A parameterized complexity view on non-preemptively scheduling interval-constrained jobs: few machines, small looseness, and small slack

TL;DR

This paper analyzes the computational complexity of non-preemptively scheduling interval-constrained jobs on multiple machines, revealing NP-hardness and W[1]-hardness results, and providing algorithms for specific parameter settings.

Contribution

It extends prior NP-hardness results by exploring parameterized complexity, offering new algorithms and hardness proofs for scheduling problems with small looseness and slack.

Findings

NP-hardness for two machines with rac{_j - t_j}{p_j} > 1

W[1]-hardness parameterized by number of machines

Pseudo-polynomial algorithm for fixed number of machines and looseness

Abstract

We study the problem of non-preemptively scheduling $n$ jobs, each job $j$ with a release time $t_j$, a deadline $d_j$, and a processing time $p_j$, on $m$ parallel identical machines. Cieliebak et al. (2004) considered the two constraints $|d_j-t_j|\leq \lambda p_j$ and $|d_j-t_j|\leq p_j +\sigma$ and showed the problem to be NP-hard for any $\lambda>1$ and for any $\sigma\geq 2$. We complement their results by parameterized complexity studies: we show that, for any $\lambda>1$, the problem remains weakly NP-hard even for $m=2$ and strongly W[1]-hard parameterized by $m$. We present a pseudo-polynomial-time algorithm for constant $m$ and $\lambda$ and a fixed-parameter tractability result for the parameter $m$ combined with $\sigma$.