# Inapproximability Results for Scheduling with Interval and Resource Restrictions

## TL;DR

This paper establishes new hardness results for scheduling problems with interval and resource restrictions, proving the non-existence of PTAS and tight approximation bounds under P≠NP.

## Contribution

It provides the first negative results for the existence of PTAS in interval restriction scheduling and tight approximation bounds for resource-restricted scheduling with multiple resources.

## Key findings

- No PTAS for interval restrictions unless P=NP.
- Approximation ratio lower bounds of 48/47 and 1.5 for resource restrictions with 2 and 4 resources.
- Results extend to Santa Claus variants maximizing minimal processing time.

## Abstract

In the restricted assignment problem, the input consists of a set of machines and a set of jobs each with a processing time and a subset of eligible machines. The goal is to find an assignment of the jobs to the machines minimizing the makespan, that is, the maximum summed up processing time any machine receives. Herein, jobs should only be assigned to those machines on which they are eligible. It is well-known that there is no polynomial time approximation algorithm with an approximation guarantee of less than 1.5 for the restricted assignment problem unless P=NP. In this work, we show hardness results for variants of the restricted assignment problem with particular types of restrictions. In the case of interval restrictions the machines can be totally ordered such that jobs are eligible on consecutive machines. We resolve the open question of whether the problem admits a polynomial time approximation scheme (PTAS) in the negative (unless P=NP). There are several special cases of this problem known to admit a PTAS. Furthermore, we consider a variant with resource restriction where each machine has capacities and each job demands for a fixed number of resources. A job is eligible on a machine if its demand is at most the capacity of the machine for each resource. For one resource, this problem is known to admit a PTAS, for two, the case of interval restrictions is contained, and in general, the problem is closely related to unrelated scheduling with a low rank processing time matrix. We show that there is no polynomial time approximation algorithm with a rate smaller than 48/47 or 1.5 for scheduling with resource restrictions with 2 or 4 resources, respectively, unless P=NP. All our results can be extended to the so called Santa Claus variants of the problems where the goal is to maximize the minimal processing time any machine receives.