A rainbow partition theorem for trees and connected maximin share allocations of chores

2026-09-29

AI Summary

This study addresses the long-standing open problem of whether connected allocations of indivisible chores on trees satisfy the maximin share fairness criterion. To resolve this, the authors integrate topological methods with the colorful KKM theorem to establish a rainbow partition theorem, demonstrating that the vertex set of a tree can be decomposed into connected subsets governed by respective partitions. Combined with a leaf-elimination labeling technique, this approach achieves fair allocation under monotone costs. This work provides the first affirmative answer to the aforementioned open question and introduces a novel combinatorial proof framework. Furthermore, it proves that the established bound on paths is tight and presents an efficient polynomial-time algorithm with a complexity of k^O(k).

Abstract

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.

Problem

• indivisible chores
• connected maximin share
• fair allocation
• rainbow partition
• trees

Innovation

• Rainbow partition theorem
• Connected maximin share
• Fair division of chores
• Colourful KKM theorem
• Tree allocation