On the Fine-grained Parameterized Complexity of Partial Scheduling to Minimize the Makespan.

We study a natural variant of scheduling that we call : in this variant an instance of a scheduling problem along with an integer  is given and one seeks an optimal schedule where not all, but only jobs, have to be processed. Specifically, we aim to determine the fine-grained parameterized complexity of partial scheduling problems parameterized by for all variants of scheduling problems that minimize the makespan and involve unit/arbitrary processing times, identical/unrelated parallel machines, release/due dates, and precedence constraints. That is, we investigate whether algorithms with runtimes of the type or exist for a function that is as small as possible. Our contribution is two-fold: First, we categorize each variant to be either in , -complete and fixed-parameter tractable by , or -hard parameterized by . Second, for many interesting cases we further investigate the runtime on a finer scale and obtain run times that are (almost) optimal assuming the Exponential Time Hypothesis. As one of our main technical contributions, we give an time algorithm to solve instances of partial scheduling problems minimizing the makespan with unit length jobs, precedence constraints and release dates, where is the graph with precedence constraints.