An Abstract Model for Branch and Cut

TL;DR

This paper introduces an abstract framework to analyze how cutting planes influence the efficiency and size of branch-and-cut algorithms in solving mathematical programming problems, bridging practical heuristics with theoretical understanding.

Contribution

It provides a novel abstract model that captures the joint effects of branching and cutting decisions on branch-and-cut performance, addressing a gap in theoretical explanations.

Findings

Model explains nonmonotonic effects of cuts on solution process

Analyzes impact of cut generation timing and frequency

Provides insights into cut strategies at root and throughout the tree

Abstract

Branch and cut is the dominant paradigm for solving a wide range of mathematical programming problems -- linear or nonlinear -- combining efficient search (via branch and bound) and relaxation-tightening procedures (via cutting planes, or cuts). While there is a wealth of computational experience behind existing cutting strategies, there is simultaneously a relative lack of theoretical explanations for these choices, and for the tradeoffs involved therein. Recent papers have explored abstract models for branching and for comparing cuts with branch and bound. However, to model practice, it is crucial to understand the impact of jointly considering branching and cutting decisions. In this paper, we provide a framework for analyzing how cuts affect the size of branch-and-cut trees, as well as their impact on solution time. Our abstract model captures some of the key characteristics of real-world phenomena in branch-and-cut experiments, regarding whether to generate cuts only at the root or throughout the tree, how many rounds of cuts to add before starting to branch, and why cuts seem to exhibit nonmonotonic effects on the solution process.