Sub-Exponential Lower Bounds for Branch-and-Bound with General Disjunctions via Interpolation

TL;DR

This paper establishes the first sub-exponential lower bounds on the size of branch-and-bound trees with general disjunctions for certain integer programs, using interpolation techniques and circuit complexity arguments.

Contribution

It introduces a novel approach linking branch-and-bound complexity with monotone real circuit lower bounds via interpolation, providing new theoretical limits.

Findings

First sub-exponential lower bound for branch-and-bound trees with general disjunctions.

Shows that monotone real circuits can perform binary search efficiently.

Implications for refuting certain CNF formulas with high probability.

Abstract

This paper investigates linear programming based branch-and-bound using general disjunctions, also known as stabbing planes, for solving integer programs. We derive the first sub-exponential lower bound (in the encoding length $L$ of the integer program) for the size of a general branch-and-bound tree for a particular class of (compact) integer programs, namely $\smash{2^{\Omega(L^{1/12 -\epsilon})}}$ for every $\epsilon >0$. This is achieved by showing that general branch-and-bound admits quasi-feasible monotone real interpolation, which allows us to utilize sub-exponential lower-bounds for monotone real circuits separating the so-called clique-coloring pair. Moreover, this also implies that refuting $\Theta(\log(n))$-CNFs requires size $2^{n^{\Omega(1)}}$ branch-and-bound trees with high probability by considering the closely related notion of infeasibility certificates introduced by Hrubes and Pudl\'ak. One important ingredient of the proof of our interpolation result is that for every general branch-and-bound tree proving integer-freeness of a product $P\times Q$ of two polytopes $P$ and $Q$, there exists a closely related branch-and-bound tree for showing integer-freeness of $P$ or one showing integer-freeness of $Q$. Moreover, we prove that monotone real circuits can perform binary search efficiently.