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

• Published in: Mathematical programming Vol. 198; no. 1; pp. 539 - 559
• Main Authors: Dey, Santanu S., Dubey, Yatharth, Molinaro, Marco
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2023; Springer
• Subjects: Calculus of Variations and Optimal Control; Optimization; Combinatorics; Full Length Paper; Management science; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Theoretical
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-022-01781-z

A general branch-and-bound tree is a branch-and-bound tree which is allowed to use general disjunctions of the form π ⊤ x ≤ π 0 ∨ π ⊤ x ≥ 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 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.