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 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 | ≤ λ p j and | d j - t j | ≤ p j + σ and showed the problem to be NP-hard for any λ > 1 and for any σ ≥ 2 . We complement their results by parameterized complexity studies: we show that, for any λ > 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 λ and a fixed-parameter tractability result for the parameter m combined with σ .