Parameterized Complexity of Partial Scheduling

TL;DR

This paper investigates the parameterized complexity of partial scheduling problems, classifying variants into tractable or hard cases and providing near-optimal algorithms for specific scenarios based on the parameter k.

Contribution

It categorizes the complexity of various partial scheduling variants and develops efficient algorithms with tight runtime bounds for key cases.

Findings

Classified scheduling variants as P, NP-complete, or W[1]-hard based on parameter k.

Provided an almost optimal algorithm with runtime O(8^k * k * (|V|+|E|)) for a specific partial scheduling case.

Achieved fine-grained runtime analyses under the Exponential Time Hypothesis.

Abstract

We study a natural variant of scheduling that we call \emph{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 $f(k)n^{\mathcal{O}(1)}$ or $n^{\mathcal{O}(f(k))}$ 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 $\mathsf{P}$, $\mathsf{NP}$-complete and fixed-parameter tractable by $k$, or $\mathsf{W}[1]$-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 $\mathcal{O}(8^kk(|V|+|E|))$ 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.