Complexity of branch-and-bound and cutting planes in mixed-integer optimization -- II

TL;DR

This paper provides a rigorous theoretical analysis demonstrating that combining cutting planes and branching in branch-and-cut algorithms can exponentially outperform using them separately in mixed-integer optimization.

Contribution

It establishes general conditions under which branch-and-cut offers provably exponential efficiency gains over pure cutting planes or branch-and-bound methods.

Findings

Branch-and-cut can be exponentially more efficient than pure methods.

Efficiency is measured by iteration count and constraint sparsity.

First rigorous proof of branch-and-cut's superiority in this context.

Abstract

We study the complexity of cutting planes and branching schemes from a theoretical point of view. We give some rigorous underpinnings to the empirically observed phenomenon that combining cutting planes and branching into a branch-and-cut framework can be orders of magnitude more efficient than employing these tools on their own. In particular, we give general conditions under which a cutting plane strategy and a branching scheme give a provably exponential advantage in efficiency when combined into branch-and-cut. The efficiency of these algorithms is evaluated using two concrete measures: number of iterations and sparsity of constraints used in the intermediate linear/convex programs. To the best of our knowledge, our results are the first mathematically rigorous demonstration of the superiority of branch-and-cut over pure cutting planes and pure branch-and-bound.