Most existing work in online stochastic combinatorial optimization assumes that inputs are drawn from independent distributions—a strong assumption that often fails in practice. At the other extreme, arbitrary correlations are equivalent to worst-case inputs via Yao’s minimax principle, making good algorithms often impossible. This motivates the study of intermediate models that capture mild correlations while still permitting nontrivial algorithms.

Online Set Cover and Load Balancing are central problems in online optimization, and there is a long line of work focusing on developing algorithms for these problems with convex objectives. Although we know optimal online algorithms with $\ell_{p}$-norm objectives, recent developments for general norms and convex objectives that rely on the online primal-dual framework apply only to fractional settings due to large integrality gaps. Our work focuses on directly designing integral online algorithms for Set Cover and Load Balancing with convex objectives, bypassing the convex-relaxation and the primal-dual technique. Some of the main implications of our approach are: 1) For Online Set Cover, we can extend the results of [1] for convex objectives and of [2] for symmetric norms from fractional to integral settings. 2) Our results for convex objectives and symmetric norms even apply to the Online Generalized Scheduling Problem, which generalizes both Set Cover and Load Balancing. Previous works could only handle the offline version of this problem with norm objectives [3]. 3) Our approach easily extends to settings involving disjointcomposition of norms. This allows us to recover or improve the norm-composition results of [4], [2] and extend our results to a large class of norms beyond the symmetric setting. Our approach involves first reducing these online problems to online packing problems, and to then design good approximation algorithms for the latter. To solve these packing problem, we use two key ideas. First, we decouple the global packing problem into a series of local packing problems on different machines. Second, we choose random activation thresholds for machines such that conditional on a machine being activated the expected number of jobs it covers is high compared to its cost. This approach may be of independent interest and could find applications to other online problems. Index Terms-online algorithms, set cover, load balancing

Abstract. A longstanding open problem in algorithmic mechanism design is to design truthful mechanisms that are computationally efficient and (approximately) maximize welfare in combinatorial auctions with submodular bidders. The first such mechanism was obtained by Dobzinski, Nisan, and Schapira [Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, ACM, New York, 2005, pp. 610–618] who gave an [Formula: see text]-approximation, where [Formula: see text] is the number of items. This problem has been studied extensively since, culminating in an [Formula: see text]-approximation mechanism by Dobzinski [Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, ACM, New York, 2016, pp. 940–948]. We present a computationally-efficient truthful mechanism with an approximation ratio that improves upon the state-of-the-art by an exponential factor. In particular, our mechanism achieves an [Formula: see text]-approximation in expectation, uses only [Formula: see text] demand queries, and has universal truthfulness guarantee. This settles an open question of Dobzinski on whether [Formula: see text] is the best approximation ratio in this setting in the negative.

In online combinatorial allocations/auctions,$n$bidders sequentially arrive, each with a combinatorial valuation (such as submodular/XOS) over subsets of$m$indivisible items. The aim is to immediately allocate a subset of the remaining items to maximize the total welfare, defined as the sum of bidder valuations. A long line of work has studied this problem when the bidder valuations come from known independent distributions. In particular, for submodular/XOS valuations, we know 2-competitive algorithms/mechanisms that set a fixed price for each item and the arriving bidders take their favorite subset of the remaining items given these prices. However, these algorithms traditionally presume the availability of the underlying distributions as part of the input to the algorithm. Contrary to this assumption, practical scenarios often require the learning of distributions, a task complicated by limited sample availability. This paper investigates the feasibility of achieving$O$(1) -competitive algorithms under the realistic constraint of having access to only a limited number of samples from the underlying bidder distributions. Our first main contribution shows that a mere single sample from each bidder distribution is sufficient to yield an$O$(1)-competitive algorithm for submodular/XOS valuations. This result leverages a novel extension of the secretary-style analysis, employing the sample to have the algorithm compete against itself. Although online, this first approach does not provide an online truthful mechanism. Our second main contribution shows that a polynomial number of samples suffices to yield a (2 + ∊) -competitive online truthful mechanism for submodular/XOS valuations and any constant ∊ > 0. This result is based on a generalization of the median-based algorithm for the single-item prophet inequality problem to combinatorial settings with multiple items.

Selling a single item to $n$ self-interested bidders is a fundamental problem in economics, where the two objectives typically considered are welfare maximization and revenue maximization. Since the optimal auctions are often impractical and do not work for sequential bidders, posted pricing auctions, where fixed prices are set for the item for different bidders, have emerged as a practical and effective alternative. This paper investigates how many samples are needed from bidders' value distributions to find near-optimal posted prices, considering both independent and correlated bidder distributions, and welfare versus revenue maximization. We obtain matching upper and lower bounds (up to logarithmic terms) on the sample complexity for all these settings.

Sequential posted pricing auctions are popular because of their simplicity in practice and their tractability in theory. A usual assumption in their study is that the Bayesian prior distributions of the buyers are known to the seller, while in reality these priors can only be accessed from historical data. To overcome this assumption, we study sequential posted pricing in the bandit learning model, where the seller interacts with n buyers over T rounds: In each round the seller posts n prices for the n buyers and the first buyer with a valuation higher than the price takes the item. The only feedback that the seller receives in each round is the revenue.

Over the past two decades, significant strides have been made in stochastic problems such as revenue-optimal auction design and prophet inequalities, traditionally modeled with n independent random variables to represent the values of n items. However, in many applications, this assumption of independence often diverges from reality. Given the strong impossibility results associated with arbitrary correlations, recent research has pivoted towards exploring these problems under models of mild dependency.

The Prophet Inequality and Pandora's Box problems are fundamental stochastic problem with applications in Mechanism Design, Online Algorithms, Stochastic Optimization, Optimal Stopping, and Operations Research. A usual assumption in these works is that the probability distributions of the n underlying random variables are given as input to the algorithm. Since in practice these distributions need to be learned under limited feedback, we initiate the study of such stochastic problems in the Multi-Armed Bandits model.

Many classical problems in theoretical computer science involve norms, even if implicitly; for example, both XOS functions and downward-closed sets are equivalent to some norms. The last decade has seen a lot of interest in designing algorithms beyond the standard ℓp norms ||· ||p. Despite notable advancements, many existing methods remain tailored to specific problems, leaving a broader applicability to general norms less understood. This paper investigates the intrinsic properties of ℓp norms that facilitate their widespread use and seeks to abstract these qualities to a more general setting. We identify supermodularity—often reserved for combinatorial set functions and characterized by monotone gradients—as a defining feature beneficial for ||·||pp. We introduce the notion of p-supermodularity for norms, asserting that a norm is p-supermodular if its pth power function exhibits supermodularity. The association of supermodularity with norms offers a new lens through which to view and construct algorithms. Our work demonstrates that for a large class of problems p-supermodularity is a sufficient criterion for developing good algorithms. This is either by reframing existing algorithms for problems like Online Load-Balancing and Bandits with Knapsacks through a supermodular lens, or by introducing novel analyses for problems such as Online Covering, Online Packing, and Stochastic Probing. Moreover, we prove that every symmetric norm can be approximated by a p-supermodular norm. Together, these recover and extend several existing results, and support p-supermodularity as a unified theoretical framework for optimization challenges centered around norm-related problems.

Abstract. The secretary and the prophet inequality problems are central to the field of stopping theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions. Kleinberg and Weinberg and Feldman, Gravin, and Lucier show that for adversarial arrival order of random variables the optimal prophet inequalities give a [Formula: see text]-approximation. For many settings, however, it is conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the [Formula: see text]-approximation and obtain [Formula: see text]-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan and of Esfandiari and colleagues who worked in the special cases where either we can fully control the arrival order or there is only a single item. Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.

Continuous submodular functions are a category of generally non-convex/non-concave functions with a wide spectrum of applications. The celebrated property of this class of functions - continuous submodularity - enables both exact minimization and approximate maximization in poly. time. Continuous submodularity is obtained by generalizing the notion of submodularity from discrete domains to continuous domains. It intuitively captures a repulsive effect amongst different dimensions of the defined multivariate function. In this paper, we systematically study continuous submodularity and a class of non-convex optimization problems: continuous submodular function maximization. We start by a thorough characterization of the class of continuous submodular functions, and show that continuous submodularity is equivalent to a weak version of the diminishing returns (DR) property. Thus we also derive a subclass of continuous submodular functions, termed continuous DR-submodular functions, which enjoys the full DR property. Then we present operations that preserve continuous (DR-)submodularity, thus yielding general rules for composing new submodular functions. We establish intriguing properties for the problem of constrained DR-submodular maximization, such as the local-global relation. We identify several applications of continuous submodular optimization, ranging from influence maximization, MAP inference for DPPs to provable mean field inference. For these applications, continuous submodularity formalizes valuable domain knowledge relevant for optimizing this class of objectives. We present inapproximability results and provable algorithms for two problem settings: constrained monotone DR-submodular maximization and constrained non-monotone DR-submodular maximization. Finally, we extensively evaluate the effectiveness of the proposed algorithms.

Vector norms play a fundamental role in computer science and optimization, so there is an ongoing effort to generalize existing algorithms to settings beyond ℓ∞ and ℓp norms. We show that many online and bandit applications for general norms admit good algorithms as long as the norm can be approximated by a function that is “gradient-stable”, a notion that we introduce. Roughly it says that the gradient of the function should not drastically decrease (multiplicatively) in any component as we increase the input vector. We prove that several families of norms, including all monotone symmetric norms, admit a gradient-stable approximation, giving us the first online and bandit algorithms for these norm families. In particular, our notion of gradient-stability gives O (log2 (dimension))-competitive algorithms for the symmetric norm generalizations of Online Generalized Load Balancing and Bandits with Knapsacks. Our techniques extend to applications beyond symmetric norms as well, e.g., to Online Vector Scheduling and to Online Generalized Assignment with Convex Costs. Some key properties underlying our applications that are implied by gradient-stable approximations are a “smooth game inequality” and an approximate converse to Jensen's inequality.

The well-known Komlós conjecture states that given n vectors in ℝ^d with Euclidean norm at most one, there always exists a ± 1 coloring such that the 𝓁_∞ norm of the signed-sum vector is a constant independent of n and d. We prove this conjecture in a smoothed analysis setting where the vectors are perturbed by adding a small Gaussian noise and when the number of vectors n = ω(d log d). The dependence of n on d is the best possible even in a completely random setting. Our proof relies on a weighted second moment method, where instead of considering uniformly randomly colorings we apply the second moment method on an implicit distribution on colorings obtained by applying the Gram-Schmidt walk algorithm to a suitable set of vectors. The main technical idea is to use various properties of these colorings, including subgaussianity, to control the second moment.

A well-known result of Banaszczyk in discrepancy theory concerns the prefix discrepancy problem (also known as the signed series problem): given a sequence of $T$ unit vectors in $\mathbb{R}^d$, find $\pm$ signs for each of them such that the signed sum vector along any prefix has a small $\ell_\infty$-norm? This problem is central to proving upper bounds for the Steinitz problem, and the popular Komlós problem is a special case where one is only concerned with the final signed sum vector instead of all prefixes. Banaszczyk gave an $O(\sqrt{\log d+ \log T})$ bound for the prefix discrepancy problem. We investigate the tightness of Banaszczyk's bound and consider natural generalizations of prefix discrepancy: We first consider a smoothed analysis setting, where a small amount of additive noise perturbs the input vectors. We show an exponential improvement in $T$ compared to Banaszczyk's bound. Using a primal-dual approach and a careful chaining argument, we show that one can achieve a bound of $O(\sqrt{\log d+ \log\!\log T})$ with high probability in the smoothed setting. Moreover, this smoothed analysis bound is the best possible without further improvement on Banaszczyk's bound in the worst case. We also introduce a generalization of the prefix discrepancy problem where the discrepancy constraints correspond to paths on a DAG on $T$ vertices. We show that an analog of Banaszczyk's $O(\sqrt{\log d+ \log T})$ bound continues to hold in this setting for adversarially given unit vectors and that the $\sqrt{\log T}$ factor is unavoidable for DAGs. We also show that the dependence on $T$ cannot be improved significantly in the smoothed case for DAGs. We conclude by exploring a more general notion of vector balancing, which we call combinatorial vector balancing. We obtain near-optimal bounds in this setting, up to poly-logarithmic factors.

We study the problem of solving Packing Integer Programs (PIPs) in the online setting, where columns in [0, 1]d of the constraint matrix are revealed sequentially, and the goal is to pick a subset of the columns that sum to at most B in each coordinate while maximizing the objective. Excellent results are known in the secretary setting, where the columns are adversarially chosen, but presented in a uniformly random order. However, these existing algorithms are susceptible to adversarial attacks: they try to “learn” characteristics of a good solution, but tend to over-fit to the model, and hence a small number of adversarial corruptions can cause the algorithm to fail. In this paper, we give the first robust algorithms for Packing Integer Programs, specifically in the recently proposed Byzantine Secretary framework [BGSZ20]. Our techniques are based on a two-level use of online learning, to robustly learn an approximation to the optimal value, and then to use this robust estimate to pick a good solution. These techniques are general and we use them to design robust algorithms for PIPs in the prophet model as well, specifically in the Prophet-with-Augmentations framework [ISW20]. We also improve known results in the Byzantine Secretary framework: we make the non-constructive results algorithmic and improve the existing bounds for single-item and matroid constraints.

The vector-balancing problem is a fundamental problem in discrepancy theory: given T vectors in [–1, 1]n, find a signing σ(a) ∊ {±1} of each vector a to minimize the discrepancy ‖ Σa σ(a) · a‖∞. This problem has been extensively studied in the static/offline setting. In this paper we initiate its study in the fully-dynamic setting with recourse: the algorithm sees a stream of T insertions and deletions of vectors, and at each time must maintain a low-discrepancy signing, while also minimizing the amortized recourse (the number of times any vector changes its sign) per update. For general vectors, we show algorithms which almost match Spencer's offline discrepancy bound, with O(n polylog T) amortized recourse per update. The crucial idea behind our algorithm is to compute a basic feasible solution to the linear relaxation in a distributed and recursive manner, which helps find a low-discrepancy signing. We bound the recourse using the distributed computation of the basic solution, and argue that only a small part of the instance needs to be re-computed at each update. Since vector balancing has also been greatly studied for sparse vectors, we then give algorithms for low-discrepancy edge orientation, where we dynamically maintain signings for 2-sparse vectors in an n-dimensional space. Alternatively, this can be seen as orienting a dynamic set of edges of an n-vertex graph to minimize the discrepancy, i.e., the absolute difference between in- and out-degrees at any vertex. We present a deterministic algorithm with O(polylog n) discrepancy and O(polylog n) amortized recourse. The core ideas are to dynamically maintain an expander-decomposition with low recourse (using a very simple approach), and then to show that, as the expanders change over time, a natural local-search algorithm converges quickly (i.e., with low recourse) to a low-discrepancy solution. We also give strong lower bounds (with some matching upper bounds) for local-search discrepancy minimization algorithms for vector balancing and edge orientation.

Our main contribution is a general framework to design efficient polynomial time approximation schemes (EPTAS) for fundamental stochastic combinatorial optimization problems. Given an error parameter ε>0, such algorithmic schemes attain a (1-ε)-approximation in t(ε)· poly(n) time, where t(·) is some function that depends only on ε. Technically speaking, our approach relies on presenting tailor-made reductions to a newly-introduced multi-dimensional load balancing problem. Even though the single-dimensional problem is already known to be APX-Hard, we prove that an EPTAS can be designed under certain structural assumptions, which hold for each of our applications. To demonstrate the versatility of our framework, we first study selection-stopping settings to derive an EPTAS for the Free-Order Prophets problem [Agrawal et al., EC'20] and for its cost-driven generalization, Pandora's Box with Commitment [Fu et al., ICALP'18]. These results constitute the first approximation schemes in the non-adaptive setting and improve on known inefficient polynomial time approximation schemes (PTAS) for their adaptive variants. Next, turning our attention to stochastic probing problems, we obtain an EPTAS for the adaptive ProbeMax problem as well as for its non-adaptive counterpart; in both cases, state-of-the-art approximability results have been inefficient PTASes [Chen et al., NIPS'16; Fu et al., ICALP'18].

We present a computationally-efficient truthful mechanism for combinatorial auctions with subadditive bidders that achieves an $O((\log\!\log{m})^3)$-approximation to the maximum welfare in expectation using $O(n)$ demand queries; here $m$ and $n$ are the number of items and bidders, respectively. This breaks the longstanding logarithmic barrier for the problem dating back to the $O(\log{m}\cdot\log\!\log{m})$-approximation mechanism of Dobzinski from 2007. Along the way, we also improve and considerably simplify the state-of-the-art mechanisms for submodular bidders.

In the stochastic online vector balancing problem, vectors v1, v2, …, vT chosen independently from an arbitrary distribution in ℝn arrive one-by-one and must be immediately given a ± sign. The goal is to keep the norm of the discrepancy vector, i.e., the signed prefix-sum, as small as possible for a given target norm. We consider some of the most well-known problems in discrepancy theory in the above online stochastic setting, and give algorithms that match the known offline bounds up to polylog(nT) factors. This substantially generalizes and improves upon the previous results of Bansal, Jiang, Singla, and Sinha (STOC' 20). In particular, for the Komlós problem where ‖vt‖2 ≤ 1 for each t, our algorithm achieves Õ(1) discrepancy with high probability, improving upon the previous Õ(n3/2) bound. For Tusnády's problem of minimizing the discrepancy of axis-aligned boxes, we obtain an O(logd+4T) bound for arbitrary distribution over points. Previous techniques only worked for product distributions and gave a weaker O(log2d+1 T) bound. We also consider the Banaszczyk setting, where given a symmetric convex body K with Gaussian measure at least 1/2, our algorithm achieves Õ(1) discrepancy with respect to the norm given by K for input distributions with sub-exponential tails. Our results are based on a new potential function approach. Previous techniques consider a potential that penalizes large discrepancy, and greedily chooses the next color to minimize the increase in potential. Our key idea is to introduce a potential that also enforces constraints on how the discrepancy vector evolves, allowing us to maintain certain anti-concentration properties. We believe that our techniques to control the evolution of states could find other applications in stochastic processes and online algorithms. For the Banaszczyk setting, we further enhance this potential by combining it with ideas from generic chaining. Finally, we also extend these results to the setting of online multicolor discrepancy.

We introduce online learning with vector costs ($OLVC_p$) where in each time step $t \in \{1,\ldots, T\}$, we need to play an action $i \in \{1,\ldots,n\}$ that incurs an unknown vector cost in $[0,1]^d$. The goal of the online algorithm is to minimize the $\ell_p$ norm of the sum of its cost vectors. This captures the classical online learning setting for $d=1$, and is interesting for general $d$ because of applications like online scheduling where we want to balance the load between different machines (dimensions). We study $OLVC_p$ in both stochastic and adversarial arrival settings, and give a general procedure to reduce the problem from $d$ dimensions to a single dimension. This allows us to use classical online learning algorithms in both full and bandit feedback models to obtain (near) optimal results. In particular, we obtain a single algorithm (up to the choice of learning rate) that gives sublinear regret for stochastic arrivals and a tight $O(\min\{p, \log d\})$ competitive ratio for adversarial arrivals. The $OLVC_p$ problem also occurs as a natural subproblem when trying to solve the popular Bandits with Knapsacks (BWK) problem. This connection allows us to use our $OLVC_p$ techniques to obtain (near) optimal results for BWK in both stochastic and adversarial settings. In particular, we obtain a tight $O(\log d \cdot \log T)$ competitive ratio algorithm for adversarial BWK, which improves over the $O(d \cdot \log T)$ competitive ratio algorithm of Immorlica et al. (2019).

We consider the online carpooling problem: given $n$ vertices, a sequence of edges arrive over time. When an edge $e_t = (u_t, v_t)$ arrives at time step $t$, the algorithm must orient the edge either as $v_t \rightarrow u_t$ or $u_t \rightarrow v_t$, with the objective of minimizing the maximum discrepancy of any vertex, i.e., the absolute difference between its in-degree and out-degree. Edges correspond to pairs of persons wanting to ride together, and orienting denotes designating the driver. The discrepancy objective then corresponds to every person driving close to their fair share of rides they participate in. In this paper, we design efficient algorithms which can maintain polylog$(n,T)$ maximum discrepancy (w.h.p) over any sequence of $T$ arrivals, when the arriving edges are sampled independently and uniformly from any given graph $G$. This provides the first polylogarithmic bounds for the online (stochastic) carpooling problem. Prior to this work, the best known bounds were $O(\sqrt{n \log n})$-discrepancy for any adversarial sequence of arrivals, or $O(\log\!\log n)$-discrepancy bounds for the stochastic arrivals when $G$ is the complete graph. The technical crux of our paper is in showing that the simple greedy algorithm, which has provably good discrepancy bounds when the arriving edges are drawn uniformly at random from the complete graph, also has polylog discrepancy when $G$ is an expander graph. We then combine this with known expander-decomposition results to design our overall algorithm.

In classical secretary problems, a sequence of n elements arrive in a uniformly random order, and we want to choose a single item, or a set of size K. The random order model allows us to escape from the strong lower bounds for the adversarial order setting, and excellent algorithms are known in this setting. However, one worrying aspect of these results is that the algorithms overfit to the model: they are not very robust. Indeed, if a few "outlier" arrivals are adversarially placed in the arrival sequence, the algorithms perform poorly. E.g., Dynkin’s popular 1/e-secretary algorithm is sensitive to even a single adversarial arrival: if the adversary gives one large bid at the beginning of the stream, the algorithm does not select any element at all. We investigate a robust version of the secretary problem. In the Byzantine Secretary model, we have two kinds of elements: green (good) and red (rogue). The values of all elements are chosen by the adversary. The green elements arrive at times uniformly randomly drawn from [0,1]. The red elements, however, arrive at adversarially chosen times. Naturally, the algorithm does not see these colors: how well can it solve secretary problems? We show that selecting the highest value red set, or the single largest green element is not possible with even a small fraction of red items. However, on the positive side, we show that these are the only bad cases, by giving algorithms which get value comparable to the value of the optimal green set minus the largest green item. (This benchmark reminds us of regret minimization and digital auctions, where we subtract an additive term depending on the "scale" of the problem.) Specifically, we give an algorithm to pick K elements, which gets within (1-ε) factor of the above benchmark, as long as K ≥ poly(ε^{-1} log n). We extend this to the knapsack secretary problem, for large knapsack size K. For the single-item case, an analogous benchmark is the value of the second-largest green item. For value-maximization, we give a poly log^* n-competitive algorithm, using a multi-layered bucketing scheme that adaptively refines our estimates of second-max over time. For probability-maximization, we show the existence of a good randomized algorithm, using the minimax principle. We hope that this work will spur further research on robust algorithms for the secretary problem, and for other problems in sequential decision-making, where the existing algorithms are not robust and often tend to overfit to the model.

Given a metric (V,d) and a root ∈ V, the classic k-TSP problem is to find a tour originating at the root of minimum length that visits at least k nodes in V. In this work, motivated by applications where the input to an optimization problem is uncertain, we study two stochastic versions of k-TSP. In Stoch-Reward k-TSP, originally defined by Ene-Nagarajan-Saket [Ene et al., 2018], each vertex v in the given metric (V,d) contains a stochastic reward R_v. The goal is to adaptively find a tour of minimum expected length that collects at least reward k; here "adaptively" means our next decision may depend on previous outcomes. Ene et al. give an O(log k)-approximation adaptive algorithm for this problem, and left open if there is an O(1)-approximation algorithm. We totally resolve their open question, and even give an O(1)-approximation non-adaptive algorithm for Stoch-Reward k-TSP. We also introduce and obtain similar results for the Stoch-Cost k-TSP problem. In this problem each vertex v has a stochastic cost C_v, and the goal is to visit and select at least k vertices to minimize the expected sum of tour length and cost of selected vertices. Besides being a natural stochastic generalization of k-TSP, this problem is also interesting because it generalizes the Price of Information framework [Singla, 2018] from deterministic probing costs to metric probing costs. Our techniques are based on two crucial ideas: "repetitions" and "critical scaling". In general, replacing a random variable with its expectation leads to very poor results. We show that for our problems, if we truncate the random variables at an ideal threshold, then their expected values form a good surrogate. Here, we rely on running several repetitions of our algorithm with the same threshold, and then argue concentration using Freedman’s and Jogdeo-Samuels' inequalities. Unfortunately, this ideal threshold depends on how far we are from achieving our target k, which a non-adaptive algorithm does not know. To overcome this barrier, we truncate the random variables at various different scales and identify a "critical" scale.

In a classical online decision problem, a decision-maker who is trying to maximize her value inspects a sequence of arriving items to learn their values (drawn from known distributions), and decides when to stop the process by taking the current item. The goal is to prove a "prophet inequality": that she can do approximately as well as a prophet with foreknowledge of all the values. In this work, we investigate this problem when the values are allowed to be correlated. Since non-trivial guarantees are impossible for arbitrary correlations, we consider a natural "linear" correlation structure introduced by Bateni et al. [ESA'15] as a generalization of the common-base value model of Chawla et al. [GEB'15].

We consider an online vector balancing question where T vectors, chosen from an arbitrary distribution over [−1,1] n , arrive one-by-one and must be immediately given a ± sign. The goal is to keep the discrepancy—the ℓ∞-norm of any signed prefix-sum—as small as possible. A concrete example of this question is the online interval discrepancy problem where T points are sampled one-by-one uniformly in the unit interval [0,1], and the goal is to immediately color them ± such that every sub-interval remains always nearly balanced. As random coloring incurs Ω(T 1/2) discrepancy, while the worst-case offline bounds are Θ(√n log(T/n)) for vector balancing and 1 for interval balancing, a natural question is whether one can (nearly) match the offline bounds in the online setting for these problems. One must utilize the stochasticity as in the worst-case scenario it is known that discrepancy is Ω(T 1/2) for any online algorithm.

Consider a two-stage matching problem, where edges of an input graph are revealed in two stages (batches) and in each stage we have to immediately and irrevocably extend our matching using the edges from that stage. The natural greedy algorithm is half competitive. Even though there is a huge literature on online matching in adversarial vertex arrival model , no positive results were previously known in adversarial edge arrival model . For two-stage bipartite matching problem, we show that the optimal competitive ratio is exactly 2/3 in both the fractional and the randomized-integral models. Furthermore, our algorithm for fractional bipartite matching is instance optimal , i.e., it achieves the best competitive ratio for any given first stage graph. We also study natural extensions of this problem to general graphs and to s stages and present randomized-integral algorithms with competitive ratio ½ + 2− O(s) . Our algorithms use a novel Instance-Optimal-LP and combine graph decomposition techniques with online primal-dual analysis.

Consider a kidney-exchange application where we want to find a max-matching in a random graph. To find whether an edge $e$ exists, we need to perform an expensive test, in which case the edge $e$ appears independently with a \emph{known} probability $p_e$. Given a budget on the total cost of the tests, our goal is to find a testing strategy that maximizes the expected maximum matching size. The above application is an example of the stochastic probing problem. In general the optimal stochastic probing strategy is difficult to find because it is \emph{adaptive}---decides on the next edge to probe based on the outcomes of the probed edges. An alternate approach is to show the \emph{adaptivity gap} is small, i.e., the best \emph{non-adaptive} strategy always has a value close to the best adaptive strategy. This allows us to focus on designing non-adaptive strategies that are much simpler. Previous works, however, have focused on Bernoulli random variables that can only capture whether an edge appears or not. In this work we introduce a multi-value stochastic probing problem, which can also model situations where the weight of an edge has a probability distribution over multiple values. Our main technical contribution is to obtain (near) optimal bounds for the (worst-case) adaptivity gaps for multi-value stochastic probing over prefix-closed constraints. For a monotone submodular function, we show the adaptivity gap is at most $2$ and provide a matching lower bound. For a weighted rank function of a $k$-extendible system (a generalization of intersection of $k$ matroids), we show the adaptivity gap is between $O(k\log k)$ and $k$. None of these results were known even in the Bernoulli case where both our upper and lower bounds also apply, thereby resolving an open question of Gupta et al.

In this paper we study how to optimally balance cheap inflexible resources with more expensive, reconfigurable resources despite uncertainty in the input problem. Specifically, we introduce the MinEMax model to study "build versus rent" problems. In our model different scenarios appear independently. Before knowing which scenarios appear, we may build rigid resources that cannot be changed for different scenarios. Once we know which scenarios appear, we are allowed to rent reconfigurable but expensive resources to use across scenarios. Although computing the objective in our model might seem to require enumerating exponentially-many possibilities, we show it is well-estimated by a surrogate objective which is representable by a polynomial-size LP. In this surrogate objective we pay for each scenario only to the extent that it exceeds a certain threshold. Using this objective we design algorithms that approximately-optimally balance inflexible and reconfigurable resources for several NP-hard covering problems. For example, we study minimum spanning and Steiner trees, minimum cuts and facility location variants. Up to constants our approximation guarantees match those of previous algorithms for the previously-studied demand-robust and stochastic two-stage models. Lastly, we demonstrate that our problem is sufficiently general to smoothly interpolate between previous demand-robust and stochastic two-stage problems.

A longstanding open problem in Algorithmic Mechanism Design is to design computationally-efficient truthful mechanisms for (approximately) maximizing welfare in combinatorial auctions with submodular bidders. The first such mechanism was obtained by Dobzinski, Nisan, and Schapira [STOC'06] who gave an O(log2m)-approximation where m is number of items. This problem has been studied extensively since, culminating in an O(√log m)-approximation mechanism by Dobzinski [STOC'16]. We present a computationally-efficient truthful mechanism with approximation ratio that improves upon the state-of-the-art by an exponential factor. In particular, our mechanism achieves an O((log log m)3)-approximation in expectation, uses only O(n) demand queries, and has universal truthfulness whether Θ(√log m) is the best approximation ratio in this guarantee. This settles an open question of Dobzinski on setting in negative.

In this paper we consider the classic matroid intersection problem: given two matroids M1= (V, I1) and M2= (V, I2) defined over a common ground set V , compute a set S ∈ I1∩ I2of largest possible cardinality, denoted by r. We consider this problem both in the setting where each Mi is accessed through an independence oracle, i.e. a routine which returns whether or not a set S ∈ Iiin Tindtime, and the setting where each Mi is accessed through a rank oracle, i.e. a routine which returns the size of the largest independent subset of S in Miin Tranktime. In each setting we provide faster exact and approximate algorithms. Given an independence oracle, we provide an exact O(nr log r · Tind) time algorithm. This improves upon previous best known running times of O(nr1.5·Tind) due to Cunningham O(n2·Tindin 1986 and + n3) due to Lee, Sidford, and Wong in 2015. We also provide two algorithms which compute a (1- ε-approximate solution to matroid intersection running in times O(n1.5/ε1.5· Tind) and O((n2r-1ε-2+ r1.5ε-4.5) · Tind), respectively. These results improve upon the O(nr/ε · Tind)time algorithm of Cunningham (noted recently by Chekuri and Quanrud). Given a rank oracle, we provide algorithms with even better dependence on n and r. We provide an O(n√r log n · Trank)time exact algorithm and an O(nε-1log n · Trank)-time algorithm which obtains a (1 - 0)-approximation to the matroid intersection problem. The former result improves over the O(nr · Trank+ n3)-time algorithm by Lee, Sidford, and Wong. The rank oracle is of particular interest as the matroid intersection problem with this oracle is a special case (via Edmond's minimax characterization of matroid intersection) of the submodular function minimization (SFM) problem with an evaluation oracle, and understanding SFM query complexity is an outstanding open question.

We consider a revenue-maximizing seller with n items facing a single buyer. We introduce the notion of symmetric menu complexity of a mechanism, which counts the number of distinct options the buyer may purchase, up to permutations of the items. Our main result is that a mechanism of quasi-polynomial symmetric menu complexity suffices to guarantee a (1 - epsilon )-approximation when the buyer is unit-demand over independent items, even when the value distribution is unbounded, and that this mechanism can be found in quasi-polynomial time. Our key technical result is a polynomial-time, (symmetric) menu-complexity-preserving black-box reduction from achieving a (1 - epsilon )-approximation for unbounded valuations that are subadditive over independent items to achieving a (1 - O(epsilon ))-approximation when the values are bounded (and still subadditive over independent items). We further apply this reduction to deduce approximation schemes for a suite of valuation classes beyond our main result. Finally, we show that selling separately (which has exponential menu complexity) can be approximated up to a (1 - epsilon ) factor with a menu of efficient-linear (f (epsilon) · n) symmetric menu complexity.

We consider the online problem of scheduling jobs on identical machines, where jobs have precedence constraints. We are interested in the demanding setting where the jobs sizes are not known up-front, but are revealed only upon completion (the non-clairvoyant setting). Such precedence-constrained scheduling problems routinely arise in map-reduce and large-scale optimization. For minimizing the total weighted completion time, we give a constant-competitive algorithm. And for total weighted flow-time, we give an O(1/epsilon^2)-competitive algorithm under (1+epsilon)-speed augmentation and a natural "no-surprises" assumption on release dates of jobs (which we show is necessary in this context). Our algorithm proceeds by assigning virtual rates to all waiting jobs, including the ones which are dependent on other uncompleted jobs. We then use these virtual rates to decide on the actual rates of minimal jobs (i.e., jobs which do not have dependencies and hence are eligible to run). Interestingly, the virtual rates are obtained by allocating time in a fair manner, using a Eisenberg-Gale-type convex program (which we can solve optimally using a primal-dual scheme). The optimality condition of this convex program allows us to show dual-fitting proofs more easily, without having to guess and hand-craft the duals. This idea of using fair virtual rates may have broader applicability in scheduling problems.

Online contention resolution schemes (OCRSs) were proposed by Feldman, Svensson, and Zenklusen as a generic technique to round a fractional solution in the matroid polytope in an online fashion. It has found applications in several stochastic combinatorial problems where there is a commitment constraint: on seeing the value of a stochastic element, the algorithm has to immediately and irrevocably decide whether to select it while always maintaining an independent set in the matroid. Although OCRSs immediately lead to prophet inequalities, these prophet inequalities are not optimal. Can we instead use prophet inequalities to design optimal OCRSs? We design the first optimal $1/2$-OCRS for matroids by reducing the problem to designing a matroid prophet inequality where we compare to the stronger benchmark of an ex-ante relaxation. We also introduce and design optimal $(1-1/e)$-random order CRSs for matroids, which are similar to OCRSs but the arrival is chosen uniformly at random.

The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg [33] and Feldman et al. [17] show that for adversarial arrival order of random variables the optimal prophet inequalities give a 1/2-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the 1/2-approximation and obtain (1 – 1/e)-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan [45] and Esfandiari et al. [15] who worked in the special cases where we can fully control the arrival order or when there is only a single item. Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.

Consider a network design application where we wish to lay down a minimum-cost spanning tree in a given graph; however, we only have stochastic information about the edge costs. To learn the precise cost of any edge, we have to conduct a study that incurs a price. Our goal is to find a spanning tree while minimizing the disutility, which is the sum of the tree cost and the total price that we spend on the studies. In a different application, each edge gives a stochastic reward value. Our goal is to find a spanning tree while maximizing the utility, which is the tree reward minus the prices that we pay. Situations such as the above two often arise in practice where we wish to find a good solution to an optimization problem, but we start with only some partial knowledge about the parameters of the problem. The missing information can be found only after paying a probing price, which we call the price of information. What strategy should we adopt to optimize our expected utility/disutility? A classical example of the above setting is Weitzman's “Pandora's box” problem where we are given probability distributions on values of n independent random variables. The goal is to choose a single variable with a large value, but we can find the actual outcomes only after paying a price. Our work is a generalization of this model to other combinatorial optimization problems such as matching, set cover, facility location, and prize-collecting Steiner tree. We give a technique that reduces such problems to their non-price counterparts, and use it to design exact/approximation algorithms to optimize our utility/disutility. Our techniques extend to situations where there are additional constraints on what parameters can be probed or when we can simultaneously probe a subset of the parameters.

Suppose we are given a submodular function f over a set of elements, and we want to maximize its value subject to certain constraints. Good approximation algorithms are known for such problems under both monotone and non-monotone submodular functions. We consider these problems in a stochastic setting, where elements are not all active and we only get value from active elements. Each element e is active independently with some known probability pe, but we don't know the element's status a priori: we find it out only when we probe the element e. Moreover, the sequence of elements we probe must satisfy a given prefix-closed constraint, e.g., matroid, orienteering, deadline, precedence, or any downward-closed constraint. In this paper we study the gap between adaptive and non-adaptive strategies for f being a submodular or a fractionally subadditive (XOS) function. If this gap is small, we can focus on finding good non-adaptive strategies instead, which are easier to find as well as to represent. We show that the adaptivity gap is a constant for monotone and non-monotone submodular functions, and logarithmic for XOS functions of small width. These bounds are nearly tight. Our techniques show new ways of arguing about the optimal adaptive decision tree for stochastic optimization problems.

We introduce a novel framework of Prophet Inequalities for combinatorial valuation functions. For a (non-monotone) submodular objective function over an arbitrary matroid feasibility constraint, we give an O(1)-competitive algorithm. For a monotone subadditive objective function over an arbitrary downward- closed feasibility constraint, we give an O(log n log2 r)- competitive algorithm (where r is the cardinality of the largest feasible subset). Inspired by the proof of our subadditive prophet inequality, we also obtain an O(log n · log2 r)-competitive algorithm for the Secretary Problem with a monotone subadditive objective function subject to an arbitrary downward-closed feasibility constraint. Even for the special case of a cardinality feasibility constraint, our algorithm circumvents an lower bound by Bateni, Hajiaghayi, and Zadimoghaddam [10] in a restricted query model. En route to our submodular prophet inequality, we prove a technical result of independent interest: we show a variant of the Correlation Gap Lemma [14, 1] for nonmonotone submodular functions.

Given an $n$-vertex graph and two straight-line planar drawings of the graph that have the same faces and the same outer face, we show that there is a morph (i.e., a continuous transformation) between the two drawings that preserves straight-line planarity and consists of $O(n)$ steps, which we prove is optimal in the worst case. Each step is a unidirectional linear morph, which means that every vertex moves at constant speed along a straight line, and the lines are parallel although the vertex speeds may differ. Thus we provide an efficient version of Cairns' 1944 proof of the existence of straight-line planarity-preserving morphs for triangulated graphs, which required an exponential number of steps.

A stochastic probing problem consists of a set of elements whose values are independent random variables. The algorithm knows the distributions of these variables, but not the actual outcomes. The only way to learn the actual outcomes is to probe these elements. However, there are constraints on which set of elements may be probed. (E.g., we may have to travel in some metric to probe elements but have limited time.) These constraints are called outer constraints. We want to develop an algorithm that picks some set of elements to maximize the (expected) value, subject to the picked subset of elements satisfying some other set of constraints, called the inner constraints. In the past, probing problems were studied for the case when both inner and outer constraints were intersections of matroids; these modeled kidney matching and Bayesian auctions applications. One limitation of past work was their reliance on linear-programming-like techniques, which made going beyond matroid-like structures difficult. In this work, we give a very general adaptivity gap result that holds for all prefix-closed outer constraints, as long as the inner constraints are intersections of matroids. The adaptivity gap is O(log n) for any constant number of inner matroid constraints. The prefix-closedness captures most “reasonable” outer constraints, like orienteering, connectivity, and precedence. Based on this we obtain the first approximation algorithms for a number of stochastic probing problems, which have applications, e.g., to path-planning and precedence-constrained scheduling.

TTL caching models have recently regained significant research interest, largely due to their ability to fit popular caching policies such as LRU. In this extended abstract we briefly describe our recent work on two exact methods to analyze TTL cache networks. The first method generalizes existing results for line networks under renewal requests to the broad class of caching policies whereby evictions are driven by stopping times. The obtained results are further generalized, using the second method, to feedforward networks with Markov arrival processes (MAP) requests. MAPs are particularly suitable for non-line networks because they are closed not only under superposition and splitting, as known, but also under input-output caching operations as proven herein for phase-type TTL distributions. The crucial benefit of the two closure properties is that they jointly enable the first exact analysis of feedforward networks of TTL caches in great generality.

In 1944, Cairns proved the following theorem: given any two straight-line planar drawings of a triangulation with the same outer face, there exists a morph (i.e., a continuous transformation) between the two drawings so that the drawing remains straight-line planar at all times. Cairns's original proof required exponentially many morphing steps. We prove that there is a morph that consists of O(n2) steps, where each step is a linear morph that moves each vertex at constant speed along a straight line. Using a known result on compatible triangulations this implies that for a general planar graph G and any two straight-line planar drawings of G with the same embedding, there is a morph between the two drawings that preserves straight-line planarity and consists of O(n4) steps.