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cs.DS updates on arXiv.org

PAC Learning with Bandit Feedback: Sharp Sample Complexity in the Realizable Setting Algorithms with Polynomially-Improved Approximation Factors for the $2 \rightarrow q$ Norm, and Applications A computational phase transition for learning-to-sample from Ising models Covering vertices by sequential stars Fermi-Dirac machines as quantizations of neurons A Comprehensive Evaluation of Vertex Elimination Algorithms for Algorithmic Differentiation A Tight Bound on Localization of Electrical Flows Optimal Dimension-Free Sampling for Regularized Classification Reducing the Randomness in Partition Oracles for Bounded Degree Minor-Free Graphs Beyond the Half-Approximation: Fair and Efficient Online Class Matching Efficient Uniform Sampling of Surjections via their Profiles Tractable Maximization of Budgeted Phylogenetic Diversity on Networks Utilizing Node Scanwidth Fairness in Aggregation: Optimal Top-$k$ and Improved Full Ranking Learning-Augmented Online Scheduling with Parsimonious Preemption Entropy Equivalence Testing Lumberjack: Better Differentially Private Random Forests through Heavy Hitter Detection in Trees The Secretary Problem with a Stochastic Precursor Polynomial-Time Robust Multiclass Linear Classification under Gaussian Marginals Efficient Banzhaf-Based Data Valuation for $k$-Nearest Neighbors Classification Block-Sphere Vector Quantization An Approximation Algorithm for Graph Label Selection Iterative Chow Filtering for Learning with Distribution Shift Complexity of Non-Log-Concave Sampling in Fisher Information Stochastic Matching via Local Sparsification Finite Sample Bounds for Learning with Score Matching What is Learnable in Valiant's Theory of the Learnable? Provable Quantization with Randomized Hadamard Transform Min-Max Optimization Requires Exponentially Many Queries Fast and Compact Graph Cuts for the Boykov-Kolmogorov Algorithm A proximal gradient algorithm for composite log-concave sampling
Prophets and Secretaries with Overbooking
Tomer Ezra, Michal Feldman, Ilan Nehama · 2018-05-14 · via cs.DS updates on arXiv.org

The prophet and secretary problems demonstrate online scenarios involving the optimal stopping theory. In a typical prophet or secretary problem, selection decisions are assumed to be immediate and irrevocable. However, many online settings accommodate some degree of revocability. To study such scenarios, we introduce the $\ell-out-of-k$ setting, where the decision maker can select up to $k$ elements immediately and irrevocably, but her performance is measured by the top $\ell$ elements in the selected set. Equivalently, the decision makes can hold up to $\ell$ elements at any given point in time, but can make up to $k-\ell$ returns as new elements arrive. We give upper and lower bounds on the competitive ratio of $\ell$-out-of-$k$ prophet and secretary scenarios. These include a single-sample prophet algorithm that gives a competitive ratio of $1-\ell\cdot e^{-Θ\left(\frac{\left(k-\ell\right)^2}{k}\right)}$, which is asymptotically tight for $k-\ell=Θ(\ell)$. For secretary settings, we devise an algorithm that obtains a competitive ratio of $1-\ell e^{-\frac{k-8\ell}{2+2\ln \ell}} - e^{-k/6}$, and show that no secretary algorithm obtains a better ratio than $1-e^{-k}$ (up to negligible terms). In passing, our results lead to an improvement of the results of Assaf et al. [2000] for $1-out-of-k$ prophet scenarios. Beyond the contribution to online algorithms and optimal stopping theory, our results have implications to mechanism design. In particular, we use our prophet algorithms to derive {\em overbooking} mechanisms with good welfare and revenue guarantees; these are mechanisms that sell more items than the seller's capacity, then allocate to the agents with the highest values among the selected agents.