Lower bounds on the size of general branch-and-bound trees

Abstract

A general branch-and-bound tree is a branch-and-bound tree which is allowed to use general disjunctions of the form \(\pi ^{\top } x \le \pi _0 \,\vee \, \pi ^{\top }x \ge \pi _0 + 1\), where \(\pi \) is an integer vector and \(\pi _0\) 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 even when we add Gaussian noise to the coefficients of the cross-polytope, thus showing that a 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. (Linear Algebra Appl 114:455–499, 1989), who proved lower bounds for the Chvátal–Gomory rank.