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Allocating Chores with Restricted Additive Costs: Achieving EFX, MMS, and Efficiency Simultaneously

Zehan Lin, Xiaowei Wu, Shengwei Zhou

Abstract

In a web-based review platform, papers from various research fields must be assigned to a group of reviewers. Each paper has an inherent cost, which represents the effort required for reading and evaluating it (e.g., the paper's length). Reviewers can bid on papers they are interested in, and if they are assigned a paper they have bid on, no cost is incurred. Otherwise, the inherent cost $c(e)$ for paper $e$ applies. We capture this with a model of restricted additive costs: every item $e$ has a cost $c(e)$, and each agent either incurs $0$ or $c(e)$ for $e$. In this work, we study how to allocate such chores fairly and efficiently. We propose an algorithm for computing allocations that are both EFX and MMS. Furthermore, we show that our algorithm achieves a $2$-approximation of the optimal social cost, and the approximation ratio is optimal. We also show that slightly weaker fairness guarantees can be obtained if one requires the algorithm to run in polynomial time.

Allocating Chores with Restricted Additive Costs: Achieving EFX, MMS, and Efficiency Simultaneously

Abstract

In a web-based review platform, papers from various research fields must be assigned to a group of reviewers. Each paper has an inherent cost, which represents the effort required for reading and evaluating it (e.g., the paper's length). Reviewers can bid on papers they are interested in, and if they are assigned a paper they have bid on, no cost is incurred. Otherwise, the inherent cost for paper applies. We capture this with a model of restricted additive costs: every item has a cost , and each agent either incurs or for . In this work, we study how to allocate such chores fairly and efficiently. We propose an algorithm for computing allocations that are both EFX and MMS. Furthermore, we show that our algorithm achieves a -approximation of the optimal social cost, and the approximation ratio is optimal. We also show that slightly weaker fairness guarantees can be obtained if one requires the algorithm to run in polynomial time.
Paper Structure (23 sections, 15 theorems, 23 equations, 2 figures, 3 tables, 3 algorithms)

This paper contains 23 sections, 15 theorems, 23 equations, 2 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. EFX Allocations for Chores.
  3. MMS Allocations for Chores.
  4. Restricted Additive Functions.
  5. Efficiency.
  6. Discussion on Existing Algorithms
  7. Our Results
  8. Other Related Works
  9. Approximate EFX Allocations.
  10. EFX Allocations for Goods.
  11. Preliminary
  12. EFX and MMS Allocations
  13. Description of the Algorithm
  14. Phase 1: Allocate items in $M^+$.
  15. Phase 2: Allocate items in $M^0$.
  16. ...and 8 more sections

Key Result

Theorem 3.1

For restricted instances of indivisible chores, there exists an algorithm that computes allocations that satisfy both EFX and MMS to all agents.

Figures (2)

  • Figure 1: Illustration for the modifications by the end of Phase 2.
  • Figure 2: Illustration of Example \ref{['Example: infty']}.

Theorems & Definitions (40)

  • Definition 2.1: Restricted Functions
  • Definition 2.2: EF
  • Definition 2.3: EF1
  • Definition 2.4: EFX
  • Definition 2.5: MMS
  • Definition 2.6: $\alpha$-MMS
  • Definition 2.7: $\alpha$-MMS-feasible
  • Definition 2.8: EFX-feasible
  • Definition 2.9: Social Cost
  • Definition 2.10: Pareto Optimal
  • ...and 30 more