Computational complexity and algorithms for two scheduling problems under linear constraints

This paper considers two different types of scheduling problems under linear constraints. The first is the single-machine scheduling problem with minimizing total completion time, while the second is the no-wait two-machine flow shop scheduling problem with minimizing makespan. For these two problems, a set of jobs is required to be scheduled to one or two machines. In contrast to the classic scheduling problems, the processing times of jobs are not fixed constants and are not predetermined. The decision-maker only knows that they should satisfy a system of given linear constraints. For both problems, the goal is to determine the processing time for each job and find the schedule that minimizes a particular criterion, namely, the total completion time or the makespan. First, we study the computational complexity and show that both the problems under linear constraints are NP-hard. These hardness results significantly differ from their traditional scheduling counterparts, as both of those are solvable in polynomial time. Then we propose polynomial time exact or approximation algorithms for various special cases. By utilizing the existing scheduling algorithms and the properties of linear programming, we demonstrate that both problems are polynomially solvable when the total number of linear constraints is a fixed constant.