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Open block scheduling in optical communication networks. (English) Zbl 1097.68011
Summary: In this paper the process of data transmission in star coupled optical communication networks is modelled as a shop-type scheduling problem, where channels (wavelengths) are treated as machines. We formulate an Open Block problem with the minimum makespan objective $$(OB||C_{\max})$$ in which a relation of a new type between the operations of each job is introduced: any two operations of a job have identical processing times and may be processed either simultaneously (in a common block) or, alternatively, at disjoint time intervals. We show that the problem is polynomially solvable for 4 machines, NP-hard for 6 machines and strongly NP-hard for a variable number of machines. For the case of a variable number of machines we present a polynomial time $$\sqrt 2$$-approximation algorithm and prove that there is no polynomial time $$\rho$$-approximation algorithm with $$\rho < 11/10$$, unless $$\text{P}=\text{NP}$$. For the case when the number of machines is fixed, we show that the problem admits a linear time PTAS. In addition, we show that the two-machine problem with release dates is NP-hard in the strong sense.
##### MSC:
 68M10 Network design and communication in computer systems 68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
##### Keywords:
scheduling; approximation; open shop; data packets
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##### References:
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