# Chore division on a graph

## TL;DR

This paper studies fair division of chores arranged on a graph, showing the inherent differences from goods division, and explores the computational complexity of achieving fairness under various conditions.

## Contribution

It demonstrates the fundamental differences between chores and goods division and analyzes the complexity of fair allocation with connected subgraph constraints.

## Key findings

- Deciding fair chores division is computationally hard on paths and stars.
- Some special cases on specific graph topologies are solvable efficiently.

## Abstract

The paper considers fair allocation of indivisible nondisposable items that generate disutility (chores). We assume that these items are placed in the vertices of a graph and each agent's share has to form a connected subgraph of this graph. Although a similar model has been investigated before for goods, we show that the goods and chores settings are inherently different. In particular, it is impossible to derive the solution of the chores instance from the solution of its naturally associated fair division instance. We consider three common fair division solution concepts, namely proportionality, envy-freeness and equitability, and two individual disutility aggregation functions: additive and maximum based. We show that deciding the existence of a fair allocation is hard even if the underlying graph is a path or a star. We also present some efficiently solvable special cases for these graph topologies.