Scheduling under Linear Constraints

TL;DR

This paper studies a novel parallel machine scheduling problem where job processing times are determined by linear constraints, focusing on complexity, algorithms, and approximation methods for different problem settings.

Contribution

It introduces the problem, analyzes its computational complexity, and provides polynomial algorithms and approximation strategies for various scenarios.

Findings

Polynomial-time solutions for fixed numbers of machines and constraints.

NP-Hardness when both the number of machines and constraints are variable.

Effective approximation algorithms for complex cases.

Abstract

We introduce a parallel machine scheduling problem in which the processing times of jobs are not given in advance but are determined by a system of linear constraints. The objective is to minimize the makespan, i.e., the maximum job completion time among all feasible choices. This novel problem is motivated by various real-world application scenarios. We discuss the computational complexity and algorithms for various settings of this problem. In particular, we show that if there is only one machine with an arbitrary number of linear constraints, or there is an arbitrary number of machines with no more than two linear constraints, or both the number of machines and the number of linear constraints are fixed constants, then the problem is polynomial-time solvable via solving a series of linear programming problems. If both the number of machines and the number of constraints are inputs of the problem instance, then the problem is NP-Hard. We further propose several approximation algorithms for the latter case.