Exploiting Structure in Integer Programs

ii The Abstract This dissertation argues the case for exploiting certain structures in integer linear programs. Integer linear programming is a well-known optimisation problem, which seeks the optimum of a linear function of variables, whose values are required to be integral as well as to satisfy certain linear equalities and inequalities. The state of the art in solvers for this problem is the " branch and bound " approach. The performance of such solvers depends crucially on four types of in-built heuristics: primal, improvement, branching, and cut-separation or, more generally, bounding heuristics. Such heuristics in general-purpose solvers have not, until recently, exploited structure in integer linear programs beyond the recognition of certain types of single-row constraints. Many alternative approaches to integer linear programming can be cast in the following, novel framework. " Structure " in any integer linear program is a class of an equivalence among triples of algorithms: deriving combinatorial objects from the input, adapting them, and transforming the adapted object to solutions of the original integer linear program. Such alternative approaches are, however, inherently incompatible with branch and bound solvers. This dissertation defines such a structure to be " useful " , only when it extracts submatrices, which allow for the implementation of more than one of the four types of heuristics required in the branch and bound approach. Three examples of such useful structures are presented: aggregations of variables, mutual-exclusion components, and precedence-constrained components. On instances from complex timetabling problems, where such structures are prevalent, a general-purpose solver, based on this approach, closes the gap between primal and dual bounds to under five percent, orders of magnitude faster than using state-of-the-art general-purpose solvers. iii Acknowledgements [C]ertain kinds of hard work are in fact [...] more fun than just about anything else. I would also like to thank many others for their informal, but no less valuable support and encouragement. Petr Hliněný introduced me to Integer Linear Programming in the fall of 2005, and has been a source of inspiration since. The organisers and attendees of Mixed Integer Programming workshops have helped shape my thinking about integer programming. In particular, I would like to acknowledge a discussion with George Nemhauser at my first MIP Workshop in the summer of 2007 and discussions with Tobias Achterberg and Roland Wunderling in subsequent years. Jacek Gondzio, Florian Jarre, and Adam Letchford have tried to help me understand convex programming. Danny Kershaw …