# Approximation Algorithms for the Open Shop Problem with Delivery Times

## TL;DR

This paper studies the open shop scheduling problem with delivery times, establishing performance bounds for algorithms and providing a near-optimal approximation scheme for fixed machine counts.

## Contribution

It proves a performance ratio bound for list scheduling and develops a PTAS for fixed number of machines, addressing the NP-hardness of the problem.

## Key findings

- Any list scheduling algorithm has a performance ratio of 2.
- A polynomial time approximation scheme (PTAS) is designed for fixed number of machines.
- The problem remains strongly NP-hard, limiting the possibility of exact solutions.

## Abstract

In this paper we consider the open shop scheduling problem where the jobs have delivery times. The minimization criterion is the maximum lateness of the jobs. This problem is known to be NP-hard, even restricted to only 2 machines. We establish that any list scheduling algorithm has a performance ratio of 2. For a fixed number of machines, we design a polynomial time approximation scheme (PTAS) which represents the best possible result due to the strong NP-hardness of the problem.