On The Parameterized Tractability of the Just-In-Time Flow-Shop Scheduling Problem

TL;DR

This paper investigates the parameterized complexity of the Just-In-Time flow-shop scheduling problem, revealing its computational hardness for certain parameters and identifying fixed-parameter tractability in specific cases.

Contribution

It provides the first complexity analysis of JIT flow-shop scheduling with respect to parameters like due dates, weights, and processing times, highlighting cases of fixed-parameter tractability and W[1]-hardness.

Findings

Two-machine problem is W[1]-hard with respect to due dates.

Problem is in XP for a parameterized number of machines.

Fixed-parameter tractability is shown for two machines with combined parameters.

Abstract

Since its development in the early 90's, parameterized complexity has been widely used to analyze the tractability of many NP-hard combinatorial optimization problems with respect to various types of problem parameters. While the generic nature of the framework allows the analysis of any combinatorial problem, the main focus along the years was on analyzing graph problems. In this paper we diverge from this trend by studying the parameterized complexity of Just-In-Time (JIT) flow-shop scheduling problems. Our analysis focuses on the case where the number of due dates is considerably smaller than the number of jobs, and can thus be considered as a parameter. We prove that the two-machine problem is W[1]-hard with respect to this parameter, even if all processing times on the second machine are of unit length, while the problem is in XP even for a parameterized number of machines. We then move on to study the tractability of the problem when combining the different number of due dates with either the number of different weights or the number of different processing times on the first machine. We prove that in both cases the problem is fixed-parameter tractable for the two machine case, and is W[1]-hard for three or more machines.