Polyhedral and Graph Theoretic Methods in Mixed Integer and Combinatorial Optimization

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This project addresses theoretical and computational aspects of integer programming and combinatorial optimization, using the tools of linear algebra and graph theory. It is focused on the approach called lift-and-project, developed in the nineties and recently incorporated into state-of-the-art software. Lift-and-project cuts are generated from a disjunction involving a combination of inequalities. One topic for investigation is how to simultaneously optimize the weighted combination of inequalities used in the disjunction underlying the cut, and the choice of multipliers in the monoidal strengthening of the resulting cut. Another topic in the same direction offers to improve the performance of Gomory cuts for mixed integer linear programs, by combining inequalities in well-defined ways. Other topics, aimed at understanding the structure of Lehman matrices and almost totally unimodular matrices, or an algorithm for finding an odd hole in a graph, are related to two important classes of problems: set packing and set covering. A better understanding of these classes of problems is a central aspect of the very active research currently occurring in the field of combinatorial optimization.To the extent that this project will be successful, it will advance the state of the art in mixed integer and combinatorial optimization, and thereby enhance our problem solving ability in a broad range of activities, from industrial production to logistics and telecommunications. The tools created here may be as useful in improving homeland security, as they may be instrumental in reinforcing our technological leadership. From the late fifties to the early nineties, integer programming was a way to formulate almost any optimization problem, yet the available computer codes could only handle toy problems of minuscule size. During the last decade the state of the art has radically changed, and today well over half of the integer programs formulated can also be solved. This project is expected to significantly accelerate this change.