🤖 AI Summary
This paper addresses the fair and efficient allocation of periodic tasks under recurring schedules. While conventional approaches optimize efficiency by minimizing the number of workers, we formally define long-term workload fairness—measuring equitable task distribution across workers over time—and characterize its fundamental trade-off with efficiency. Contribution/Results: We prove that achieving fairness incurs at most one additional worker beyond the optimal efficient solution, and this bound is tight; we further derive necessary and sufficient conditions for joint fairness-efficiency feasibility. Method: We propose an exact *O*(*n* log *n*) algorithm and an efficient nearest-neighbor heuristic, both constructing one-to-one fair allocations without resorting to aperiodic scheduling. Experiments confirm that the theoretical fairness-cost upper bound is attainable and demonstrate that our approach guarantees optimal efficiency while ensuring strict long-term fairness.
📝 Abstract
We study the periodic assignment problem, in which a set of periodically repeating tasks must be assigned to workers within a repeating schedule. The classical efficiency objective is to minimize the number of workers required to operate the schedule. We propose a O(n log n) algorithm to solve this problem. Next, we formalize a notion of fairness among workers, and impose that each worker performs the same work over time. We analyze the resulting trade-off between efficiency and fairness, showing that the price of fairness is at most one extra worker, and that such a fair solution can always be found using the Nearest Neighbor heuristic. We characterize all instances that admit a solution that is both fair and efficient, and use this result to develop a O(n log n) exact algorithm for the fair periodic assignment problem. Finally, we show that allowing aperiodic schedules never reduces the price of fairness.