On the Parameterized Complexity of Interval Scheduling with Eligible Machine Sets

We provide new parameterized complexity results for Interval Scheduling on Eligible Machines. In this problem, a set of n jobs is given to be processed non-preemptively on a set of m machines. Each job has a processing time, a deadline, a weight, and a set of eligible machines that can process it. The goal is to find a maximum weight subset of jobs that can each be processed on one of its eligible machines such that it completes exactly at its deadline.We focus on two parameters: The number m of machines, and the largest processing time pmax. Our main contribution is showing W[1]-hardness when parameterized by m. This answers Open Problem 8 of Mnich and van Bevern's list of 15 open problems in parameterized complexity of scheduling problems [Computers & Operations Research, 2018]. Furthermore, we show NP-hardness even when pmax=O(1) and present an FPT-algorithm with for the combined parameter m+pmax.