Complexity of interval minmax regret scheduling on parallel identical machines with total completion time criterion

TL;DR

This paper investigates the computational complexity of scheduling jobs on parallel identical machines to minimize total completion time under uncertain processing times, revealing the problem is strongly NP-hard.

Contribution

It proves that the minmax regret scheduling problem on parallel identical machines is strongly NP-hard, extending known complexity results.

Findings

The problem is polynomially solvable in the deterministic case.

Minmax regret version is weakly NP-hard for a single machine.

The paper establishes strong NP-hardness for parallel identical machines.

Abstract

In this paper, we consider the problem of scheduling jobs on parallel identical machines, where the processing times of jobs are uncertain: only interval bounds of processing times are known. The optimality criterion of a schedule is the total completion time. In order to cope with the uncertainty, we consider the maximum regret objective and we seek a schedule that performs well under all possible instantiations of processing times. Although the deterministic version of the considered problem is solvable in polynomial time, the minmax regret version is known to be weakly NP-hard even for a single machine, and strongly NP-hard for parallel unrelated machines. In this paper, we show that the problem is strongly NP-hard also in the case of parallel identical machines.