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

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 $|d_j-t_j|\le \lambda p_j$ and $|d_j-t_j|\le p_j+\sigma$ and showed the problem to be NP-hard for any $\lambda >1$ and for any $\sigma \ge 2$. 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 $\sigma$.

Version accepted at Journal of Scheduling

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