Open block scheduling in optical communication networks

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||Cmax) 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 -approximation algorithm and prove that there is no polynomial time ρ-approximation algorithm with ρ<11/10, unless P=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.