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

We study the problem of non-preemptively scheduling $n$ jobs, each job $j$ with a release time $tj$, a deadline $dj$, and a processing time $pj$, on $m$ parallel identical machines. Cieliebak et al. (2004) considered the two constraints $|dj-tj|\leq \lambda pj$ and $|dj-tj|\leq p_j +\sigma$ 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 $\sigma$.