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The bulk synchronous task scheduling problem (BSSPO) is known as an effective task scheduling problem for distributed memory machines. We present a proof of NP-completeness of the decision counterpart of BSSPO, even in the case of makespan of at most five. This implies nonapproximability of BSSPO, meaning that there is no approximation algorithm with performance guarantee smaller than 6/5 unless P = NP. We also give an approximation algorithm with a performance guarantee of two for BSSPO in several restricted cases.