Fair Interval Scheduling of Indivisible Chores

Scheduling chores fairly when time conflicts exist is a hard problem, but a new paper from six researchers delivers provable guarantees. Their core result is a polynomial-time algorithm for two agents with monotone valuations (where more of any chore is never worse) and an arbitrary interval graph of conflict constraints. The method uses a coloring technique that yields an EF1 schedule—meaning no agent envies another after removing at most one chore—and is maximal (no unallocated chore can be assigned to any agent without breaking feasibility). This bridges algorithmic game theory with practical scheduling needs.

For larger groups, the team shows that with identical additive valuations on a path graph (a linear sequence of time slots), an EF1 and maximal schedule always exists for any number of agents, via a clever reduction to the 'cycle-plus-triangles' theorem. With dichotomous (like/dislike) valuations and four or more agents, a separate efficient algorithm works. On the downside, they prove that stronger fairness like EFX with maximality or EF1 with Pareto optimality cannot be guaranteed for interval scheduling, defining clear limits on what's theoretically possible.