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.

翻译：我们研究了在最大化份额（MMS）公平概念下将不可分割家务分配给智能体的问题。这些家务被嵌入在一个图中，且每个智能体分配到的家务包应具有连通性。尽管在树上存在一种简单的商品MMS分配算法，但树上家务的MMS分配是否总是存在仍是一个未解决问题，这是家务分配中一个简单但令人困扰的问题。本文提出了一种新的具有连通性约束的家务分配方法，称为群体满意法，该方法能够证明在树的若干子类上家务的MMS分配的存在性。即使这些子情况也非平凡，我们的结果可被视为向该开放问题迈出的重要一步。我们还考虑了环上家务的MMS分配，并获得了三个智能体时的紧逼近比率。该结果通过线性规划（LP）方法获得，该方法不仅使我们能够计算近似MMS分配，还能在无需复杂组合分析的情况下构造MMS分配不存在的紧示例。所提出的这两种方法——群体满意法和线性规划方法——具有解决更多相关问题的潜力。