%0 Journal Article
%T A parameterized complexity view on non-preemptively scheduling interval-constrained jobs: few machines, small looseness, and small slack
%A Ren¨¦ van Bevern
%A Rolf Niedermeier
%A OndŁżej Suchy
%J Computer Science
%D 2015
%I arXiv
%X 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 a minimum number $m$ of 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 $\sigma$.
%U http://arxiv.org/abs/1508.01657v1