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Improved Maximin Share Approximations for Chores by Bin Packing

Jugal Garg, Xin Huang, Erel Segal-Halevi

TL;DR

The existence of 1-out-of-9n/11 MMS allocations is shown, which improves the state-of-the-art factor of 1-out-of-3n/4 and a 15/13-MMS allocation for personalized bivalued instances is provided, improving the state-of-the-art factor of 13/11.

Abstract

We study fair division of indivisible chores among $n$ agents with additive cost functions using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist for more than two agents, the goal has been to improve its approximations and identify interesting special cases where MMS allocations exists. We show the existence of 1) 1-out-of-$\lfloor \frac{9}{11}n\rfloor$ MMS allocations, which improves the state-of-the-art factor of 1-out-of-$\lfloor \frac{3}{4}n\rfloor$. 2) MMS allocations for factored instances, which resolves an open question posed by Ebadian et al. (2021). 3) $15/13$-MMS allocations for personalized bivalued instances, improving the state-of-the-art factor of $13/11$. We achieve these results by leveraging the HFFD algorithm of Huang and Lu (2021). Our approach also provides polynomial-time algorithms for computing an MMS allocation for factored instances and a $15/13$-MMS allocation for personalized bivalued instances.

Improved Maximin Share Approximations for Chores by Bin Packing

TL;DR

The existence of 1-out-of-9n/11 MMS allocations is shown, which improves the state-of-the-art factor of 1-out-of-3n/4 and a 15/13-MMS allocation for personalized bivalued instances is provided, improving the state-of-the-art factor of 13/11.

Abstract

We study fair division of indivisible chores among agents with additive cost functions using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist for more than two agents, the goal has been to improve its approximations and identify interesting special cases where MMS allocations exists. We show the existence of 1) 1-out-of- MMS allocations, which improves the state-of-the-art factor of 1-out-of-. 2) MMS allocations for factored instances, which resolves an open question posed by Ebadian et al. (2021). 3) -MMS allocations for personalized bivalued instances, improving the state-of-the-art factor of . We achieve these results by leveraging the HFFD algorithm of Huang and Lu (2021). Our approach also provides polynomial-time algorithms for computing an MMS allocation for factored instances and a -MMS allocation for personalized bivalued instances.

Paper Structure

This paper contains 29 sections, 12 theorems, 8 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. $\alpha$-MMS
  3. $1$-out-of-$d$-MMS
  4. Our contributions
  5. Further related work
  6. Goods
  7. Preliminaries
  8. IDO instances
  9. FFD algorithm
  10. MultiFit algorithm
  11. HFFD algorithm
  12. First Fit Valid
  13. 1-out-of-$\lfloor\frac{9}{11}n\rfloor$ MMS
  14. Reduction: Factored & Bivalued Instances
  15. Polynomial-time computation
  16. ...and 14 more sections

Key Result

lemma 1

Let $n\geq 2$ be an integer, $\mathcal{M}$ a set of chores, and $v$ any cost function, Let $d := \lfloor\frac{9}{11}n\rfloor$, and suppose If the tuple $(\mathcal{M}, \mathbf A, v,1)$ is First Fit Valid, then the allocation $\mathbf A$ must contain all chores in $\mathcal{M}$.

Figures (1)

  • Figure 1: Columns A--D describe all possible cases for numbers of large and small chores in $P_k$ and $Q_k$, that satisfy Lemma \ref{['lem:twosmallchores']}. Column E describes a lower bound on $l/s$ derived from Lemma \ref{['lem:PkQk']}. Column F describes a lower bound on $\mu/s$ derived from substituting column E in $\mu \geq a_q l + b_q s$. Columns G and H describe lower bounds on $\tau$ derived from substituting column E in $\tau \geq (15/13)\cdot(a_q l + b_q s)$.

Theorems & Definitions (37)

  • Definition 1: Lexicographic order
  • Definition 2: Lexicographically maximal subset
  • Definition 3: Benchmark bundle
  • proof
  • Definition 4: First Fit Valid (FFV)
  • proof
  • lemma 1
  • proof
  • Definition 5: Suitable reduced chore
  • Definition 6: Fit-in space huang2023reduction
  • ...and 27 more