In the standard model of fair allocation of resources to agents, every agent has some utility for every resource, and the goal is to assign resources to agents so that the agents' welfare is maximized. Motivated by job scheduling, interest in this problem dates back to the work of Deuermeyer et al. [SIAM J. on Algebraic Discrete Methods'82]. Recent works consider the compatibility between resources and assign only mutually compatible resources to an agent. We study a fair allocation problem in which we are given a set of agents, a set of resources, a utility function for every agent over a set of resources, and a {\it conflict graph} on the set of resources (where an edge denotes incompatibility). The goal is to assign resources to the agents such that $(i)$ the set of resources allocated to an agent are compatible with each other, and $(ii)$ the minimum satisfaction of an agent is maximized, where the satisfaction of an agent is the sum of the utility of the assigned resources. Chiarelli et al. [Algorithmica'22] explore this problem from the classical complexity perspective to draw the boundary between the cases that are polynomial-time solvable and those that are \NP-hard. In this article, we study the parameterized complexity of the problem (and its variants) by considering several natural and structural parameters.

翻译：在标准公平资源分配模型中，每个代理对每种资源具有某种效用，目标是合理分配资源以最大化代理的福利。受作业调度问题的启发，对该问题的研究可追溯至Deuermeyer等人[SIAM J. on Algebraic Discrete Methods'82]的工作。近期研究关注资源间的相容性，仅将互容资源分配给同一代理。本文研究一个公平分配问题：给定一组代理、一组资源、每个代理对资源集合的效用函数，以及资源集合上的冲突图（边表示不相容性）。目标是将资源分配给代理，使得：（i）分配给同一代理的资源集合彼此相容；（ii）最小化代理满意度（代理满意度定义为所分配资源效用之和）最大化。Chiarelli等人[Algorithmica'22]从经典复杂度视角探讨该问题，划定了多项式时间可解与NP难问题的边界。本文通过考虑多个自然参数与结构参数，研究该问题（及其变体）的参数化复杂度。