We study the problem of fairly assigning a set of discrete tasks (or chores) among a set of agents with additive valuations. Each chore is associated with a start and finish time, and each agent can perform at most one chore at any given time. The goal is to find a fair and efficient schedule of the chores, where fairness pertains to satisfying envy-freeness up to one chore (EF1) and efficiency pertains to maximality (i.e., no unallocated chore can be feasibly assigned to any agent). Our main result is a polynomial-time algorithm for computing an EF1 and maximal schedule for two agents under monotone valuations when the conflict constraints constitute an arbitrary interval graph. The algorithm uses a coloring technique in interval graphs that may be of independent interest. For an arbitrary number of agents, we provide an algorithm for finding a fair schedule under identical dichotomous valuations when the constraints constitute a path graph. We also show that stronger fairness and efficiency properties, including envy-freeness up to any chore (EFX) along with maximality and EF1 along with Pareto optimality, cannot be achieved.

翻译：我们研究了在具有加性估值的智能体之间公平分配一组离散任务（或家务）的问题。每个任务都关联一个开始时间和结束时间，且每个智能体在任意时刻最多只能执行一个任务。目标是找到一个既公平又高效的任务调度方案，其中公平性涉及满足至多一个任务的无嫉妒性（EF1），效率则涉及最大性（即，任何未分配的任务都无法被可行地分配给任何智能体）。我们的主要结果是：当冲突约束构成任意区间图时，针对两个智能体在单调估值下，提出了一个计算EF1且最大调度的多项式时间算法。该算法使用了区间图中的着色技术，这一技术可能具有独立的研究价值。对于任意数量的智能体，当约束构成路径图时，我们在相同二分类估值下提供了一种寻找公平调度的算法。我们还证明了更强的公平性和效率属性（包括至多任意任务的无嫉妒性（EFX）与最大性，以及EF1与帕累托最优性）是无法实现的。