A new polyhedral approach to combinatorial designs

A New Polyhedral Approach to Combinatorial Designs. (May 2004) Ivette Arámbula Mercado, B.S., Tecnológico de Monterrey (ITESM); M.S., Tecnológico de Monterrey (ITESM) Chair of Advisory Committee: Dr. Illya V. Hicks We consider combinatorial t-design problems as discrete optimization problems. Our motivation is that only a few studies have been done on the use of exact optimization techniques in designs, and that classical methods in design theory have still left many open existence questions. Roughly defined, t-designs are pairs of discrete sets that are related following some strict properties of size, balance, and replication. These highly structured relationships provide optimal solutions to a variety of problems in computer science like error-correcting codes, secure communications, network interconnection, design of hardware; and are applicable to other areas like statistics, scheduling, games, among others. We give a new approach to combinatorial t-designs that is useful in constructing t-designs by polyhedral methods. The first contribution of our work is a new result of equivalence of t-design problems with a graph theory problem. This equivalence leads to a novel integer programming formulation for t-designs, which we call GDP. We analyze the polyhedral properties of GDP and conclude, among other results, the associated polyhedron dimension. We generate new classes of valid inequalities to aim at approximating this integer program by a linear program that has the same optimal solution. Some new classes of valid inequalities are generated as Chvátal-Gomory cuts, other classes are generated by graph complements and combinatorial arguments, and others are generated by the use of