On the NP-hardness of scheduling with time restrictions

In a recent paper, Braun et al. (2014) have addressed a single-processor scheduling problem with time restrictions. Given a fixed integer B≥2, there is a set of jobs to be processed by a single processor subject to the following B-constraint. For any real x, no unit time interval [x,x+1) is allowed to intersect more than B jobs. The makespan minimization problem has been shown to be NP-hard when B is a part of input and left as an open question whether it remains NP-hard or not if B is fixed (Braun et al., 2014; 2016; Zhang, 2017). This paper contributes to answering this question that we prove the problem is NP-hard even when B=2. A PTAS is also presented for any constant B≥2.