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The Algorithmic Landscape of Fair and Efficient Distribution of Delivery Orders in the Gig Economy

Hadi Hosseini, Šimon Schierreich

TL;DR

This work addresses fair and efficient distribution of delivery tasks on a hub-and-tree topology under a submodular cost model by studying MMS fairness and introducing non-wastefulness as a verifiable efficiency criterion. It develops polynomial-time verification and conversion to non-wasteful allocations, and provides a detailed fixed-parameter and topology-based complexity landscape for MMS+NW, including FPT results by leaves, internal nodes, and central-path structures. The results reveal sharp tractability boundaries: for certain restricted topologies (e.g., stars, caterpillars) and small parameters, MMS+NW can be achieved efficiently, while general weighted trees remain NP-hard, with hardness proofs mediated by 3-Partition and related problems. The paper also explores envy-based fairness notions and their compatibility with non-wastefulness, highlighting inherent trade-offs and offering insights into how NW interacts with EF/EF1 in graph-constrained settings. Overall, the findings advance both theory and potential practice for fair, transferrable-salary-free task allocation in gig-economy delivery scenarios, across weighted and unweighted topologies.

Abstract

Distributing services, goods, and tasks in the gig economy heavily relies upon on-demand workers (aka agents), leading to new challenges varying from logistics optimization to the ethical treatment of gig workers. We focus on fair and efficient distribution of delivery tasks -- placed on the vertices of a graph -- among a fixed set of agents. We consider the fairness notion of minimax share (MMS), which aims to minimize the maximum (submodular) cost among agents and is particularly appealing in applications without monetary transfers. We propose a novel efficiency notion -- namely non-wastefulness -- that is desirable in a wide range of scenarios and, more importantly, does not suffer from computational barriers. Specifically, given a distribution of tasks, we can, in polynomial time, i) verify whether the distribution is non-wasteful and ii) turn it into an equivalent non-wasteful distribution. Moreover, we investigate several fixed-parameter tractable and polynomial-time algorithms and paint a complete picture of the (parameterized) complexity of finding fair and efficient distributions of tasks with respect to both the structure of the topology and natural restrictions of the input. Finally, we highlight how our findings shed light on computational aspects of other well-studied fairness notions, such as envy-freeness and its relaxations.

The Algorithmic Landscape of Fair and Efficient Distribution of Delivery Orders in the Gig Economy

TL;DR

This work addresses fair and efficient distribution of delivery tasks on a hub-and-tree topology under a submodular cost model by studying MMS fairness and introducing non-wastefulness as a verifiable efficiency criterion. It develops polynomial-time verification and conversion to non-wasteful allocations, and provides a detailed fixed-parameter and topology-based complexity landscape for MMS+NW, including FPT results by leaves, internal nodes, and central-path structures. The results reveal sharp tractability boundaries: for certain restricted topologies (e.g., stars, caterpillars) and small parameters, MMS+NW can be achieved efficiently, while general weighted trees remain NP-hard, with hardness proofs mediated by 3-Partition and related problems. The paper also explores envy-based fairness notions and their compatibility with non-wastefulness, highlighting inherent trade-offs and offering insights into how NW interacts with EF/EF1 in graph-constrained settings. Overall, the findings advance both theory and potential practice for fair, transferrable-salary-free task allocation in gig-economy delivery scenarios, across weighted and unweighted topologies.

Abstract

Distributing services, goods, and tasks in the gig economy heavily relies upon on-demand workers (aka agents), leading to new challenges varying from logistics optimization to the ethical treatment of gig workers. We focus on fair and efficient distribution of delivery tasks -- placed on the vertices of a graph -- among a fixed set of agents. We consider the fairness notion of minimax share (MMS), which aims to minimize the maximum (submodular) cost among agents and is particularly appealing in applications without monetary transfers. We propose a novel efficiency notion -- namely non-wastefulness -- that is desirable in a wide range of scenarios and, more importantly, does not suffer from computational barriers. Specifically, given a distribution of tasks, we can, in polynomial time, i) verify whether the distribution is non-wasteful and ii) turn it into an equivalent non-wasteful distribution. Moreover, we investigate several fixed-parameter tractable and polynomial-time algorithms and paint a complete picture of the (parameterized) complexity of finding fair and efficient distributions of tasks with respect to both the structure of the topology and natural restrictions of the input. Finally, we highlight how our findings shed light on computational aspects of other well-studied fairness notions, such as envy-freeness and its relaxations.

Paper Structure

This paper contains 28 sections, 32 theorems, 9 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Non-Wasteful Allocations.
  4. Algorithms for MMS and Non-wasteful Allocations.
  5. The Impact of Topology Structure.
  6. Envy-Based Fairness
  7. Related Work
  8. Preliminaries
  9. Graph Theory
  10. Distribution of Delivery Orders.
  11. Fairness.
  12. Efficiency.
  13. Non-wasteful Allocations
  14. Algorithmics of Non-wastefulness
  15. Relations to Other Efficiency Notions
  16. ...and 13 more sections

Key Result

Proposition 1

A non-wasteful allocation is guaranteed to exist and can be found in linear time.

Figures (4)

  • Figure 1: An illustration of non-wastefulness. On the top, we depict an allocation that is not non-wasteful: while both the green (circle, $1$) agent and yellow (diamond, $3$) agents service a vertex if and only if they service a leaf in the respective sub-tree, the red (square, $2$) agent services the order of the top branch even though it is not servicing any leaf of this sub-tree. On the bottom, we depict a non-wasteful allocation for the same instance. Observe that in this case, the non-wasteful allocation even strictly improved the cost for the red agent.
  • Figure 2: An example of a non-wasteful and MMS allocation that is not Pareto optimal. We use green circles to highlight orders serviced by the agent $1$, red squares to highlight orders serviced by the agent $2$, and yellow diamonds for those serviced by the agent $3$. Observe that if we move the order $u$ from $\pi_3$ to $\pi_1$ and simultaneously the order $v$ and $\operatorname{parent}(v)$ from $\pi_1$ to $\pi_3$, the cost of bundle $\pi_3$ remains the same, while the cost of bundle $\pi_1$ decreases. That is, the depicted allocation is not Pareto optimal.
  • Figure 3: An illustration of the construction used to prove \ref{['thm:MMS:NPh:depthDiameter4pvc']}.
  • Figure 4: An instance used to prove \ref{['thm:EF1:standardAlgorithmsDoNotWorkForNonWastefulness']}. Highlighted is an EF1 and non-wasteful allocation.

Theorems & Definitions (68)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Proposition 1
  • Theorem 1
  • proof
  • ...and 58 more