Non-Existence of PMMS Allocations and a $4/3$-PMMS Guarantee for Additive Chores

2026-09-09

AI Summary

研究了不可分物品的PMMS公平性问题，通过多项式时间归约方法证明了其在某些情况下的非存在性，并给出了一种4/3-PMMS近似解法。

Abstract

We study pairwise maximin share (PMMS) fairness for indivisible items with additive preferences. We give a polynomial-time reduction from chores to goods that preserves the existence of a PMMS allocation. Together with known nonexistence results for chores, this yields nonexistence for additive goods. In addition, we show that deciding if a given instance admits a PMMS allocation is NP-hard. We also give explicit instances whose PMMS factors are $226/227$ for goods and $1.102065$ for chores, certified by exact enumeration. Complementing these impossibility results, we prove that every additive-chore instance admits a $4/3$-PMMS allocation.

Problem

• PMMS
• additive preferences
• indivisible items
• NP-hard
• fairness

Innovation

• pairwise maximin share (PMMS)
• polynomial-time reduction
• additive preferences
• $4/3$-PMMS guarantee
• NP-hard