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Scheduling with Time Dependent Utilities: Fairness and Efficiency

Gaia Nicosia, Andrea Pacifici, Ulrich Pferschy

Abstract

A new class of multi agent single machine scheduling problems is introduced, where each job is associated with a self interested agent with a utility function decreasing in completion time. We aim to achieve a fair solution by maximizing the minimum utility across all agents. We study the problem's complexity and propose solution methods for several variants. For the general case, we present a binary search procedure to find the largest possible minimum utility, as well as an exact greedy based alternative. Variants with release and due dates are analyzed, showing strong NP hardness for arbitrary release dates, but weak NP hardness for a single release date job, and polynomial solvability when all jobs share processing times. For all these cases we also study the corresponding problem of finding efficient solutions where the sum of utilities is maximized. We also examine settings where linear utility functions can be adjusted within budget constraints, exploring the impact on optimal schedules when intercepts or slopes are modified. From a single agent perspective, we investigate the effect of improving one agent's utility in the overall solution. Adding a new job to be inserted with the best possible utility gives rise to rescheduling problems, where different lower bounds depending on the utilities of the original fair schedule are imposed. Finally, we consider a bi level setting where a leader wants to enforce a certain target schedule by modifying utility functions while the follower computes a fair solution for the modified instance. Our work contributes to scheduling theory, multi agent systems, and algorithmic fairness, highlighting fairness oriented objectives in competitive scheduling.

Scheduling with Time Dependent Utilities: Fairness and Efficiency

Abstract

A new class of multi agent single machine scheduling problems is introduced, where each job is associated with a self interested agent with a utility function decreasing in completion time. We aim to achieve a fair solution by maximizing the minimum utility across all agents. We study the problem's complexity and propose solution methods for several variants. For the general case, we present a binary search procedure to find the largest possible minimum utility, as well as an exact greedy based alternative. Variants with release and due dates are analyzed, showing strong NP hardness for arbitrary release dates, but weak NP hardness for a single release date job, and polynomial solvability when all jobs share processing times. For all these cases we also study the corresponding problem of finding efficient solutions where the sum of utilities is maximized. We also examine settings where linear utility functions can be adjusted within budget constraints, exploring the impact on optimal schedules when intercepts or slopes are modified. From a single agent perspective, we investigate the effect of improving one agent's utility in the overall solution. Adding a new job to be inserted with the best possible utility gives rise to rescheduling problems, where different lower bounds depending on the utilities of the original fair schedule are imposed. Finally, we consider a bi level setting where a leader wants to enforce a certain target schedule by modifying utility functions while the follower computes a fair solution for the modified instance. Our work contributes to scheduling theory, multi agent systems, and algorithmic fairness, highlighting fairness oriented objectives in competitive scheduling.

Paper Structure

This paper contains 21 sections, 13 theorems, 31 equations, 9 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related literature
  3. Problem statement
  4. Problem with fixed parameters
  5. Variable linear utility functions
  6. Rescheduling from the perspective of a new agent
  7. Leader-follower utility adjustment
  8. Summary of contributions
  9. Problems with fixed parameters
  10. General solution approach
  11. Greedy-type algorithm
  12. Release dates
  13. Due dates
  14. Adjustable linear utility functions
  15. Modifications of the intercepts $b_j$
  16. ...and 6 more sections

Key Result

Theorem 2

If, for all $j\in J$, $u_j$ is any non-increasing function of the completion time $C_j$ of job $j$, then MaxMinGreedy computes an optimal solution of $1|u_j|u_{\min}$ in $O(n^2)$ time.

Figures (9)

  • Figure 1: Algorithm MaxMinGreedy applied to an instance of $1|u_j|u_{\min}$ with three jobs and linear utility functions.
  • Figure 2: Proof of Theorem \ref{['th:release_weakNPHsum']}: Non-balanced schedule $\sigma^*$ and balanced schedule $\tilde{\sigma}$ obtained by swapping blocks $E$ and $L$.
  • Figure 3: Example to illustrate that idle times may be useful. Schedule $\langle 1,2 \rangle$ without idle times in red, optimal schedule $\langle 2,1 \rangle$ with idle time in blue.
  • Figure 4: Optimal schedules corresponding to a yes-instance (above) and a no-instance of 3-Partition with $n=3$.
  • Figure 5: Function $\tilde{u}_j(C_j(\sigma))$ as defined in (\ref{['eq:utilarea']}).
  • ...and 4 more figures

Theorems & Definitions (13)

  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 9
  • Theorem 11
  • Theorem 12
  • Theorem 13
  • ...and 3 more