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A \emph{general branch-and-bound tree} is a branch-and-bound tree which isallowed to use general disjunctions of the form $\pi^{\top} x \leq \pi_0\,\vee\, \pi^{\top}x \geq \pi_0 + 1$, where $\pi$ is an integer vector and$\pi_0$ is an integer scalar, to create child nodes. We construct a packinginstance, a set covering instance, and a Traveling Salesman Problem instance,such that any general branch-and-bound tree that solves these instances must beof exponential size. We also verify that an exponential lower bound on the sizeof general branch-and-bound trees persists when we add Gaussian noise to thecoefficients of the cross polytope, thus showing that polynomial-size "smoothedanalysis" upper bound is not possible. The results in this paper can be viewedas the branch-and-bound analog of the seminal paper by Chv\'atal et al.\cite{chvatal1989cutting}, who proved lower bounds for the Chv\'atal-Gomoryrank.

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