Makespan Minimization with Machine Availability Constraints

We investigate the problems of scheduling n jobs tom = m1+ m2 identical machines where m1 machines are always available, m2 machines have some specified unavailable intervals.The objective is tominimize the makespan. We assume that if a job is interrupted by the unavailable interval, it can be resumed after the machine becomes available. We show that if at least one machine is always available, i.e. m1 > 0, then the PTAS for Multiple Subset Sum problem given by Kellerer [3] can be applied to get a PTAS; otherwise,m = m2, every machine has some unavailable intervals, we show that if (m− 1) machines each of which has unavailable intervalswith total length boundedbyα(n)·Psum/mwherePsum is the total processing time of all jobs and α(n) can be any non-negative function, we can develop a (1 + α(n) + )−approximation algorithm for any constant 0 < < 1; finally we show that there does not exist any polynomial time (1 + α(n)− o(1))−approximation unless P=NP.