Approximating the $2$-Machine Flow Shop Problem with Exact Delays Taking Two Values

TL;DR

This paper investigates the 2-machine flow shop scheduling problem with exact delays limited to two values, proving the difficulty of approximation and providing a 2-approximation algorithm.

Contribution

It establishes the NP-hardness of better than 1.25-approximation and introduces a 2-approximation algorithm for the problem with two delay values.

Findings

Proves that approximating within (1.25 - ε) is NP-hard.

Develops a 2-approximation algorithm for the problem.

Highlights the computational complexity of scheduling with limited delay values.

Abstract

In the $2$-Machine Flow Shop problem with exact delays the operations of each job are separated by a given time lag (delay). Leung et al. (2007) established that the problem is strongly NP-hard when the delays may have at most two different values. We present further results for this case: we prove that the existence of $(1.25-\varepsilon)$-approximation implies P$=$NP and develop a $2$-approximation algorithm.