We study the problem of non-preemptively scheduling n independent sequential jobs on a system of m identical parallel machines in the presence of reservations, where m is constant. This setting is practically relevant because for various reasons, some machines may not be available during specified time intervals. The objective is to minimize the makespan C
max, which is the maximum completion time. The general case of the problem is inapproximable unless P=NP; hence, we study a suitable strongly NP-hard restriction, namely the case where at least one machine is always available. For this setting we contribute approximation schemes, complemented by inapproximability results. The approach is based on algorithms for multiple subset sum problems; our technique yields a PTAS (Polynomial-Time Approximation Scheme) which is best possible in the sense that an FPTAS (Non-Polynomial-Time Approximation Scheme) is ruled out unless P=NP. The PTAS presented here is the first one for the problem under consideration; so far, not even for well-known special cases approximation schemes have been proposed. Furthermore we derive a low cost algorithm with a constant approximation ratio and discuss FPTASes for special cases as well as the complexity of the problem if m is part of the input.
