# MMS Allocations of Chores with Connectivity Constraints: New Methods and New Results

## TL;DR

This paper introduces new methods for allocating indivisible chores with connectivity constraints, establishing existence results on certain tree subclasses and providing tight approximation ratios on cycles for three agents.

## Contribution

It presents the group-satisfied method for MMS chores allocation on trees and the LP method for approximation and nonexistence proofs, advancing understanding of chores allocation problems.

## Key findings

- Existence of MMS allocations on some tree subclasses proved.
- Tight approximation ratio for three agents on cycles established.
- LP method enables both allocation computation and nonexistence examples.

## Abstract

We study the problem of allocating indivisible chores to agents under the Maximin share (MMS) fairness notion. The chores are embedded on a graph and each bundle of chores assigned to an agent should be connected. Although there is a simple algorithm for MMS allocations of goods on trees, it remains open whether MMS allocations of chores on trees always exist or not, which is a simple but annoying problem in chores allocation. In this paper, we introduce a new method for chores allocation with connectivity constraints, called the group-satisfied method, that can show the existence of MMS allocations of chores on several subclasses of trees. Even these subcases are non-trivial and our results can be considered as a significant step to the open problem. We also consider MMS allocations of chores on cycles where we get the tight approximation ratio for three agents. Our result was obtained via the linear programming (LP) method, which enables us not only to compute approximate MMS allocations but also to construct tight examples of the nonexistence of MMS allocations without complicated combinatorial analysis. These two proposed methods, the group-satisfied method and the LP method, have the potential to solve more related problems.