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

TL;DR

This paper establishes exponential lower bounds on the size of general branch-and-bound trees for certain combinatorial optimization problems, demonstrating that even with advanced disjunctions, polynomial-size trees are unlikely.

Contribution

It introduces exponential lower bounds for general branch-and-bound trees on specific instances, extending the understanding of their limitations.

Findings

Any general branch-and-bound tree solving the constructed instances must be exponentially large.

Exponential lower bounds persist under Gaussian noise, ruling out polynomial-size smoothed bounds.

Results are analogous to lower bounds in Chvátal-Gomory cutting-plane methods.

Abstract

A \emph{general branch-and-bound tree} is a branch-and-bound tree which is allowed 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 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 Chv\'atal et al. \cite{chvatal1989cutting}, who proved lower bounds for the Chv\'atal-Gomory rank.