Lower Bounds on the Size of General Branch-and-Bound Trees

A \emph{general branch-and-bound tree} is a branch-and-bound tree which is allowed to use general disjunctions of the form $π^{\top} x \leq π_0 \,\vee\, π^{\top}x \geq π_0 + 1$, where $π$ is an integer vector and $π_0$ is an integer scalar, to create child nodes. We construct a packing instance, a set covering instance, and a Traveling Salesman Problem instance, such that any general branch-and-bound tree that solves these instances must be of exponential size. We also verify that an exponential lower bound on the size of general branch-and-bound trees persists when we add Gaussian noise to the coefficients of the cross polytope, thus showing that polynomial-size "smoothed analysis" upper bound is not possible. The results in this paper can be viewed as the branch-and-bound analog of the seminal paper by Chvátal et al. \cite{chvatal1989cutting}, who proved lower bounds for the Chvátal-Gomory rank.

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