Open block scheduling in optical communication networks 

[[abstract]]In this paper the process of data transmission in optical communication networks is modeled as a shop-type scheduling problem, where channels (wavelengths) are treated as machines. We formulate an Open Block problem with the minimum makespan objective (an OB\\C(max) problem) 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 completely simultaneously (in a common block) or, alternatively, with full diversity in time. We show that the problem is polynomially solvable for 4 machines, binary NP-hard for 6 machines and strongly NP-hard for a variable number of machines. Adding release dates to the two-machine problem also leads to the NP-hardness in strong sense. For the case of a variable number of machines we present a polynomial time root2-approximation algorithm and prove that there is no polynomial time rho-approximation algorithm with rho < 11/10, unless P=NP. For the case when the number of machines is fixed, we show that the problem can be solved by a linear time PTAS and by a few linear time statistically optimal algorithms (generating optimal schedules for almost all instances).[[note]]SC