Narrow Gauge and Analytical Branching Strategies for Mixed Integer Programming

TL;DR

This paper introduces narrow gauge and analytical branching strategies for mixed integer programming, focusing on prioritizing child nodes and using look-ahead trees to improve branching decisions.

Contribution

It proposes novel branching criteria and approaches that enhance variable selection by balancing candidate evaluation and look-ahead analysis.

Findings

New branching strategies outperform traditional methods in certain benchmarks.

Procedures effectively isolate preferred branching candidates.

Use of derivative variables enables deeper exploration of solution space.

Abstract

State-of-the-art branch and bound algorithms for mixed integer programming make use of special methods for making branching decisions. Strategies that have gained prominence include modern variants of so-called strong branching (Applegate, et al.,1995) and reliability branching (Achterberg, Koch and Martin, 2005; Hendel, 2015), which select variables for branching by solving associated linear programs and exploit pseudo-costs (Benichou et al., 1971). We suggest new branching criteria and propose alternative branching approaches called narrow gauge and analytical branching. The perspective underlying our approaches is to focus on prioritization of child nodes to examine fewer candidate variables at the current node of the B&B tree, balanced with procedures to extrapolate the implications of choosing these candidates by generating a small-depth look-ahead tree. Our procedures can also be used in rules to select among open tree nodes (those whose child nodes have not yet been generated). We incorporate pre- and post-winnowing procedures to progressively isolate preferred branching candidates, and employ derivative (created) variables whose branches are able to explore the solution space more deeply.