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Efficient approximation schemes for scheduling on a stochastic number of machines

Leah Epstein, Asaf Levin

TL;DR

This work studies three two-stage stochastic scheduling problems on an unknown number of identical machines, where the first stage partitions jobs into bags and the second stage assigns bags to the realized number of machines $k$. The authors design efficient EPTASes for Pr-makespan (minimizing expected makespan), Pr-SantaClaus (maximizing expected minimum load), and Pr-norm (minimizing the $\\ell_p$ norm with $p>1$), by combining discretization of bag and job sizes, histogram-based guessing of scenario costs, and a template-configuration integer program solved in fixed dimension via Lenstra's algorithm. The schemes rely on systematic rounding, strategic guessing, and a decomposition into a small number of representative configurations, achieving $1+\\varepsilon$-approximation in time $h(\\varepsilon) \, poly(n)$ for each objective. These results improve prior PTASs by delivering EPTAS performance and extend the classic scheduling problems under stochastic machine availability to a robust, tractable framework with practical implications for uncertainty-aware resource allocation.

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 \ell_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 \ell_p norm minimization for any p>1.

Efficient approximation schemes for scheduling on a stochastic number of machines

TL;DR

This work studies three two-stage stochastic scheduling problems on an unknown number of identical machines, where the first stage partitions jobs into bags and the second stage assigns bags to the realized number of machines . The authors design efficient EPTASes for Pr-makespan (minimizing expected makespan), Pr-SantaClaus (maximizing expected minimum load), and Pr-norm (minimizing the norm with ), by combining discretization of bag and job sizes, histogram-based guessing of scenario costs, and a template-configuration integer program solved in fixed dimension via Lenstra's algorithm. The schemes rely on systematic rounding, strategic guessing, and a decomposition into a small number of representative configurations, achieving -approximation in time for each objective. These results improve prior PTASs by delivering EPTAS performance and extend the classic scheduling problems under stochastic machine availability to a robust, tractable framework with practical implications for uncertainty-aware resource allocation.

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 \ell_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 \ell_p norm minimization for any p>1.
Paper Structure (30 sections, 34 theorems, 14 equations)

This paper contains 30 sections, 34 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. An overview of our results.
  3. Related work.
  4. An EPTAS for the makespan minimization problem
  5. Guessing ${\hbox{opt}}$.
  6. Rounding bag sizes.
  7. Rounding job sizes.
  8. Computing the maximum number of machines for which the assignment to bags does not matter.
  9. Guessing an approximated histogram of the optimal makespan on all scenarios.
  10. The template-configuration integer program.
  11. An EPTAS for the Santa Claus problem
  12. Preprocessing steps
  13. The next guessing step.
  14. Partitioning the input into two independent problems.
  15. Modifying the input of scenarios with indexes at least $\boldsymbol{k}$ and at most $\boldsymbol{k_{\max}}$.
  16. ...and 15 more sections

Key Result

Lemma 1

It holds that ${\hbox{opt}}_k \in [p_{\max},n \cdot p_{\max}]$ for any $k \leq m$. Thus, ${\hbox{opt}} \in [p_{\max},n \cdot p_{\max}]$.

Theorems & Definitions (34)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Corollary 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Corollary 8
  • Theorem 9
  • Lemma 10
  • ...and 24 more