Compressing Branch-and-Bound Trees

TL;DR

This paper introduces the tree compression problem for branch-and-bound trees, exploring how to create smaller trees with the same or better bounds through disjunctions or leaf removal, supported by complexity analysis and experiments.

Contribution

It formally defines the tree compression problem, analyzes its computational complexity, and evaluates practical compressibility of BB trees with algorithms.

Findings

Compression can significantly reduce tree size without losing bounds

Heuristic methods effectively approximate optimal compression

Certain trees are inherently resistant to compression

Abstract

A branch-and-bound (BB) tree certifies a dual bound on the value of an integer program. In this work, we introduce the tree compression problem (TCP): Given a BB tree T that certifies a dual bound, can we obtain a smaller tree with the same (or stronger) bound by either (1) applying a different disjunction at some node in T or (2) removing leaves from T? We believe such post-hoc analysis of BB trees may assist in identifying helpful general disjunctions in BB algorithms. We initiate our study by considering computational complexity and limitations of TCP. We then conduct experiments to evaluate the compressibility of realistic branch-and-bound trees generated by commonly-used branching strategies, using both an exact and a heuristic compression algorithm.