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An FPTAS for 7/9-Approximation to Maximin Share Allocations

Xin Huang, Shengwei Zhou

TL;DR

This work advances MMS allocations for indivisible goods with additive valuations by achieving a $ rac{7}{9}$-approximation and, more importantly, introducing an FPTAS that delivers a $( rac{7}{9}-oldsymbol{ extvarepsilon})$-MMS allocation in time $ rac{1}{oldsymbol{ extvarepsilon}} ext{poly}(n,m)$ without requiring explicit MMS estimation. The core innovation is the witness allocation framework, which acts as a dynamic analytical certificate that evolves with the allocation process, decoupling feasibility analysis from the real allocation. Adaptive reductions are embedded into the algorithmic process, allowing reductions to be interleaved with allocation while preserving key invariants. The results improve the state-of-the-art beyond the previous $ rac{10}{13}$-approximation, and the framework paves the way for potential further improvements by refining the structural invariants (e.g., the witness, shortages, and canonical allocations). Overall, the paper provides both a conceptually simpler algorithm and a practically efficient method for near-optimal MMS allocations, with implications for fair division in settings with indivisible goods.

Abstract

We present a new algorithm that achieves a $\frac{7}{9}$-approximation for the maximin share (MMS) allocation of indivisible goods under additive valuations, improving the current best ratio of $\frac{10}{13}$ (Heidari et al., SODA 2026). Building on a new analytical framework, we further obtain an FPTAS that achieves a $\frac{7}{9}-\varepsilon$ approximation in $\tfrac{1}{\varepsilon} \cdot \mathrm{poly}(n,m)$ time. Compared with prior work (Heidari et al., SODA 2026), our algorithm is substantially simpler.

An FPTAS for 7/9-Approximation to Maximin Share Allocations

TL;DR

This work advances MMS allocations for indivisible goods with additive valuations by achieving a -approximation and, more importantly, introducing an FPTAS that delivers a -MMS allocation in time without requiring explicit MMS estimation. The core innovation is the witness allocation framework, which acts as a dynamic analytical certificate that evolves with the allocation process, decoupling feasibility analysis from the real allocation. Adaptive reductions are embedded into the algorithmic process, allowing reductions to be interleaved with allocation while preserving key invariants. The results improve the state-of-the-art beyond the previous -approximation, and the framework paves the way for potential further improvements by refining the structural invariants (e.g., the witness, shortages, and canonical allocations). Overall, the paper provides both a conceptually simpler algorithm and a practically efficient method for near-optimal MMS allocations, with implications for fair division in settings with indivisible goods.

Abstract

We present a new algorithm that achieves a -approximation for the maximin share (MMS) allocation of indivisible goods under additive valuations, improving the current best ratio of (Heidari et al., SODA 2026). Building on a new analytical framework, we further obtain an FPTAS that achieves a approximation in time. Compared with prior work (Heidari et al., SODA 2026), our algorithm is substantially simpler.

Paper Structure

This paper contains 37 sections, 22 theorems, 15 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Computation Efficiency.
  3. Our Results and Technical Contributions
  4. Other Related Works
  5. Chore Allocation.
  6. Other Settings.
  7. Preliminary
  8. Reduction Rules
  9. Algorithm and Technical Overview
  10. The Algorithm.
  11. Remark.
  12. Witness Allocation
  13. Pebble, Ice, and Water
  14. Fully Polynomial-Time Approximation Scheme
  15. Constructing the FPTAS
  16. ...and 22 more sections

Key Result

Lemma 2.4

For any instance $\mathcal{I}$, there exists an ordered instance $\hat{\mathcal{I}}$ such that any $\alpha$-MMS allocation for $\hat{\mathcal{I}}$ can be convert into an $\alpha$-MMS allocation for $\mathcal{I}$. Moreover, such a conversion can be done in polynomial time.

Figures (2)

  • Figure 1: Illustration for the shortage block. During some round, we implement a swap by exchange two pebbles with a combination of pebble and water. The resulted shortage is contains the combination of pebble and water, and the shortage will be regarded as a complete unit during the following swap.
  • Figure 2: Illustration for 1-shortage block. Different from 2-shortage block, 1-shortage block only contains water.

Theorems & Definitions (59)

  • Definition 2.1
  • Definition 2.2: $\alpha$-MMS
  • Definition 2.3: Ordered instance
  • Lemma 2.4: journals/teco/BarmanK20
  • Definition 2.5
  • Proposition 2.6
  • Theorem 3.1
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • ...and 49 more