Parallel branch-and-cut for set partitioning

This thesis investigates three major steps in the solution process of the Set Partitioning Problem (SPP): problem size reduction techniques, LP-based feasible solution heuristics and Branch-and-Cut solution methodology. SPPs arise in many practical applications (airline crew scheduling, vehicle routing, circuit partitioning). Theoretical aspects of this problem have been studied for a long time, but only recently have computers become powerful enough to attack practical instances. Problem size reduction methods reduce the set of variables and/or constraints through logical implications without eliminating optimal solutions to the original problem. We show that the reduction operations well-known in the literature, applied in any order to an SPP instance until no further reduction is possible, always produce the same reduced problem. Finding good feasible solutions is essential for upper bounding in a Branch-and-Cut framework. Our LP-based feasible solution heuristic iterates a heuristic fixing phase with reduced cost fixing to improve the quality of the feasible solution. Our heuristic procedure is somewhat more conservative than earlier approaches in that it eliminates unnecessary variables instead of forcing variables into the solution. Our parallel Branch-and-Cut procedure was implemented using the COMPSys framework. COMPSys provides the user with the necessary infrastructure to implement an efficient Branch-and-Cut application by handling tasks common for parallel Branch-and-Cut (search tree management, message passing, LP interface). To interface with COMPSys we implemented procedures particular to the SPP. We generate cuts both algorithmically and manually through a graphical user interface. Our experiments were carried out on the IBM RS/6000 Scalable POWERparallel System of the Cornell Theory Center. Our test set included problems from airline crew scheduling and vehicle routing applications. Our computational results demonstrate our implementation to be an effective approach for solving SPPs of moderately large size.