New complexity results for parallel identical machine scheduling problems with preemption, release dates and regular criteria

TL;DR

This paper investigates the complexity of parallel identical machine scheduling with preemption and release dates, providing new insights into problem solvability and refining the boundary between polynomial solvability and NP-hardness.

Contribution

It establishes conditions under which solutions have a permutation flow shop structure and identifies subclasses with ordered completion times, advancing complexity classification.

Findings

Solutions with permutation flow shop structure are dominant under certain conditions.

Identifies subclasses where all jobs' completion times can be ordered in optimal solutions.

Refines the boundary between polynomial solvability and NP-hardness for these scheduling problems.

Abstract

In this paper, we are interested in parallel identical machine scheduling problems with preemption and release dates in case of a regular criterion to be minimized. We show that solutions having a permutation flow shop structure are dominant if there exists an optimal solution with completion times scheduled in the same order as the release dates, or if there is no release date. We also prove that, for a subclass of these problems, the completion times of all jobs can be ordered in an optimal solution. Using these two results, we provide new results on polynomially solvable problems and hence refine the boundary between P and NP for these problems.