The Branch-and-Bound Tree Closure

This paper investigates the a-posteriori analysis of Branch-and-Bound~(BB) trees to extract structural information about the feasible region of mixed-binary linear programs. We introduce three novel outer approximations of the feasible region, systematically constructed from a BB tree. These are: a tight formulation based on disjunctive programming, a branching-based formulation derived from the tree's branching logic, and a mixing-set formulation derived from the on-off properties inside the tree. We establish an inclusion hierarchy, which ranks the approximations by their theoretical strength \wrt to the original feasible region. The analysis is extended to the generation of valid inequalities, revealing a separation-time hierarchy that mirrors the inclusion hierarchy in reverse. This highlights a trade-off between the tightness of an approximation and the computational cost of generating cuts from it. Motivated by the computational expense of the stronger approximations, we introduce a new family of valid inequalities called star tree inequalities. Although their closure forms the weakest of the proposed approximations, their practical appeal lies in an efficient, polynomial-time combinatorial separation algorithm. A computational study on multi-dimensional knapsack and set-covering problems empirically validates the theoretical findings. Moreover, these experiments confirm that computationally useful valid inequalities can be generated from BB trees obtained by solving optimization problems considered in practice.