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Constant Weighted Maximin Share Approximations for Chores

Bo Li, Fangxiao Wang, Shiji Xing

TL;DR

The paper addresses fair division of indivisible chores among weighted agents by WMMS. It introduces a modular framework with canonical instances, delegating agents, and proxy costs, yielding a 3-WMMS algorithm on canonical instances and a polynomial-time method achieving $24+\epsilon$-WMMS, underpinned by a canonical-reduction analysis that inflates the approximation by at most a factor of 4. It proves a matching near-tight lower bound of $(2-\epsilon)$-WMMS, demonstrating the existence of allocations that resist better constant-factor WMMS guarantees. Collectively, the results establish the first constant-factor WMMS approximation for asymmetric chores, with a clear path from existential constructions to practical, polynomial-time algorithms, and identify a substantial gap between achievable upper and lower bounds for future work.

Abstract

We study the fair allocation of indivisible chores among agents with asymmetric weights. Among the various fairness notions, weighted maximin share (WMMS) stands out as particularly compelling. However, whether WMMS admits a constant-factor approximation has remained unknown and is one of the important open problems in weighted fair division [ALMW22, Suk25]. So far, the best known approximation ratio is O(log n), where n is the number of agents. In this paper, we advance the state of the art and present the first constant-factor approximate WMMS algorithm. To this end, we introduce canonical instance reductions and different bounds of agents' valuations. We also prove that guaranteeing better than 2-approximation is not possible, which improves the best-known lower bound of 1.366.

Constant Weighted Maximin Share Approximations for Chores

TL;DR

The paper addresses fair division of indivisible chores among weighted agents by WMMS. It introduces a modular framework with canonical instances, delegating agents, and proxy costs, yielding a 3-WMMS algorithm on canonical instances and a polynomial-time method achieving -WMMS, underpinned by a canonical-reduction analysis that inflates the approximation by at most a factor of 4. It proves a matching near-tight lower bound of -WMMS, demonstrating the existence of allocations that resist better constant-factor WMMS guarantees. Collectively, the results establish the first constant-factor WMMS approximation for asymmetric chores, with a clear path from existential constructions to practical, polynomial-time algorithms, and identify a substantial gap between achievable upper and lower bounds for future work.

Abstract

We study the fair allocation of indivisible chores among agents with asymmetric weights. Among the various fairness notions, weighted maximin share (WMMS) stands out as particularly compelling. However, whether WMMS admits a constant-factor approximation has remained unknown and is one of the important open problems in weighted fair division [ALMW22, Suk25]. So far, the best known approximation ratio is O(log n), where n is the number of agents. In this paper, we advance the state of the art and present the first constant-factor approximate WMMS algorithm. To this end, we introduce canonical instance reductions and different bounds of agents' valuations. We also prove that guaranteeing better than 2-approximation is not possible, which improves the best-known lower bound of 1.366.

Paper Structure

This paper contains 21 sections, 7 theorems, 47 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Definitions and Technical Overview
  4. Solution Concepts
  5. Technical Overview
  6. Canonical Instances and Polynomial-time Implementation
  7. A 3-WMMS Algorithm for Canonical Instances
  8. Lower Bound
  9. The Design of the Instance
  10. Proof of Theorem \ref{['the:lb']}
  11. Optimization 1
  12. Optimization 2
  13. Conclusion and Discussion
  14. Proof of Theorem \ref{['the:can']}
  15. $\mathcal{I} \to \mathcal{I}^1$
  16. ...and 6 more sections

Key Result

Theorem 1

If there is an $\alpha$-WMMS allocation for all canonical instances, there is a $4\alpha$-WMMS allocation for arbitrary instances.

Figures (2)

  • Figure 1: An illustration of delegating agents $\mathcal{N}^*$. For each interval $a_{r_i}$, the left dashed boundary means that $a_i$ becomes active at item $d_{r_i}+1$, and the right double lines mean $a_i$ becomes inactive after receiving this item.
  • Figure 2: An Illustration of $v_i(\cdot)$.

Theorems & Definitions (33)

  • Definition 1: $\alpha$-WMMS
  • Example 1
  • Definition 2: Canonical Instances
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Claim 1
  • proof
  • Claim 2
  • proof
  • ...and 23 more