Equitable Scheduling for the Total Completion Time Objective

We investigate a novel scheduling problem where we have $n$ clients, each associated with a single job on each of a set of $m$ different days. On each day, a single machine is available to process the $n$ jobs non-preemptively. The goal is provide an equitable set of schedules for all $m$ days such that the sum of completion times of each client over all days is not greater than some specified equability parameter $k$. The $1\mid\mid\max_j\sum_i C_{i,j}$ problem, as we refer to it in this paper, fits nicely into a new model introduced by Heeger et al. [AAAI '21] that aims at capturing a generic notion of fairness in scheduling settings where the same set of clients repeatedly submit scheduling requests over a fixed period of time. We show that the $1\mid\mid\max_j\sum_i C_{i,j}$ problem is NP-hard even under quite severe restrictions. This leads us to investigating two natural special cases: One where we assume the number of days to be small and one where we consider the number of clients to be small. We present several tractability results for both cases.