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Efficient Approximation Schemes for Scheduling on a Stochastic Number of Machines

Authors Leah Epstein , Asaf Levin

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Author Details

Leah Epstein
  • Department of Mathematics, University of Haifa, Israel
Asaf Levin
  • Faculty of Data and Decisions Science, The Technion, Haifa, Israel

Funding

  • Levin, Asaf: Partially supported by ISF - Israel Science Foundation grant number 1467/22.

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Leah Epstein and Asaf Levin. Efficient Approximation Schemes for Scheduling on a Stochastic Number of Machines. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 31:1-31:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.31

Abstract

We study three two-stage optimization problems with a similar structure and different objectives. In the first stage of each problem, the goal is to assign input jobs of positive sizes to unsplittable bags. After this assignment is decided, the realization of the number of identical machines that will be available is revealed. Then, in the second stage, the bags are assigned to machines. The probability vector of the number of machines in the second stage is known to the algorithm as part of the input before making the decisions of the first stage. Thus, the vector of machine completion times is a random variable. The goal of the first problem is to minimize the expected value of the makespan of the second stage schedule, while the goal of the second problem is to maximize the expected value of the minimum completion time of the machines in the second stage solution. The goal of the third problem is to minimize the 𝓁_𝔭 norm for a fixed 𝔭 > 1, where the norm is applied on machines' completion times vectors. Each one of the first two problems admits a PTAS as Buchem et al. showed recently. Here we significantly improve all their results by designing an EPTAS for each one of these problems. We also design an EPTAS for 𝓁_𝔭 norm minimization for any 𝔭 > 1.

Subject Classification

ACM Subject Classification
  • Theory of computation → Scheduling algorithms
Keywords
  • Approximation algorithms
  • Approximation schemes
  • Two-stage stochastic optimization problems
  • Multiprocessor scheduling

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