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Ordering Collective Unit Tasks: from Scheduling to Computational Social Choice

Martin Durand, Fanny Pascual

TL;DR

The collective schedules problem, which consists in computing a one machine schedule of a set of tasks, knowing that a set of individuals have preferences regarding the order of the execution of the tasks, is studied and a consensus schedule is returned.

Abstract

We study the collective schedules problem, which consists in computing a one machine schedule of a set of tasks, knowing that a set of individuals (also called voters) have preferences regarding the order of the execution of the tasks. Our aim is to return a consensus schedule. We consider the setting in which all tasks have the same length -- such a schedule can therefore also be viewed as a ranking. We study two rules, one based on a distance criterion, and another one based one a binary criterion, and we show that these rules extend classic scheduling criteria. We also consider time constraints and precedence constraints between the tasks, and focus on two cases: the preferences of the voters fulfill these constraints, or they do not fulfill these constraints (but the collective schedule should fulfill them). In each case, either we show that the problem is NP-hard, or we provide a polynomial time algorithm which solves it. We also provide an analysis of a heuristic, which appears to be a 2 approximation of the Spearman's rule.

Ordering Collective Unit Tasks: from Scheduling to Computational Social Choice

TL;DR

The collective schedules problem, which consists in computing a one machine schedule of a set of tasks, knowing that a set of individuals have preferences regarding the order of the execution of the tasks, is studied and a consensus schedule is returned.

Abstract

We study the collective schedules problem, which consists in computing a one machine schedule of a set of tasks, knowing that a set of individuals (also called voters) have preferences regarding the order of the execution of the tasks. Our aim is to return a consensus schedule. We consider the setting in which all tasks have the same length -- such a schedule can therefore also be viewed as a ranking. We study two rules, one based on a distance criterion, and another one based one a binary criterion, and we show that these rules extend classic scheduling criteria. We also consider time constraints and precedence constraints between the tasks, and focus on two cases: the preferences of the voters fulfill these constraints, or they do not fulfill these constraints (but the collective schedule should fulfill them). In each case, either we show that the problem is NP-hard, or we provide a polynomial time algorithm which solves it. We also provide an analysis of a heuristic, which appears to be a 2 approximation of the Spearman's rule.
Paper Structure (14 sections, 22 theorems, 9 equations, 15 figures)

This paper contains 14 sections, 22 theorems, 9 equations, 15 figures.

Table of Contents

  1. Preliminaries
  2. Definitions and notations
  3. Generalization of classical scheduling criteria.
  4. An analysis of the EMD rule
  5. Additional results in voting theory
  6. Scheduling tasks with time constraints
  7. Getting optimal solutions with time constraints
  8. Axiomatic study of rules with inferred time constraints
  9. Release dates and deadlines consistencies.
  10. Temporal unanimity
  11. Precedence constraints
  12. Inferred precedence constraints
  13. Imposed precedence graph
  14. Conclusion

Key Result

Proposition 1

The EMD rule is $2$-approximate for the total tardiness criterion.

Figures (15)

  • Figure 1: Schedule $S^*$ and the swap performed to obtain $S$.
  • Figure 2: Schedule $S^*$ and a preliminary swap (case 1) ensuring that the final swap of task $l$ will strictly decrease the total distance.
  • Figure 3: Preferred schedules of the $3$ voters generated for task $j$.
  • Figure 4: Schedules $S$ and $S'$. The first $dum$ task to be scheduled just before a $t$ task in $S$ is $dum_i$.
  • Figure 5: Preferred schedules of the $n+1$ voters generated for task $i$.
  • ...and 10 more figures

Theorems & Definitions (44)

  • Example 1
  • Proposition 1
  • proof
  • Corollary 1
  • Corollary 2
  • Proposition 2
  • proof
  • Definition 1
  • Definition 2
  • Proposition 3
  • ...and 34 more