Fair Interval Scheduling of Indivisible Chores

TL;DR

This work investigates fair and efficient scheduling of indivisible chores under conflict constraints captured by interval graphs. It targets envy-freeness up to one chore ($EF1$) combined with maximality, and establishes a polynomial-time algorithm for two agents under monotone valuations on any interval graph, using a novel adjacent-schedules coloring approach. For more than two agents, positive results hold only under restricted valuations (identical valuations) and graph structure (paths or components of size at most $n$), leaving the general three-or-more-agent case open. The paper also presents non-existence results showing that stronger fairness notions (such as $EFX$ with maximality) or combining $EF1$ with Pareto optimality can be impossible, even in simple conflict graphs, highlighting intrinsic trade-offs in constrained fair division of chores. Overall, it develops new tools (coloring techniques and adjacent-schedules sequences) to achieve $EF1$+maximality in the presence of temporal conflicts and delineates the boundaries of what is algorithmically feasible.

Abstract

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.

Figures (2)

• Figure 1: Summary of our results. The arrows denote logical implications between fairness and efficiency notions. The positive and negative results are shown in green and red, respectively.
• Figure 2: (a) A scheduling instance and (b) its conflict graph.