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Improved Maximin Share Guarantee for Additive Valuations

Ehsan Heidari, Alireza Kaviani, Masoud Seddighin, AmirMohammad Shahrezaei

TL;DR

This work advances fair division for indivisible goods with additive valuations by achieving a $( frac{10}{13})$-approximation to the maximin share (MMS). It introduces a sophisticated framework of dynamic reductions, deferred matching, bag-filling, and calibration functions to tightly bound agents’ MMS during every stage. The core contributions include a novel reduction family extending prior rules, a deferred, priority-aware matching scheme, and calibration techniques that preserve MMS under transformations, culminating in a polynomial-time $( frac{10}{13}- ext{ε})$-MMS allocation for any constant ε>0. The result narrows the MMS-approximation gap and provides practical algorithmic machinery with implications for equitable resource distribution in settings with indivisible goods. The methodology blends combinatorial reductions with a moving-knife-inspired bag-filling process, yielding improvements with clear theoretical and potential applied impact.

Abstract

The maximin share ($\textsf{MMS}$) is the most prominent share-based fairness notion in the fair allocation of indivisible goods. Recent years have seen significant efforts to improve the approximation guarantees for $\textsf{MMS}$ for different valuation classes, particularly for additive valuations. For the additive setting, it has been shown that for some instances, no allocation can guarantee a factor better than $1-\tfrac{1}{n^4}$ of maximin share value to all agents. However, the best currently known algorithm achieves an approximation guarantee of $\tfrac{3}{4} + \tfrac{3}{3836}$ for $\textsf{MMS}$. In this work, we narrow this gap and improve the best-known approximation guarantee for $\textsf{MMS}$ to $\tfrac{10}{13}$.

Improved Maximin Share Guarantee for Additive Valuations

TL;DR

This work advances fair division for indivisible goods with additive valuations by achieving a -approximation to the maximin share (MMS). It introduces a sophisticated framework of dynamic reductions, deferred matching, bag-filling, and calibration functions to tightly bound agents’ MMS during every stage. The core contributions include a novel reduction family extending prior rules, a deferred, priority-aware matching scheme, and calibration techniques that preserve MMS under transformations, culminating in a polynomial-time -MMS allocation for any constant ε>0. The result narrows the MMS-approximation gap and provides practical algorithmic machinery with implications for equitable resource distribution in settings with indivisible goods. The methodology blends combinatorial reductions with a moving-knife-inspired bag-filling process, yielding improvements with clear theoretical and potential applied impact.

Abstract

The maximin share () is the most prominent share-based fairness notion in the fair allocation of indivisible goods. Recent years have seen significant efforts to improve the approximation guarantees for for different valuation classes, particularly for additive valuations. For the additive setting, it has been shown that for some instances, no allocation can guarantee a factor better than of maximin share value to all agents. However, the best currently known algorithm achieves an approximation guarantee of for . In this work, we narrow this gap and improve the best-known approximation guarantee for to .

Paper Structure

This paper contains 32 sections, 17 theorems, 85 equations, 17 figures, 4 tables, 6 algorithms.

Table of Contents

  1. Table of Frequently Used Notation
  2. Examples
  3. Bounds on $\textsf{MMS}$ Values for Calibrated Valuations
  4. Introduction
  5. Further Related Work
  6. Basic Notations
  7. Algorithm Structure and Notation
  8. Highlights of Techniques
  9. Algorithmic Overview
  10. Techniques
  11. Techniques
  12. Reductions
  13. Our reductions.
  14. Deferred matching.
  15. Bag-filling
  16. ...and 17 more sections

Key Result

Lemma 1

Let $\hat{M}$ be a set of goods, $d$ be a constant, and let $\hat{v}$ be a valuation function such that $\Psi^d_{\hat{v}}(\hat{M}) \ge 1$, and for all $\hat{g} \in \hat{M}$ we have $\hat{v}(\{\hat{g}\})\le 1$. Then for every $0\le\lambda \le \tfrac{4\alpha}{3}-1$ we have $\Psi^{d}_{({f_\lambda}\sta

Figures (17)

  • Figure 1: Comparison of classical and our modified reductions. Red boxes indicate the bundle chosen by the classical reductions, while blue boxes indicate the bundle chosen by our reductions.
  • Figure 2: The final result of primary reductions.
  • Figure 3: Recent progress on approximation guarantees for $\textsf{MMS}$ in the additive setting.
  • Figure 4: A flowchart of our algorithm.
  • Figure 5: Bipartite graph demonstrating: (1) Reduction matching (solid blue) covering all bundles $\{B_i\}_{i=1}^5$, and (2) A matching (dashed magenta) prioritizing red over green agents. In both matchings, $B_1$ is paired with agent $a_1$ and $B_5$ is paired with agent $a_9$.
  • ...and 12 more figures

Theorems & Definitions (51)

  • Example 1: Reductions
  • Example 2: Primary Reductions
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 41 more