Integer and Combinatorial Optimization: Polyhedral and Graph Theoretic Methods

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This project addresses basic theoretical and computational aspects of integer programming and combinatorial optimization, using tools of linear algebra and graph theory. In the nineties the principal investigators have developed a computationally successful approach to mixed 0-1 programming known as lift-and-project. A central theme of the research is to develop this approach in new directions that seem computationally even more promising. One of these directions uses a one to one correspondence recently established by the investigators' team between basic solutions to the higher dimensional linear program used to generate lift-and-project cuts, and certain basic solutions of the LP relaxation of the mixed integer program itself. This correspondence can be used to generate "deepest cuts" in the lift-and-project sense without explicitly generating the higher dimensional linear program. Another important topic is the creation of bridges from integer programming to the branch of computer science known as constraint programming. A recently discovered linear characterization of cardinality rules and similar logical constructs, along with the linear time separability of the inequalities involved, makes it possible to develop symbolic constraints usable in an integer programming context that may significantly enhance the power of algorithms dealing with problems involving logical conditions. A third line of research pursued under this project investigates properties of a 0-1 matrix that make the set packing problem or the set covering problem (or both) defined by it have only integer basic solutions. Structural properties of balanced matrices were obtained under previous NSF grants.; ideal and perfect matrices are currently under investigation.Decision makers often face problems that have a combinatorial aspect: choose one among a very large number of possible decisions. A standard approach is to formulate such problems as integer programs and to use a "solver" to find the best solution. Although integer programming solvers have improved significantly over the last decade, they are still unable to solve many large scale problems to optimality. This project lays the theoretical and analytical foundation for a new generation of solvers for integer programs. The lift-and-project approach developed by the principal investigators has proved well suited to solve hard integer programming problems. Speed remains an issue however. This research project addresses the speed issue by investigating new, faster ways of computing the cuts, lift-and-project as well as other. The potential benefits of the project are significant since cut generators are already being implemented in commercial integer programming solvers and, obviously, the performance of these solvers would be improved by better cut generators. Based on the recent increase in the use of solvers by managers in most fields of business, solvers with improved performance have the potential to increase productivity in the industries where these solvers are used (manufacturing, airline, financial and other industries).