A Note on Hardness of Multiprocessor Scheduling with Scheduling Solution Space Tree

TL;DR

This paper investigates the complexity of non-preemptive multiprocessor scheduling, introduces the Scheduling Solution Space Tree data structure, and presents the first non-deterministic polynomial-time algorithm for the problem.

Contribution

It defines the SSST data structure, proves NP-hardness using SSST and WSSST, and develops the first non-deterministic polynomial-time scheduling algorithm.

Findings

Proves the problem is NP-hard using SSST and WSSST.

Introduces the Magic Scheduling (MS) algorithm.

Defines a new scheduling variant with an additional user parameter.

Abstract

We study the computational complexity of the non-preemptive scheduling problem of a list of independent jobs on a set of identical parallel processors with a makespan minimization objective. We make a maiden attempt to explore the combinatorial structure showing the exhaustive solution space of the problem by defining the \textit{Scheduling Solution Space Tree (SSST)} data structure. The properties of the \textit{SSST} are formally defined and characterized through our analytical results. We develop a unique technique to show the problem $\mathcal{NP}$ using the SSST and the \textit{Weighted Scheduling Solution Space Tree (WSSST)} data structures. We design the first non-deterministic polynomial-time algorithm named \textit{Magic Scheduling (MS)} for the problem based on the reduction framework. We also define a new variant of multiprocessor scheduling by including the user as an additional input parameter. We formally establish the complexity class of the variant by the reduction principle. Finally, we conclude the article by exploring several interesting open problems for future research investigation.