π€ AI Summary
This work addresses the problem of fairly allocating vertices of a graph among multiple agents with identical preferences under conflict constraints, where adjacent vertices cannot be assigned to the same agent. The authors introduce, for the first time, a hierarchical framework based on the strong chromatic number to unify the modeling of fair allocation under three fairness criteria: SD-EF1, EF1, and EF[1,1]. Leveraging techniques from strong graph coloring, they design a deterministic polynomial-time algorithm whose performance depends on the graphβs maximum degree Ξ. They prove that for any graph with maximum degree Ξ, a fair allocation satisfying all three criteria exists whenever the number of agents is at least 3Ξβ1. Moreover, when the number of agents is at least (3+Ξ΅)Ξ for any constant Ξ΅>0, such an allocation can be efficiently constructed.
π Abstract
In the fair allocation problem under conflict constraints, the goal is to partition the vertices of a graph among agents in a fair manner, such that no two adjacent vertices are assigned to the same agent. We study this problem for agents with common preferences through the lens of three fairness criteria: stochastic-dominance envy-freeness up to one item for preference orders (SD-EF1), envy-freeness up to one item for monotone additive valuations (EF1), and envy-freeness up to one item from each side for general additive valuations (EF[1,1]). To do so, we introduce a hierarchy of variants of the strong chromatic number, a graph quantity introduced independently by Alon and Fellows in the early nineties. Our results reveal a close connection between fair allocation under conflict constraints and the first two levels of this hierarchy, providing a unified route to both existential and algorithmic results.
For SD-EF1, we fully characterize the number of agents needed to guarantee a fair allocation of a given graph for every common preference order. For EF1 and EF[1,1], we provide analogous sufficient conditions, extending a result on path graphs due to Equbal, Gurjar, Igarashi, Kumar, Manurangsi, Nath, Saxena, Vaish, and Yoneda. We also show that, unlike in the SD-EF1 setting, the sufficient conditions for EF1 and EF[1,1] are not necessary in general. Our framework yields existential and algorithmic consequences in terms of the maximum degree. We obtain that every graph with maximum degree $Ξ$ admits SD-EF1, EF1, and EF[1,1] allocations for common preferences whenever the number of agents is at least $3Ξ-1$. We further provide, for any $\varepsilon>0$, deterministic polynomial-time algorithms that find such allocations whenever the number of agents is at least $(3+\varepsilon)Ξ$. These guarantees strengthen earlier work by Barman and Viswanathan on equitable colorings.