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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
Linear Time Algorithms for Multiple Cluster Scheduling an...
Klaus Jansen, Malin Rau · 2019-02-09 · via cs.DS updates on arXiv.org

We study the Multiple Cluster Scheduling problem and the Multiple Strip Packing problem. For both problems, there is no algorithm with approximation ratio better than $2$ unless $P = NP$. In this paper, we present an algorithm with approximation ratio $2$ and running time $O(n)$ for both problems. While a $2$ approximation was known before, the running time of the algorithm is at least $Ω(n^{256})$ in the worst case. Therefore, an $O(n)$ algorithm is surprising and the best possible. We archive this result by calling an AEPTAS with approximation guarantee $(1+\varepsilon)OPT +p_{\max}$ and running time of the form $O(n\log(1/\varepsilon)+ f(1/\varepsilon))$ with a constant $\varepsilon$ to schedule the jobs on a single cluster. This schedule is then distributed on the $N$ clusters in $O(n)$. Moreover, this distribution technique can be applied to any variant of of Multi Cluster Scheduling for which there exists an AEPTAS with additive term $p_{\max}$. While the above result is strong from a theoretical point of view, it might not be very practical due to a large hidden constant caused by calling an AEPTAS with a constant $\varepsilon \geq 1/8$ as subroutine. Nevertheless, we point out that the general approach of finding first a schedule on one cluster and then distributing it onto the other clusters might come in handy in practical approaches. We demonstrate this by presenting a practical algorithm with running time $O(n\log(n))$, with out hidden constants, that is a $9/4$-approximation for one third of all possible instances, i.e, all instances where the number of clusters is dividable by $3$, and has an approximation ratio of at most $2.3$ for all instances with at least $9$ clusters.