# Structural Parameters for Scheduling with Assignment Restrictions

## TL;DR

This paper investigates scheduling problems with assignment restrictions, analyzing how structural graph parameters like tree- and rankwidth influence computational complexity and approximation algorithms.

## Contribution

It introduces three graphs based on assignment restrictions and studies their properties to identify cases with efficient algorithms, extending prior results.

## Key findings

- Certain graph structures allow polynomial-time approximation schemes.
- Tree- and rankwidth parameters are key to understanding problem complexity.
- New algorithms are developed for specific graph classes.

## Abstract

We consider scheduling on identical and unrelated parallel machines with job assignment restrictions. These problems are NP-hard and they do not admit polynomial time approximation algorithms with approximation ratios smaller than $1.5$ unless P$=$NP. However, if we impose limitations on the set of machines that can process a job, the problem sometimes becomes easier in the sense that algorithms with approximation ratios better than $1.5$ exist. We introduce three graphs, based on the assignment restrictions and study the computational complexity of the scheduling problem with respect to structural properties of these graphs, in particular their tree- and rankwidth. We identify cases that admit polynomial time approximation schemes or FPT algorithms, generalizing and extending previous results in this area.