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

We study a natural variant of scheduling that we call partial scheduling : in this variant an instance of a scheduling problem along with an integer  k is given and one seeks an optimal schedule where not all, but only k jobs, have to be processed. Specifically, we aim to determine the fine-grained parameterized complexity of partial scheduling problems parameterized by k 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 f ( k ) n O ( 1 ) or n O ( f ( k ) ) exist for a function f that is as small as possible. Our contribution is two-fold: First, we categorize each variant to be either in P , N P -complete and fixed-parameter tractable by k , or W [ 1 ] -hard parameterized by k . 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 O ( 8 k k ( | V | + | E | ) ) time algorithm to solve instances of partial scheduling problems minimizing the makespan with unit length jobs, precedence constraints and release dates, where G = ( V , E ) is the graph with precedence constraints.