# Almost Group Envy-free Allocation of Indivisible Goods and Chores

## TL;DR

This paper introduces new fairness concepts for allocating indivisible goods and chores among agents with varying utilities, proposing algorithms for GEF1 allocations and analyzing their computational complexity.

## Contribution

It presents a taxonomy of fairness concepts, introduces GEF1, and provides polynomial algorithms for certain utility classes while proving complexity results.

## Key findings

- Polynomial-time algorithms for GEF1 allocation in specific utility domains
- Existence guarantees for certain fairness concepts under particular preferences
- coNP-completeness of checking GEF1 in general cases

## Abstract

We consider a multi-agent resource allocation setting in which an agent's utility may decrease or increase when an item is allocated. We take the group envy-freeness concept that is well-established in the literature and present stronger and relaxed versions that are especially suitable for the allocation of indivisible items. Of particular interest is a concept called group envy-freeness up to one item (GEF1). We then present a clear taxonomy of the fairness concepts. We study which fairness concepts guarantee the existence of a fair allocation under which preference domain. For two natural classes of additive utilities, we design polynomial-time algorithms to compute a GEF1 allocation. We also prove that checking whether a given allocation satisfies GEF1 is coNP-complete when there are either only goods, only chores or both.