Cyclical scheduling and multi-shift scheduling: Complexity and approximation algorithms

We consider the multiple shift scheduling problem modelled as a covering problem. Such problems are characterized by a constraint matrix that has in every column blocks of consecutive ones, each corresponding to a shift. We focus on three type of shift scheduling problems classified according to the column structure in the constraint matrix: consecutive ones columns, cyclical ones columns and k consecutive blocks columns. In particular the complexity of the cyclical scheduling problem, where the matrix satisfies the cyclical 1’s property in each column was noted recently by Hochbaum and Tucker to be open. They further showed that the unit demand case is polynomially solvable. Here we extend this result to the uniform requirements case, and provide a 2-approximation algorithm for the non-uniform case. We also establish that the cyclical scheduling problem’s complexity is equivalent to that of the exact matching problem – a problem the complexity status of which is known to be randomized polynomial, RP. We then investigate the three types of shift scheduling problems and show that while the consecutive ones version is polynomial and the k-block columns is NP-hard (for k ≥ 2), For the k-blocks problem we give a simple k-approximation algorithm, which is the first approximation algorithm determined for the problem.