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

Abstract

AbstractWe 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 $$f(k)n^{{\mathcal {O}}(1)}$$ f ( k ) n O ( 1 ) or $$n^{{\mathcal {O}}(f(k))}$$ 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 $${\mathsf {P}}$$ P , $${{\mathsf {N}}}{{\mathsf {P}}}$$ N P -complete and fixed-parameter tractable by k, or $${\mathsf {W}}[1]$$ 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 $${\mathcal {O}}(8^kk(|V|+|E|))$$ 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 $$G=(V,E)$$ G = ( V , E ) is the graph with precedence constraints.