Beyond matroids: secretary problem and prophet inequality with general constraints

Authors

Aviad Rubinstein

Type: Article

Publication Date: 2016-06-10

Citations: 57

DOI: https://doi.org/10.1145/2897518.2897540

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Abstract

We study generalizations of the ``Prophet Inequality'' and ``Secretary Problem'', where the algorithm is restricted to an arbitrary downward-closed set system. For 0,1 values, we give O(n)-competitive algorithms for both problems. This is close to the Omega(n/log n) lower bound due to Babaioff, Immorlica, and Kleinberg. For general values, our results translate to O(log(n) log(r))-competitive algorithms, where r is the cardinality of the largest feasible set. This resolves (up to the O(loglog(n) log(r)) factor) an open question posed to us by Bobby Kleinberg.

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arXiv (Cornell University)
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2020-11-01

Vladimir Braverman , Petros Drineas , Cameron Musco +4 more

We initiate the study of numerical linear algebra in the sliding window model, where only the most recent W updates in a stream form the underlying data set. Although many … We initiate the study of numerical linear algebra in the sliding window model, where only the most recent W updates in a stream form the underlying data set. Although many existing algorithms in the sliding window model use or borrow elements from the smooth histogram framework (Braverman and Ostrovsky, FOCS 2007), we show that many interesting linear-algebraic problems, including spectral and vector induced matrix norms, generalized regression, and lowrank approximation, are not amenable to this approach in the row-arrival model. To overcome this challenge, we first introduce a unified row-sampling based framework that gives randomized algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and ℓ 1-subspace embeddings in the sliding window model, which often use nearly optimal space and achieve nearly input sparsity runtime. Our algorithms are based on "reverse online" versions of offline sampling distributions such as (ridge) leverage scores, ℓ 1 sensitivities, and Lewis weights to quantify both the importance and the recency of a row; our structural results on these distributions may be of independent interest for future algorithmic design. Although our techniques initially address numerical linear algebra in the sliding window model, our row-sampling framework rather surprisingly implies connections to the well-studied online model; our structural results also give the first sample optimal (up to lower order terms) online algorithm for low-rank approximation/projection-cost preservation. Using this powerful primitive, we give online algorithms for column/row subset selection and principal component analysis that resolves the main open question of Bhaskara et al. (FOCS 2019). We also give the first online algorithm for ℓ 1-subspace embeddings. We further formalize the connection between the online model and the sliding window model by introducing an additional unified framework for deterministic algorithms using a merge and reduce paradigm and the concept of online coresets, which we define as a weighted subset of rows of the input matrix that can be used to compute a good approximation to some given function on all of its prefixes. Our sampling based algorithms in the row-arrival online model yield online coresets, giving deterministic algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and ℓ 1-subspace embeddings in the sliding window model that use nearly optimal space.

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Robust Algorithms for the Secretary Problem

2019-01-01

Domagoj Bradač , Anupam Gupta , Sahil Singla +1 more

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 … 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 fails with even a single adversarial arrival. 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 give algorithms which get value comparable to the value of the optimal green set minus the largest green item. Specifically, we give an algorithm to pick $K$ elements that gets within $(1-\varepsilon)$ factor of the above benchmark, as long as $K \geq \mathrm{poly}(\varepsilon^{-1} \log n)$. 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 $\mathrm{poly} \log^* n$-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.

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Competitive Online Optimization under Inventory Constraints

2019-03-26

Qiulin Lin , Hanling Yi , J.H.L. Pang +4 more

This paper studies online optimization under inventory (budget) constraints. While online optimization is a well-studied topic, versions with inventory constraints have proven difficult. We consider a formulation of inventory-constrained optimization … This paper studies online optimization under inventory (budget) constraints. While online optimization is a well-studied topic, versions with inventory constraints have proven difficult. We consider a formulation of inventory-constrained optimization that is a generalization of the classic one-way trading problem and has a wide range of applications. We present a new algorithmic framework, \textsfCR-Pursuit, and prove that it achieves the minimal competitive ratio among all deterministic algorithms (up to a problem-dependent constant factor) for inventory-constrained online optimization. Our algorithm and its analysis not only simplify and unify the state-of-the-art results for the standard one-way trading problem, but they also establish novel bounds for generalizations including concave revenue functions. For example, for one-way trading with price elasticity, the \textsfCR-Pursuit algorithm achieves a competitive ratio that is within a small additive constant (i.e., 1/3) to the lower bound of ln 0+1, where 0 is the ratio between the maximum and minimum base prices.

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Competitive Online Optimization under Inventory Constraints

2019-12-17

Qiulin Lin , Hanling Yi , John Z. F. Pang +4 more

This paper studies online optimization under inventory (budget) constraints. While online optimization is a well-studied topic, versions with inventory constraints have proven difficult. We consider a formulation of inventory-constrained optimization … This paper studies online optimization under inventory (budget) constraints. While online optimization is a well-studied topic, versions with inventory constraints have proven difficult. We consider a formulation of inventory-constrained optimization that is a generalization of the classic one-way trading problem and has a wide range of applications. We present a new algorithmic framework, CR-Pursuit, and prove that it achieves the optimal competitive ratio among all deterministic algorithms (up to a problem-dependent constant factor) for inventory-constrained online optimization. Our algorithm and its analysis not only simplify and unify the state-ofthe- art results for the standard one-way trading problem, but they also establish novel bounds for generalizations including concave revenue functions. For example, for one-way trading with price elasticity, CR-Pursuit achieves a competitive ratio within a small additive constant (i.e., 1/3) to the lower bound of ln θ + 1, where θ is the ratio between the maximum and minimum base prices.

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Choosing Behind the Veil: Tight Bounds for Identity-Blind Online Algorithms

2024-07-08

Tomer Ezra , Michal Feldman , Zhihao Gavin Tang

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The Stationary Prophet Inequality Problem

2022-07-12

Kristen Kessel , Ali Shameli , Amin Saberi +1 more

We study a continuous and infinite time horizon counterpart to the classic prophet inequality, which we term the stationary prophet inequality problem. Here, copies of a good arrive and perish … We study a continuous and infinite time horizon counterpart to the classic prophet inequality, which we term the stationary prophet inequality problem. Here, copies of a good arrive and perish according to Poisson point processes. Buyers arrive similarly and make take-it-or-leave-it offers for unsold items. The objective is to maximize the (infinite) time average revenue of the seller. Our main results are pricing-based policies which (i) achieve a 1/2-approximation of the optimal offline policy, which is best possible, and (ii) achieve a better than (1-1/e)-approximation of the optimal online policy. Result (i) improves upon bounds implied by recent work of Collina et al. (WINE'20), and is the first optimal prophet inequality for a stationary problem. Result (ii) improves upon a 1-1/e bound implied by recent work of Aouad and Sarita (EC'20), and shows that this prevalent bound in online algorithms is not optimal for this problem.

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Secretary Matching with General Arrivals

2020-01-01

Tomer Ezra , Michal Feldman , Nick Gravin +1 more

We provide online algorithms for secretary matching in general weighted graphs, under the well-studied models of vertex and edge arrivals. In both models, edges are associated with arbitrary weights that … We provide online algorithms for secretary matching in general weighted graphs, under the well-studied models of vertex and edge arrivals. In both models, edges are associated with arbitrary weights that are unknown from the outset, and are revealed online. Under vertex arrival, vertices arrive online in a uniformly random order; upon the arrival of a vertex $v$, the weights of edges from $v$ to all previously arriving vertices are revealed, and the algorithm decides which of these edges, if any, to include in the matching. Under edge arrival, edges arrive online in a uniformly random order; upon the arrival of an edge $e$, its weight is revealed, and the algorithm decides whether to include it in the matching or not. We provide a $5/12$-competitive algorithm for vertex arrival, and show it is tight. For edge arrival, we provide a $1/4$-competitive algorithm. Both results improve upon state of the art bounds for the corresponding settings. Interestingly, for vertex arrival, secretary matching in general graphs outperforms secretary matching in bipartite graphs with 1-sided arrival, where $1/e$ is the best possible guarantee.

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On Multi-Dimensional Gains from Trade Maximization

2020-01-01

Yang Cai , Kira Goldner , Steven Ma +1 more

We study gains from trade in multi-dimensional two-sided markets. Specifically, we focus on a setting with $n$ heterogeneous items, where each item is owned by a different seller $i$, and … We study gains from trade in multi-dimensional two-sided markets. Specifically, we focus on a setting with $n$ heterogeneous items, where each item is owned by a different seller $i$, and there is a constrained-additive buyer with feasibility constraint $\mathcal{F}$. Multi-dimensional settings in one-sided markets, e.g. where a seller owns multiple heterogeneous items but also is the mechanism designer, are well-understood. In addition, single-dimensional settings in two-sided markets, e.g. where a buyer and seller each seek or own a single item, are also well-understood. Multi-dimensional two-sided markets, however, encapsulate the major challenges of both lines of work: optimizing the sale of heterogeneous items, ensuring incentive-compatibility among both sides of the market, and enforcing budget balance. We present, to the best of our knowledge, the first worst-case approximation guarantee for gains from trade in a multi-dimensional two-sided market. Our first result provides an $O(\log (1/r))$-approximation to the first-best gains from trade for a broad class of downward-closed feasibility constraints (such as matroid, matching, knapsack, or the intersection of these). Here $r$ is the minimum probability over all items that a buyer's value for the item exceeds the seller's cost. Our second result removes the dependence on $r$ and provides an unconditional $O(\log n)$-approximation to the second-best gains from trade. We extend both results for a general constrained-additive buyer, losing another $O(\log n)$-factor en-route.

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Tight Revenue Gaps among Simple Mechanisms

2019-01-01

Yaonan Jin , Pinyan Lu , Zhihao Gavin Tang +1 more

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 … 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.

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Online Stochastic Max-Weight Matching: prophet inequality for vertex and edge arrival models

2020-02-23

Tomer Ezra , Michal Feldman , Nick Gravin +1 more

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 … 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 scheme (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 the vertex arrival, our result is tight. Interestingly, a pricing-based prophet inequality with comparable competitive ratios is unknown.

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Unknown I.I.D. prophets: better bounds, streaming algorithms, and a new impossibility

2020-07-12

José R. Correa , Paul Dütting , Felix Fischer +2 more

A prophet inequality states, for some α ∈ [0, 1], that the expected value achievable by a gambler who sequentially observes random variables X1, . . . , Xn and … A prophet inequality states, for some α ∈ [0, 1], that the expected value achievable by a gambler who sequentially observes random variables X1, . . . , Xn and selects one of them is at least an α fraction of the maximum value in the sequence. We obtain three distinct improvements for a setting that was first studied by Correa et al. (EC, 2019) and is particularly relevant to modern applications in algorithmic pricing. In this setting, the random variables are i.i.d. from an unknown distribution and the gambler has access to an additional βn samples for some β ≥ 0. We first give improved lower bounds on α for a wide range of values of β; specifically, α ≥ (1 + β)/e when β ≤ 1/(e − 1), which is tight, and α ≥ 0.648 when β = 1, which improves on a bound of around 0.635 due to Correa et al. (SODA, 2020). Adding to their practical appeal, specifically in the context of algorithmic pricing, we then show that the new bounds can be obtained even in a streaming model of computation and thus in situations where the use of relevant data is complicated by the sheer amount of data available. We finally establish that the upper bound of 1/e for the case without samples is robust to additional information about the distribution, and applies also to sequences of i.i.d. random variables whose distribution is itself drawn, according to a known distribution, from a finite set of known candidate distributions. This implies a tight prophet inequality for exchangeable sequences of random variables, answering a question of Hill and Kertz (Contemporary Mathematics, 1992), but leaves open the possibility of better guarantees when the number of candidate distributions is small, a setting we believe is of strong interest to applications.

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Prophet Inequalities with Linear Correlations and Augmentations

2020-01-01

Nicole Immorlica , Sahil Singla , Bo Waggoner

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 … 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 2015] as a generalization of the common-base value model of Chawla et al. [GEB 2015]. A key challenge is that threshold-based algorithms, which are commonly used for prophet inequalities, no longer guarantee good performance for linear correlations. We relate this roadblock to another "augmentations" challenge that might be of independent interest: many existing prophet inequality algorithms are not robust to slight increase in the values of the arriving items. We leverage this intuition to prove bounds (matching up to constant factors) that decay gracefully with the amount of correlation of the arriving items. We extend these results to the case of selecting multiple items by designing a new $(1+o(1))$ approximation ratio algorithm that is robust to augmentations.

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Prophet Inequalities with Linear Correlations and Augmentations

2020-07-09

Nicole Immorlica , Sahil Singla , Bo Waggoner

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Strong Algorithms for the Ordinal Matroid Secretary Problem

2021-02-23

José A. Soto , Abner Turkieltaub , Víctor Verdugo

In the ordinal matroid secretary problem (MSP), candidates do not reveal numerical weights, but the decision maker can still discern if a candidate is better than another. An algorithm [Formula: … In the ordinal matroid secretary problem (MSP), candidates do not reveal numerical weights, but the decision maker can still discern if a candidate is better than another. An algorithm [Formula: see text] is probability-competitive if every element from the optimum appears with probability [Formula: see text] in the output. This measure is stronger than the standard utility competitiveness. Our main result is the introduction of a technique based on forbidden sets to design algorithms with strong probability-competitive ratios on many matroid classes. We improve upon the guarantees for almost every matroid class considered in the MSP literature. In particular, we achieve probability-competitive ratios of 4 for graphic matroids and of [Formula: see text] for laminar matroids. Additionally, we modify Kleinberg’s [Formula: see text] utility-competitive algorithm for uniform matroids of rank [Formula: see text] in order to obtain a [Formula: see text] probability-competitive algorithm. We also contribute algorithms for the ordinal MSP on arbitrary matroids.

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Matroid Secretary is Equivalent to Contention Resolution.

2021-03-06

Shaddin Dughmi

We show that the matroid secretary problem is equivalent to correlated contention resolution in the online random-order model. Specifically, the matroid secretary conjecture is true if and only if every … We show that the matroid secretary problem is equivalent to correlated contention resolution in the online random-order model. Specifically, the matroid secretary conjecture is true if and only if every matroid admits an online random-order contention resolution scheme which, given an arbitrary (possibly correlated) prior distribution over subsets of the ground set, matches the balance ratio of the best offline scheme for that distribution up to a constant. We refer to such a scheme as universal. Our result indicates that the core challenge of the matroid secretary problem lies in resolving contention for positively correlated inputs, in particular when the positive correlation is benign in as much as offline contention resolution is concerned. Our result builds on our previous work which establishes one direction of this equivalence, namely that the secretary conjecture implies universal random-order contention resolution, as well as a weak converse, which derives a matroid secretary algorithm from a random-order contention resolution scheme with only partial knowledge of the distribution. It is this weak converse that we strengthen in this paper: We show that universal random-order contention resolution for matroids, in the usual setting of a fully known prior distribution, suffices to resolve the matroid secretary conjecture in the affirmative. Our proof is the composition of three reductions. First, we use duality arguments to reduce the matroid secretary problem to the matroid prophet secretary problem with arbitrarily correlated distributions. Second, we introduce a generalization of contention resolution we term labeled contention resolution, to which we reduce the correlated matroid prophet secretary problem. Finally, we combine duplication of elements with limiting arguments to reduce labeled contention resolution to classical contention resolution.

Single-Sample Prophet Inequalities via Greedy-Ordered Selection

2022-01-01

Constantine Caramanis , Paul Dütting , Matthew Faw +6 more

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Single-Sample Prophet Inequalities via Greedy-Ordered SelectionConstantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Single-Sample Prophet Inequalities via Greedy-Ordered SelectionConstantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip Lazos, Stefano Leonardi, Orestis Papadigenopoulos, Emmanouil Pountourakis, and Rebecca ReiffenhäuserConstantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip Lazos, Stefano Leonardi, Orestis Papadigenopoulos, Emmanouil Pountourakis, and Rebecca Reiffenhäuserpp.1298 - 1325Chapter DOI:https://doi.org/10.1137/1.9781611977073.54PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771

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Robust Secretary and Prophet Algorithms for Packing Integer Programs

2022-01-01

C. J. Argue , Anupam Gupta , Marco Molinaro +1 more

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Robust Secretary and Prophet Algorithms for Packing Integer ProgramsC.J. Argue, Anupam Gupta, Marco Molinaro, … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Robust Secretary and Prophet Algorithms for Packing Integer ProgramsC.J. Argue, Anupam Gupta, Marco Molinaro, and Sahil SinglaC.J. Argue, Anupam Gupta, Marco Molinaro, and Sahil Singlapp.1273 - 1297Chapter DOI:https://doi.org/10.1137/1.9781611977073.53PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract 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. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771

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Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Non-Stochastic Inputs

2017-10-01

Paul Dütting , Michal Feldman , Thomas Keßelheim +1 more

We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold … We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms, and is used to derive new and improved results for combinatorial markets (with and without complements), multi-dimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.

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The Outer Limits of Contention Resolution on Matroids and Connections to the Secretary Problem

2019-01-01

Shaddin Dughmi

Contention resolution schemes have proven to be a useful and unifying abstraction for a variety of constrained optimization problems, in both offline and online arrival models. Much of prior work … Contention resolution schemes have proven to be a useful and unifying abstraction for a variety of constrained optimization problems, in both offline and online arrival models. Much of prior work restricts attention to product distributions for the input set of elements, and studies contention resolution for increasingly general packing constraints, both offline and online. In this paper, we instead focus on generalizing the input distribution, restricting attention to matroid constraints in both the offline and online random arrival models. In particular, we study contention resolution when the input set is arbitrarily distributed, and may exhibit positive and/or negative correlations between elements. We characterize the distributions for which offline contention resolution is possible, and establish some of their basic closure properties. Our characterization can be interpreted as a distributional generalization of the matroid covering theorem. For the online random arrival model, we show that contention resolution is intimately tied to the secretary problem via two results. First, we show that a competitive algorithm for the matroid secretary problem implies that online contention resolution is essentially as powerful as offline contention resolution for matroids, so long as the algorithm is given the input distribution. Second, we reduce the matroid secretary problem to the design of an online contention resolution scheme of a particular form.

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Optimal Single-Choice Prophet Inequalities from Samples

2019-11-18

Aviad Rubinstein , Jack Z. Wang , S. Matthew Weinberg

We study the single-choice Prophet Inequality problem when the gambler is given access to samples. We show that the optimal competitive ratio of $1/2$ can be achieved with a single … We study the single-choice Prophet Inequality problem when the gambler is given access to samples. We show that the optimal competitive ratio of $1/2$ can be achieved with a single sample from each distribution. When the distributions are identical, we show that for any constant $\varepsilon > 0$, $O(n)$ samples from the distribution suffice to achieve the optimal competitive ratio ($\approx 0.745$) within $(1+\varepsilon)$, resolving an open problem of Correa, Dutting, Fischer, and Schewior.

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Near Optimal Linear Algebra in the Online and Sliding Window Models

2018-01-01

Vladimir Braverman , Petros Drineas , Cameron Musco +4 more

We initiate the study of numerical linear algebra in the sliding window model, where only the most recent $W$ updates in a stream form the underlying data set. We first … We initiate the study of numerical linear algebra in the sliding window model, where only the most recent $W$ updates in a stream form the underlying data set. We first introduce a unified row-sampling based framework that gives randomized algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and $\ell_1$-subspace embeddings in the sliding window model, which often use nearly optimal space and achieve nearly input sparsity runtime. Our algorithms are based on "reverse online" versions of offline sampling distributions such as (ridge) leverage scores, $\ell_1$ sensitivities, and Lewis weights to quantify both the importance and the recency of a row. Our row-sampling framework rather surprisingly implies connections to the well-studied online model; our structural results also give the first sample optimal (up to lower order terms) online algorithm for low-rank approximation/projection-cost preservation. Using this powerful primitive, we give online algorithms for column/row subset selection and principal component analysis that resolves the main open question of Bhaskara et. al.,(FOCS 2019). We also give the first online algorithm for $\ell_1$-subspace embeddings. We further formalize the connection between the online model and the sliding window model by introducing an additional unified framework for deterministic algorithms using a merge and reduce paradigm and the concept of online coresets. Our sampling based algorithms in the row-arrival online model yield online coresets, giving deterministic algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and $\ell_1$-subspace embeddings in the sliding window model that use nearly optimal space.

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On optimal ordering in the optimal stopping problem

2019-01-01

Shipra Agrawal , Jay Sethuraman , Xingyu Zhang

In the classical optimal stopping problem, a player is given a sequence of random variables $X_1\ldots X_n$ with known distributions. After observing the realization of $X_i$, the player can either … In the classical optimal stopping problem, a player is given a sequence of random variables $X_1\ldots X_n$ with known distributions. After observing the realization of $X_i$, the player can either accept the observed reward from $X_i$ and stop, or reject the observed reward from $X_i$ and continue to observe the next variable $X_{i+1}$ in the sequence. Under any fixed ordering of the random variables, an optimal stopping policy, one that maximizes the player's expected reward, is given by the solution of a simple dynamic program. In this paper, we investigate the relatively less studied question of selecting the order in which the random variables should be observed so as to maximize the expected reward at the stopping time. To demonstrate the benefits of order selection, we prove a novel prophet inequality showing that, when the support of each random variable has size at most 2, the optimal ordering can achieve an expected reward that is within a factor of 1.25 of the expected hindsight maximum; this is an improvement over the corresponding factor of 2 for the worst-case ordering. We also provide a simple $O(n^2)$ algorithm for finding an optimal ordering in this case. Perhaps surprisingly, we demonstrate that a slightly more general case - each random variable $X_i$ is restricted to have 3-point support of form $\{0, m_i, 1\}$ - is NP-hard, and provide an FPTAS for that case.

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Bayesian and Randomized Clock Auctions

2022-07-12

Michal Feldman , Vasilis Gkatzelis , Nick Gravin +1 more

In a single-parameter mechanism design problem, a provider is looking to sell some service to a group of potential buyers. Each buyer i has a private value vi for receiving … In a single-parameter mechanism design problem, a provider is looking to sell some service to a group of potential buyers. Each buyer i has a private value vi for receiving this service, and some feasibility constraint restricts which subsets of buyers can be served simultaneously. Recent work in economics introduced (deferred-acceptance) clock auctions as a superior class of auctions for this problem, due to their transparency, simplicity, and very strong incentive guarantees. Subsequent work in computer science focused on evaluating these auctions with respect to their social welfare approximation guarantees, leading to strong impossibility results: in the absence of prior information regarding the buyers' values, no deterministic clock auction can achieve a bounded approximation, even for simple feasibility constraints with only two maximal feasible sets.

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Prophet Inequalities with Cancellation Costs

2024-06-10

Farbod Ekbatani , Rad Niazadeh , Pranav Nuti +1 more

Most of the literature on online algorithms and sequential decision-making focuses on settings with "irrevocable decisions" where the algorithm's decision upon arrival of the new input is set in stone … Most of the literature on online algorithms and sequential decision-making focuses on settings with "irrevocable decisions" where the algorithm's decision upon arrival of the new input is set in stone and can never change in the future. One canonical example is the classic prophet inequality problem, where realizations of a sequence of independent random variables X1, X2,… with known distributions are drawn one by one and a decision maker decides when to stop and accept the arriving random variable, with the goal of maximizing the expected value of their pick. We consider "prophet inequalities with recourse" in the linear buyback cost setting, where after accepting a variable Xi, we can still discard Xi later and accept another variable Xj, at a buyback cost of f × Xi. The goal is to maximize the expected net reward, which is the value of the final accepted variable minus the total buyback cost. Our first main result is an optimal prophet inequality in the regime of f ≥ 1, where we prove that we can achieve an expected reward 1+f/1+2f times the expected offline optimum. The problem is still open for 0<f<1 and we give some partial results in this regime. In particular, as our second main result, we characterize the asymptotic behavior of the competitive ratio for small f and provide almost matching upper and lower bounds that show a factor of 1−Θ(flog(1/f)). Our results are obtained by two fundamentally different approaches: One is inspired by various proofs of the classical prophet inequality, while the second is based on combinatorial optimization techniques involving LP duality, flows, and cuts.

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Packing Returning Secretaries.

2018-10-26

Martin Hoefer , Lisa Wilhelmi

We study online secretary problems with returns in combinatorial packing domains with $n$ candidates that arrive sequentially over time in random order. The goal is to accept a feasible packing … We study online secretary problems with returns in combinatorial packing domains with $n$ candidates that arrive sequentially over time in random order. The goal is to accept a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All $2n$ arrivals occur in random order. We propose a simple 0.5-competitive algorithm that can be combined with arbitrary approximation algorithms for the packing domain, even when the total value of candidates is a subadditive function. For bipartite matching, we obtain an algorithm with competitive ratio at least $0.5721 - o(1)$ for growing $n$, and an algorithm with ratio at least $0.5459$ for all $n \ge 1$. We extend all algorithms and ratios to $k \ge 2$ arrivals per candidate. In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed (returned into the pool). We mainly focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of $\Theta(n \log n)$ is always sufficient. For matroids, we show that the expected number can be reduced to $O(r \log (n/r))$, where $r \le n/2$ is the minimum of the ranks of matroid and dual matroid. For bipartite matching, we show a bound of $O(r \log n)$, where $r$ is the size of the optimum matching. For general packing, we show a lower bound of $\Omega(n \log \log n)$, even when the size of the optimum is $r = \Theta(\log n)$.

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Supermodular Approximation of Norms and Applications

2024-06-10

Thomas Keßelheim , Marco Molinaro , Sahil Singla

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 … 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.

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An $O(\log \log m)$ Prophet Inequality for Subadditive Combinatorial Auctions

2022-05-19

Paul Dütting , Thomas Keßelheim , Brendan Lucier

Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case … Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here $n$ agents with subadditive valuation functions compete for the assignment of $m$ items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of $O(\log m)$. We make major progress on this question by providing an $O(\log \log m)$ prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an $O(\log \log m)$ approximation to the optimal revenue for subadditive valuations under an item-independence assumption.

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Hiring Secretaries over Time: The Benefit of Concurrent Employment

2019-08-28

Yann Disser , John Fearnley , Martin Gairing +5 more

We consider a stochastic online problem where n applicants arrive over time, one per time step. Upon the arrival of each applicant, their cost per time step is revealed, and … We consider a stochastic online problem where n applicants arrive over time, one per time step. Upon the arrival of each applicant, their cost per time step is revealed, and we have to fix the duration of employment, starting immediately. This decision is irrevocable; that is, we can neither extend a contract nor dismiss a candidate once hired. In every time step, at least one candidate needs to be under contract, and our goal is to minimize the total hiring cost, which is the sum of the applicants’ costs multiplied with their respective employment durations. We provide a competitive online algorithm for the case that the applicants’ costs are drawn independently from a known distribution. Specifically, the algorithm achieves a competitive ratio of 2.965 for the case of uniform distributions. For this case, we give an analytical lower bound of 2 and a computational lower bound of 2.148. We then adapt our algorithm to stay competitive even in settings with one or more of the following restrictions: (i) at most two applicants can be hired concurrently; (ii) the distribution of the applicants’ costs is unknown; (iii) the total number n of time steps is unknown. On the other hand, we show that concurrent employment is a necessary feature of competitive algorithms by proving that no algorithm has a competitive ratio better than [Formula: see text] if concurrent employment is forbidden.

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References (7)

Polymatroid Prophet Inequalities

2015-01-01

Paul Dütting , Robert Kleinberg

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On Talagrand's deviation inequalities for product measures

1997-01-01

Michel Ledoux

We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M. Talagrand in the recent years. Our method is based on … We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M. Talagrand in the recent years. Our method is based on functional inequalities of Poincaré and logarithmic Sobolev type and iteration of these inequalities. In particular, we establish with theses tools sharp deviation inequalities from the mean on norms of sums of independent random vectors and empirical processes. Concentration for the Hamming distance may also be deduced from this approach.

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O(log log Rank) Competitive Ratio for the Matroid Secretary Problem

2014-10-01

Oded Lachish

In the Matroid Secretary Problem (MSP), the elements of the ground set of a Matroid are revealed on-line one by one, each together with its value. An algorithm for the … In the Matroid Secretary Problem (MSP), the elements of the ground set of a Matroid are revealed on-line one by one, each together with its value. An algorithm for the MSP is called Matroid-Unknown if, at every stage of its execution, it only knows (i) the elements that have been revealed so far and their values and (ii) an oracle for testing whether or not a subset the elements that have been revealed so far forms an independent set. An algorithm is called Known-Cardinality if it knows (i), (ii) and also knows from the start the cardinality n of the ground set of the Matroid. We present here a Known-Cardinality algorithm with a competitive-ratio of order log log the rank of the Matroid. The prior known results for a OC algorithm are a competitive-ratio of log the rank of the Matroid, by Babaioff et al. (2007), and a competitive-ratio of square root of log the rank of the Matroid, by Chakraborty and Lachish (2012).

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Secretary Problems with Non-Uniform Arrival Order

2015-06-03

Thomas Keßelheim , Robert Kleinberg , Rad Niazadeh

For a number of problems in the theory of online algorithms, it is known that the assumption that elements arrive in uniformly random order enables the design of algorithms with … For a number of problems in the theory of online algorithms, it is known that the assumption that elements arrive in uniformly random order enables the design of algorithms with much better performance guarantees than under worst-case assumptions. The quintessential example of this phenomenon is the secretary problem, in which an algorithm attempts to stop a sequence at the moment it observes the maximum value in the sequence. As is well known, if the sequence is presented in uniformly random order there is an algorithm that succeeds with probability 1/e, whereas no non-trivial performance guarantee is possible if the elements arrive in worst-case order.

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Matroid prophet inequalities

2012-05-19

Robert Kleinberg , S. Matthew Weinberg

Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent … Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of $p$ matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most $O(p)$, and this factor is also tight.

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On Tail Probabilities for Martingales

1975-02-01

David A. Freedman

Watch a martingale with uniformly bounded increments until it first crosses the horizontal line of height $a$. The sum of the conditional variances of the increments given the past, up … Watch a martingale with uniformly bounded increments until it first crosses the horizontal line of height $a$. The sum of the conditional variances of the increments given the past, up to the crossing, is an intrinsic measure of the crossing time. Simple and fairly sharp upper and lower bounds are given for the Laplace transform of this crossing time, which show that the distribution is virtually the same as that for the crossing time of Brownian motion, even in the tail. The argument can be adapted to extend inequalities of Bernstein and Kolmogorov to the dependent case, proving the law of the iterated logarithm for martingales. The argument can also be adapted to prove Levy's central limit theorem for martingales. The results can be extended to martingales whose increments satisfy a growth condition.

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The Submodular Secretary Problem Goes Linear

2015-07-30

Moran Feldman , Rico Zenklusen

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