eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-12-04
25:1
25:17
10.4230/LIPIcs.IPEC.2020.25
article
On the Fine-Grained Parameterized Complexity of Partial Scheduling to Minimize the Makespan
Nederlof, Jesper
1
Swennenhuis, Céline M. F.
2
Utrecht University, Algorithms and Complexity Group, The Netherlands
Eindhoven University of Technology, Combinatorial Optimization Group, The Netherlands
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, NP-complete and fixed-parameter tractable by k, or 𝖶[1]-hard parameterized by k. Second, for many interesting cases we further investigate the run time 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol180-ipec2020/LIPIcs.IPEC.2020.25/LIPIcs.IPEC.2020.25.pdf
Fixed-Parameter Tractability
Scheduling
Precedence Constraints