Maximin Shares Under Cardinality Constraints

TL;DR

This paper introduces new algorithms for fair division of indivisible items under category limits, achieving improved approximation guarantees for maximin share fairness in both goods and chores.

Contribution

It presents a polynomial-time algorithm that guarantees a 1/2-approximate MMS for goods, improving previous guarantees, and extends these results to chores and single-category cases.

Findings

A 1/2-approximate MMS algorithm for goods under cardinality constraints.

A 2/3-approximate MMS guarantee for single-category instances.

Existence of a /-approximate MMS allocation for goods.

Abstract

We study the problem of fair allocation of a set of indivisible items among agents with additive valuations, under cardinality constraints. In this setting, the items are partitioned into categories, each with its own limit on the number of items it may contribute to any bundle. We consider the fairness measure known as the maximin share (MMS) guarantee, and propose a novel polynomial-time algorithm for finding $1/2$-approximate MMS allocations for goods -- an improvement from the previously best available guarantee of $11/30$. For single-category instances, we show that a modified variant of our algorithm is guaranteed to produce $2/3$-approximate MMS allocations. Among various other existence and non-existence results, we show that a $(\sqrt{n}/(2\sqrt{n} - 1))$-approximate MMS allocation always exists for goods. For chores, we show similar results as for goods, with a $2$-approximate algorithm in the general case and a $3/2$-approximate algorithm for single-category instances. We extend the notions and algorithms related to ordered and reduced instances to work with cardinality constraints, and combine these with bag filling style procedures to construct our algorithms.