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Collective schedules: axioms and algorithms

Martin Durand, Fanny Pascual

TL;DR

This work studies how to compute a consensus schedule when each voter prefers a full order over tasks with distinct durations. It introduces three aggregation rules—PTA Kemeny (weighted Kendall tau), SigmaD (sum of absolute deviations), and SigmaT (sum tardiness)—and conducts an axiomatic analysis that reveals incompatibilities among neutral, distance-based, and duration-aware properties. The authors prove NP-hardness for the two main rules, propose a linear-programming approach for small instances, and develop a fast heuristic (Lowest Median Time) with local search that performs near-optimally on larger instances. Experimental results show PTA Kemeny and SigmaT are well-suited for EB settings, while SigmaD remains competitive for non-EB scenarios; the work also introduces and validates new axioms like length reduction monotonicity for duration-aware scheduling.

Abstract

The collective schedules problem consists in computing a schedule of tasks shared between individuals. Tasks may have different duration, and individuals have preferences over the order of the shared tasks. This problem has numerous applications since tasks may model public infrastructure projects, events taking place in a shared room, or work done by co-workers. Our aim is, given the preferred schedules of individuals (voters), to return a consensus schedule. We propose an axiomatic study of the collective schedule problem, by using classic axioms in computational social choice and new axioms that take into account the duration of the tasks. We show that some axioms are incompatible, and we study the axioms fulfilled by three rules: one which has been studied in the seminal paper on collective schedules (Pascual et al. 2018), one which generalizes the Kemeny rule, and one which generalizes Spearman's footrule. From an algorithmic point of view, we show that these rules solve NP-hard problems, but that it is possible to solve optimally these problems for small but realistic size instances, and we give an efficient heuristic for large instances. We conclude this paper with experiments.

Collective schedules: axioms and algorithms

TL;DR

This work studies how to compute a consensus schedule when each voter prefers a full order over tasks with distinct durations. It introduces three aggregation rules—PTA Kemeny (weighted Kendall tau), SigmaD (sum of absolute deviations), and SigmaT (sum tardiness)—and conducts an axiomatic analysis that reveals incompatibilities among neutral, distance-based, and duration-aware properties. The authors prove NP-hardness for the two main rules, propose a linear-programming approach for small instances, and develop a fast heuristic (Lowest Median Time) with local search that performs near-optimally on larger instances. Experimental results show PTA Kemeny and SigmaT are well-suited for EB settings, while SigmaD remains competitive for non-EB scenarios; the work also introduces and validates new axioms like length reduction monotonicity for duration-aware scheduling.

Abstract

The collective schedules problem consists in computing a schedule of tasks shared between individuals. Tasks may have different duration, and individuals have preferences over the order of the shared tasks. This problem has numerous applications since tasks may model public infrastructure projects, events taking place in a shared room, or work done by co-workers. Our aim is, given the preferred schedules of individuals (voters), to return a consensus schedule. We propose an axiomatic study of the collective schedule problem, by using classic axioms in computational social choice and new axioms that take into account the duration of the tasks. We show that some axioms are incompatible, and we study the axioms fulfilled by three rules: one which has been studied in the seminal paper on collective schedules (Pascual et al. 2018), one which generalizes the Kemeny rule, and one which generalizes Spearman's footrule. From an algorithmic point of view, we show that these rules solve NP-hard problems, but that it is possible to solve optimally these problems for small but realistic size instances, and we give an efficient heuristic for large instances. We conclude this paper with experiments.
Paper Structure (14 sections, 21 theorems, 16 equations, 4 figures, 3 tables)

This paper contains 14 sections, 21 theorems, 16 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Axiomatic properties
  4. Neutrality and PTA neutrality.
  5. Distance.
  6. PTA Condorcet consistency.
  7. Incompatibilities between axioms and properties.
  8. Length reduction monotonicity.
  9. Reinforcement.
  10. Unanimity.
  11. Summary.
  12. Computational complexity and algorithms
  13. Experiments
  14. Discussion and conclusion

Key Result

Proposition 1

The $\Sigma$D rule is not neutral even if it does not apply any tie-breaking mechanism.

Figures (4)

  • Figure 1: Voters associated with a task $i'$
  • Figure 2: Structure of an existing optimal solution
  • Figure 3: Voters associated with a task $i'$
  • Figure 4: Structure of an existing optimal solution

Theorems & Definitions (47)

  • Example 1
  • Definition 1
  • Proposition 1
  • proof
  • Definition 2
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Definition 3: PTA Condorcet consistency pascual2018collective
  • ...and 37 more