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Approximation algorithms for Job Scheduling with reconfigurable resources

Pierre Bergé, Mari Chaikovskaia, Jean-Philippe Gayon, Alain Quilliot

TL;DR

This work tackles Multi_Bot, a high-multiplicity scheduling problem with reconfigurable resources, by minimizing the horizon-wide resource maximum $H$ while meeting all type demands. The authors connect Multi_Bot to IMS and develop Bot-approx, a polynomial-time $\frac{4}{3}$-approximation that rests on a carefully constructed, small collection of packings $\Pi$ and a transformation of packings into IMS instances solved by the Longest Processing Time first heuristic. They design a dynamic-programming backbone to generate packings with bounded volume and scale, enabling a polynomial-size IMS reduction and guaranteeing the overall approximation bound. This approach yields a practically efficient method for reconfigurable-robot production and transportation settings, with potential extensions to PTAS improvements and alternative IMS heuristics. The main contribution is both a concrete, provable approximation algorithm and new structural insights into optimal resource configurations for reconfigurable scheduling problems, connecting theory to industrial robotics applications.

Abstract

We consider here the MultiBot problem for the scheduling and the resource parametrization of jobs related to the production or the transportation of different products inside a given time horizon. Those jobs must meet known in advance demands. The time horizon is divided into several discrete identical periods representing each the time needed to proceed a job. The objective is to find a parametrization and a schedule for the jobs in such a way they require as less resources as possible. Though this problem derived from the applicative context of reconfigurable robots, we focus here on fundamental issues. We show that the resulting strongly NP-hard Multibot problem may be handled in a greedy way with an approximation ratio of $\frac{4}{3}$.

Approximation algorithms for Job Scheduling with reconfigurable resources

TL;DR

This work tackles Multi_Bot, a high-multiplicity scheduling problem with reconfigurable resources, by minimizing the horizon-wide resource maximum while meeting all type demands. The authors connect Multi_Bot to IMS and develop Bot-approx, a polynomial-time -approximation that rests on a carefully constructed, small collection of packings and a transformation of packings into IMS instances solved by the Longest Processing Time first heuristic. They design a dynamic-programming backbone to generate packings with bounded volume and scale, enabling a polynomial-size IMS reduction and guaranteeing the overall approximation bound. This approach yields a practically efficient method for reconfigurable-robot production and transportation settings, with potential extensions to PTAS improvements and alternative IMS heuristics. The main contribution is both a concrete, provable approximation algorithm and new structural insights into optimal resource configurations for reconfigurable scheduling problems, connecting theory to industrial robotics applications.

Abstract

We consider here the MultiBot problem for the scheduling and the resource parametrization of jobs related to the production or the transportation of different products inside a given time horizon. Those jobs must meet known in advance demands. The time horizon is divided into several discrete identical periods representing each the time needed to proceed a job. The objective is to find a parametrization and a schedule for the jobs in such a way they require as less resources as possible. Though this problem derived from the applicative context of reconfigurable robots, we focus here on fundamental issues. We show that the resulting strongly NP-hard Multibot problem may be handled in a greedy way with an approximation ratio of .
Paper Structure (18 sections, 11 theorems, 27 equations, 4 figures, 1 algorithm)

This paper contains 18 sections, 11 theorems, 27 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem description and notations
  3. ILP formulation for Multi_Bot
  4. Schedules and packings
  5. Preliminaries
  6. Approximation algorithms
  7. Identical-machines scheduling
  8. Optimal configurations
  9. Structure and analysis of the approximation algorithm
  10. Presentation of the algorithm
  11. Transformation of a packing into polynomial IMS
  12. Approximation analysis
  13. Generation of packings with small volume
  14. Optimal packings with limited volume and scale
  15. Dynamic programming for the single-period problems
  16. ...and 3 more sections

Key Result

Lemma 1

There is an optimum solution $\mathbf{x}^*$ for which, for any $k \in \mathcal{K}$, and $p \neq p_0(k)$, $x_{pkt}^* \le p_0(k) \le P$.

Figures (4)

  • Figure 1: An example of schedule $\mathbf{x} = (x_{pkt})_{p,k,t}$ with $H = \max_t H_t = 6$, meaning that at most $6$ resources are used per period.
  • Figure 2: Output of Lpt-first with item sizes $(5,3,3,3,2,2,1)$
  • Figure 3: The result of Lpt-first after filling periods with big and medium items first. Items of sizes 1 and 2 ($p \le \frac{\lambda}{3}$) have still to be treated.
  • Figure 4: 2D-projection of the 4D-vector Table illustrating the different recursive calls

Theorems & Definitions (27)

  • Definition 1: Volume
  • Definition 2: Maximum
  • Definition 3: Scale
  • Definition 4: Optimal configuration for $k \in \mathcal{K}$
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2: Approximation ratio of Bot-approx
  • proof
  • ...and 17 more