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

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Abstract

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 kk(|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.

Original languageEnglish
Title of host publication15th International Symposium on Parameterized and Exact Computation (IPEC 2020)
EditorsYixin Cao, Marcin Pilipczuk
PublisherSchloss Dagstuhl - Leibniz-Zentrum für Informatik
Pages25:1-25:17
ISBN (Electronic)9783959771726
ISBN (Print)978-3-95977-172-6
DOIs
Publication statusPublished - Dec 2020

Publication series

NameLeibniz International Proceedings in Informatics (LIPIcs)
Publisher1868-8969
Volume180

Bibliographical note

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Keywords

  • Fixed-Parameter Tractability
  • Precedence Constraints
  • Scheduling

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