发表机构
Poznań University of Economics and Business; Jagiellonian University; Warsaw University of Technology（波兹南经济大学; 雅盖隆大学; 华沙理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了树上不可分割家务的连通分配存在性，即使成本单调，并给出等价组合定理及k^{O(k)}时间算法。
AI 中文摘要
Xiao、Qiu和Huang（AAMAS 2023）以及独立的Lonc（个人通信）提出了一个问题：位于树的顶点上的不可分割家务是否总能以连通束的形式分配给n个智能体，使得每个智能体的成本至多为其连通极大最小份额；对于商品，这是Bouveret、Cechlárová、Elkind、Igarashi和Peters的一个定理。我们肯定地回答了这个问题，即使对于单调成本也是如此。该答案源于一个组合定理：如果P_1,…,P_k是有限树的顶点集的划分，每个划分至多包含k个连通部分，那么顶点集可以被分割成不相交的集合B_1,…,B_k，其中一些可能为空，使得每个非空的B_i是连通的并且包含在P_i的一个部分中。等价地，如果k种颜色中的每一种在树中至多出现在k-1条边上，那么顶点可以被划分成由不同颜色标记的连通集合，其中没有一个包含其自身颜色的边；特别地，对于每种颜色，可以选择通过删除该颜色获得的森林的一个分量，使得所选分量覆盖整棵树。对于路径，该界已经是最优的；对于加性成本，该定理等价于公平分配陈述。证明将问题归结为有向树的内向划分，我们通过边权单纯形上的多彩KKM定理获得，使用一种叶消除标记，该标记在权重消失时仍然兼容。我们还给出了一种运行时间为k^{O(k)}加上多项式时间的算法。
英文摘要
Xiao, Qiu, and Huang (AAMAS 2023) and independently Lonc (personal communication) asked whether indivisible chores located at the vertices of a tree can always be allocated to $n$ agents in connected bundles so that the cost of every agent is at most its connected maximin share; for goods, this is a theorem of Bouveret, Cechlárová, Elkind, Igarashi, and Peters. We answer the question affirmatively, even for monotone costs. The answer follows from a combinatorial theorem: if $\mathcal P_1,\ldots,\mathcal P_k$ are partitions of the vertex set of a finite tree, each into at most $k$ connected parts, then the vertex set can be split into disjoint sets $B_1,\ldots,B_k$, some possibly empty, such that each nonempty $B_i$ is connected and contained in a part of $\mathcal P_i$. Equivalently, if each of $k$ colours occurs on at most $k-1$ edges of a tree, then the vertices can be partitioned into connected sets labelled by distinct colours, none containing an edge of its own colour; in particular, one can choose for every colour a component of the forest obtained by deleting that colour so that the chosen components cover the tree. The bound is already best possible for paths, and for additive costs, the theorem is equivalent to the fair-division statement. The proof reduces the problem to inward partitions of oriented trees, which we obtain from the colourful KKM theorem on a simplex of edge weights, using a leaf-elimination labelling that remains compatible when weights vanish. We also give an algorithm running in time $k^{O(k)}$ plus polynomial time.
