wide range of applications, stochastic matching problems have been studied in different models, including prophet inequality, Query-Commit, and Price-of-Information. While there have been recent breakthroughs in all these settings for bipartite graphs, few non-trivial results are known for general graphs. In this paper, we study the random order vertex arrival contention resolution scheme for matching in general graphs, which is inspired by the recent work of Ezra et al. (EC 2020). We design an 8/15-selectable batched RCRS for matching and apply it to achieve 8/15-competitive/approximate algorithms for all the three models. Our results are the first non-trivial results for random order prophet matching and Price-of-Information matching in general graphs. For the Query-Commit model, our result substantially improves upon the 0.501 approximation ratio by Tang et al. (STOC 2020). We also show that no batched RCRS for matching can be better than 1/2+1/(2e²) ≈ 0.567-selectable.

Online bipartite matching with edge arrivals remained a major open question for a long time until a recent negative result by Gamlath et al., who showed that no online policy is better than the straightforward greedy algorithm, i.e., no online algorithm has a worst-case competitive ratio better than 0.5. In this work, we consider the bipartite matching problem with edge arrivals in a natural stochastic framework, i.e., Bayesian setting where each edge of the graph is independently realized according to a known probability distribution. We focus on a natural class of prune & greedy online policies motivated by practical considerations from a multitude of online matching platforms. Any prune & greedy algorithm consists of two stages: first, it decreases the probabilities of some edges in the stochastic instance and then runs greedy algorithm on the pruned graph. We propose prune & greedy algorithms that are 0.552-competitive on the instances that can be pruned to a 2-regular stochastic bipartite graph, and 0.503-competitive on arbitrary stochastic bipartite graphs. The algorithms and our analysis significantly deviate from the prior work. We first obtain analytically manageable lower bound on the size of the matching, which leads to a non-linear optimization problem. We further reduce this problem to a continuous optimization with a constant number of parameters that can be solved using standard software tools.

Inspired by a recent line of work in online algorithms with predictions, we study the constrained adversary model that utilizes predictions from a different perspective. Prior works mostly focused on designing simultaneously robust and consistent algorithms, without making assumptions on the quality of the predictions. In contrary, our model assumes the adversarial instance is consistent with the predictions and aim to design algorithms that have best worst-case performance against all such instances. We revisit classical online selection problems under the constrained adversary model. For the single item selection problem, we design an optimal algorithm in the adversarial arrival model and an improved algorithm in the random arrival model (a.k.a., the secretary problem). For the online edge-weighted bipartite matching problem, we extend the classical Water-filling and Ranking algorithms and achieve improved competitive ratios.

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).

We provide prophet inequality algorithms for online weighted matching in general (non-bipartite) graphs, under two well-studied arrival models, namely edge arrival and vertex arrival. The weight of each edge is drawn independently from an a-priori known probability distribution. Under edge arrival, the weight of each edge is revealed upon arrival, and the algorithm decides whether to include it in the matching or not. Under vertex arrival, the weights of all edges from the newly arriving vertex to all previously arrived vertices are revealed, and the algorithm decides which of these edges, if any, to include in the matching. To study these settings, we introduce a novel unified framework of batched prophet inequalities that captures online settings where elements arrive in batches; in particular it captures matching under the two aforementioned arrival models. Our algorithms rely on the construction of suitable online contention resolution schemes (OCRS). We first extend the framework of OCRS to batched-OCRS, we then establish a reduction from batched prophet inequality to batched OCRS, and finally we construct batched OCRSs with selectable ratios of 0.337 and 0.5 for edge and vertex arrival models, respectively. Both results improve the state of the art for the corresponding settings. For vertex arrival, our result is tight. Interestingly, pricing-based prophet inequalities with comparable competitive ratios are unknown.

There has been a long history for studying randomized greedy matching algorithms since the work by Dyer and Frieze(RSA 1991). We follow this trend and consider the problem formulated in the oblivious setting, in which the algorithm makes (random) decisions that are essentially oblivious to the input graph. We revisit the Modified Randomized Greedy (MRG) algorithm by Aronson et al.(RSA 1995) which is proved to be (0.5+epsilon)-approximate. In particular, we study a weaker version of the algorithm named Random Decision Order (RDO) that in each step, randomly picks an unmatched vertex and matches it to an arbitrary neighbor if exists. We prove the RDO algorithm is 0.639-approximate and 0.531-approximate for bipartite graphs and general graphs respectively. As a corollary, we substantially improve the approximation ratio of MRG. Furthermore, we generalize the RDO algorithm to the edge-weighted case and prove that it achieves a 0.501 approximation ratio. This result solves the open question by Chan et al.(SICOMP 2018) about the existence of an algorithm that beats greedy in this setting. As a corollary, it also solves the open questions by Gamlath et al.(SODA 2019) in the stochastic setting.

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.

We consider a fundamental problem in microeconomics: selling a single item to a number of potential buyers, whose values are drawn from known independent and regular (not necessarily identical) distributions. There are four widely used and widely studied mechanisms in the literature: Myerson Auction (OPT), Sequential Posted-Pricing (SPM), Second-Price Auction with Anonymous Reserve (AR), and Anonymous Pricing (AP). OPT is revenue-optimal but complicated and also experiences several issues in practice such as fairness; AP is the simplest mechanism but also generates the lowest revenue among these four mechanisms; SPM and AR are of intermediate complexity and revenue. We explore revenue gaps among these mechanisms, each of which is defined as the largest ratio between revenues from a pair of mechanisms. We establish two tight bounds and one tighter bound: 1. SPM vs. AP: this ratio studies the power of discrimination in pricing schemes. We obtain the tight ratio of constant ${\cal{C}}^* \approx {2.62}$, closing the gap between $[\frac{e}{e - 1}, e]$ left before. 2. AR vs. AP: this ratio measures the relative power of auction scheme vs. pricing scheme, when no discrimination is allowed. We attain the tight ratio of $\frac{\pi^2}{6} \approx 1.64$, closing the previously known bounds $[\frac{e}{e - 1}, e]$. 3. OPT vs. AR: this ratio quantifies the power of discrimination in auction schemes and is previously known to be somewhere between [2, e]. The lower bound of 2 was conjectured to be tight by Hartline and Roughgarden [ Proceedings of the 10th ACM Conference on Electronic Commerce, 2009, pp. 225--234] and Alaei et al. [ Games Econom. Behav., 118 (2019), pp. 494--510]. We acquire a better lower bound of 2.15 and thus disprove this conjecture.

We revisit semi-supervised learning on hypergraphs. Same as previous approaches, our method uses a convex program whose objective function is not everywhere differentiable. We exploit the non-uniqueness of the optimal solutions, and consider confidence intervals which give the exact ranges that unlabeled vertices take in any optimal solution. Moreover, we give a much simpler approach for solving the convex program based on the subgradient method. Our experiments on real-world datasets confirm that our confidence interval approach on hypergraphs outperforms existing methods, and our subgradient method gives faster running times when the number of vertices is much larger than the number of edges. Our experiments also support that using directed hypergraphs to capture causal relationships can improve the prediction accuracy. Furthermore, our model can be readily extended to capture multiclass learning.

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).

We investigate the problem of revenue maximization in single-item auction within the new correlation-robust framework proposed by Carroll [2017] and further developed by Gravin and Lu [2018]. In this framework the auctioneer is assumed to have only partial information about marginal distributions, but does not know the dependency structure of the joint distribution. The auctioneer's revenue is evaluated in the worst-case over the uncertainty of possible joint distribution. For the problem of optimal auction design in the correlation robust-framework we observe that in most cases the optimal auction does not admit a simple form like the celebrated Myerson's auction for independent valuations. We analyze and compare performances of several DSIC mechanisms used in practice. Our main set of results concern the sequential posted-price mechanism (SPM). We show that SPM achieves a constant (4.78) approximation to the optimal correlation-robust mechanism. We also show that in the symmetric (anonymous) case when all bidders have the same marginal distribution, (i) SPM has almost matching worst-correlation revenue as any second price auction with common reserve price, and (ii) when the number of bidders is large, SPM converges to optimum. In addition, we extend some results on approximation and computational tractability for lookahead auctions to the correlation-robust framework.

We consider a fundamental problem in microeconomics: Selling a single item among a number of buyers whose values are drawn from known independent and regular distributions. There are four widely-used and widely-studied mechanisms in this literature: Anonymous Posted-Pricing (AP), Second-Price Auction with Anonymous Reserve (AR), Sequential Posted-Pricing (SPM), and Myerson Auction (OPT). Myerson Auction is optimal but complicated, which also suffers a few issues in practice such as fairness; AP is the simplest mechanism, but its revenue is also the lowest among these four; AR and SPM are of intermediate complexity and revenue. We study the revenue gaps among these four mechanisms, which is defined as the largest ratio between revenues from two mechanisms. We establish two tight ratios and one tighter bound: 1. SPM/AP. This ratio studies the power of discrimination in pricing schemes. We obtain the tight ratio of roughly 2.62, closing the previous known bounds [e/(e – 1), e]. 2. AR/AP. This ratio studies the relative power of auction vs. pricing schemes, when no discrimination is allowed. We get the tight ratio of π2/6 ≈ 1.64, closing the previous known bounds [e/(e – 1), e]. 3. OPT/AR. This ratio studies the power of discrimination in auctions. Previously, the revenue gap is known to be in interval [2, e], and the lower-bound of 2 is conjectured to be tight [38, 37, 4]. We disprove this conjecture by obtaining a better lower-bound of 2.15.

This paper considers two canonical Bayesian mechanism design settings. In the single-item setting, the tight approximation ratio of Anonymous Pricing is obtained: (1) compared to Myerson Auction, Anonymous Pricing always generates at least a 1/2.62-fraction of the revenue; (2) there is a matching lower-bound instance.

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.

We consider a revenue optimizing seller selling a single item to a buyer, on whose private value the seller has a noisy signal. We show that, when the signal is kept private, arbitrarily more revenue could potentially be extracted than if the signal is leaked or revealed. We then show that, if the seller is not allowed to make payments to the buyer and if the value distribution conditioning on each signal is regular, the gap between the two is bounded by a multiplicative factor of 3. We give examples showing that both conditions are necessary for a constant bound on the gap to hold. We connect this scenario to multi-bidder single-item auctions where bidders’ values are correlated. Similarly to the setting above, we show that the revenue of a Bayesian incentive compatible, ex post individually rational auction can be arbitrarily larger than that of a dominant strategy incentive compatible auction, whereas the two are no more than a factor of 5 apart if the auctioneer never pays the bidders and if the distribution is jointly regular. The upper bounds in both settings degrade gracefully when the distribution is a mixture of a small number of regular distributions.

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.

The celebrated Cheeger’s Inequality (Alon and Milman 1985; Alon 1986) establishes a bound on the edge expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency matrix (and other related matrices) of graphs. It has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph. In this article, we introduce a new hypergraph Laplacian operator generalizing the Laplacian matrix of graphs. In particular, the operator is induced by a diffusion process on the hypergraph, such that within each hyperedge, measure flows from vertices having maximum weighted measure to those having minimum. Since the operator is nonlinear, we have to exploit other properties of the diffusion process to recover the Cheeger’s Inequality that relates hyperedge expansion with the “second eigenvalue” of the resulting Laplacian. However, we show that higher-order spectral properties cannot hold in general using the current framework. Since higher-order spectral properties do not hold for the Laplacian operator, we instead use the concept of procedural minimizers to consider higher-order Cheeger-like inequalities. For any k ∈ N, we give a polynomial-time algorithm to compute an O (log r )-approximation to the k th procedural minimizer, where r is the maximum cardinality of a hyperedge. We show that this approximation factor is optimal under the SSE hypothesis (introduced by Raghavendra and Steurer (2010)) for constant values of k . Moreover, using the factor-preserving reduction from vertex expansion in graphs to hypergraph expansion, we show that all our results for hypergraphs extend to vertex expansion in graphs.

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 online vector packing problem in which we have a d dimensional knapsack and items u with weight vectors w_u in R_+^d arrive online in an arbitrary order. Upon the arrival of an item, the algorithm must decide immediately whether to discard or accept the item into the knapsack. When item u is accepted, w_u(i) units of capacity on dimension i will be taken up, for each i in [d]. To satisfy the knapsack constraint, an accepted item can be later disposed of with no cost, but discarded or disposed of items cannot be recovered. The objective is to maximize the utility of the accepted items S at the end of the algorithm, which is given by f(S) for some non-negative monotone submodular function f. For any small constant epsilon > 0, we consider the special case that the weight of an item on every dimension is at most a (1- epsilon) fraction of the total capacity, and give a polynomial-time deterministic O(k / epsilon^2)-competitive algorithm for the problem, where k is the (column) sparsity of the weight vectors. We also show several (almost) tight hardness results even when the algorithm is computationally unbounded. We first show that under the epsilon-slack assumption, no deterministic algorithm can obtain any o(k) competitive ratio, and no randomized algorithm can obtain any o(k / log k) competitive ratio. We then show that for the general case (when epsilon = 0), no randomized algorithm can obtain any o(k) competitive ratio. In contrast to the (1+delta) competitive ratio achieved in Kesselheim et al. [STOC 2014] for the problem with random arrival order of items and under large capacity assumption, we show that in the arbitrary arrival order case, even when |w_u|_infinity is arbitrarily small for all items u, it is impossible to achieve any o(log k / log log k) competitive ratio.

We revisit semi-supervised learning on hypergraphs. Same as previous approaches, our method uses a convex program whose objective function is not everywhere differentiable. We exploit the non-uniqueness of the optimal solutions, and consider confidence intervals which give the exact ranges that unlabeled vertices take in any optimal solution. Moreover, we give a much simpler approach for solving the convex program based on the subgradient method. Our experiments on real-world datasets confirm that our confidence interval approach on hypergraphs outperforms existing methods, and our sub-gradient method gives faster running times when the number of vertices is much larger than the number of edges.

We consider the edge partitioning problem that partitions the edges of an input graph into multiple balanced components, while minimizing the total number of vertices replicated (one vertex might appear in more than one partition). This problem is critical in minimizing communication costs and running time for several large-scale distributed graph computation platforms (e.g., PowerGraph, Spark GraphX). We first prove that this problem is NP-hard, and then present a new partitioning heuristic with polynomial running time. We provide a worst-case upper bound of replication factor for our heuristic on general graphs. To our knowledge, we are the first to provide such bound for edge partitioning algorithms on general graphs. Applying this bound to random power-law graphs greatly improves the previous bounds of expected replication factor. Extensive experiments demonstrated that our partitioning algorithm consistently produces much smaller replication factors on various benchmark data sets than the state-of-the-art. When deployed in the production graph engine, PowerGraph, in average it reduces replication factor, communication, and running time by 54%, 66%, and 21%, respectively.

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.

We study the max-min fair allocation problem in which a set of m indivisible items are to be distributed among n agents such that the minimum utility among all agents is maximized. In the restricted setting, the utility of each item j on agent i is either 0 or some non-negative weight w_j. For this setting, Asadpour et al. [TALG, 2012] showed that a certain configuration-LP can be used to estimate the optimal value within a factor of 4 + delta, for any delta > 0, which was recently extended by Annamalai et al. [SODA 2015] to give a polynomial-time 13-approximation algorithm for the problem. For hardness results, Bezáková and Dani [SIGecom Exch., 2005] showed that it is NP-hard to approximate the problem within any ratio smaller than 2. In this paper we consider the (1, epsilon)-restricted max-min fair allocation problem, in which for some parameter epsilon in (0, 1), each item j is either heavy (w_j = 1) or light (w_j = epsilon). We show that the (1, epsilon)-restricted case is also NP-hard to approximate within any ratio smaller than 2. Hence, this simple special case is still algorithmically interesting. Using the configuration-LP, we are able to estimate the optimal value of the problem within a factor of 3 + delta, for any delta > 0. Extending this idea, we also obtain a quasi-polynomial time (3 + 4 epsilon)-approximation algorithm and a polynomial time 9-approximation algorithm. Moreover, we show that as epsilon tends to 0, the approximation ratio of our polynomial-time algorithm approaches 3 + 2 sqrt{2} approx 5.83.