A Theoretical and Computational Analysis of Full Strong-Branching

TL;DR

This paper analyzes the performance of the strong-branching variable selection rule in branch-and-bound algorithms, providing theoretical bounds and computational experiments showing it often produces near-optimal trees, especially for vertex cover instances.

Contribution

It offers a combined theoretical and computational analysis of strong-branching, identifying classes of instances where it performs well or poorly, and quantifying its effectiveness relative to optimal solutions.

Findings

Strong-branching yields provably small trees for vertex cover.

In some cases, strong-branching leads to exponentially larger trees.

Empirical results show strong-branching trees are within a factor of two of optimal.

Abstract

Full strong-branching is a well-known variable selection rule that is known experimentally to produce significantly smaller branch-and-bound trees in comparison to all other known variable selection rules. In this paper, we attempt an analysis of the performance of the strong-branching rule both from a theoretical and a computational perspective. On the positive side for strong-branching we identify vertex cover as a class of instances where this rule provably works well. In particular, for vertex cover we present an upper bound on the size of the branch-and-bound tree using strong-branching as a function of the additive integrality gap, show how the Nemhauser-Trotter property of persistency which can be used as a pre-solve technique for vertex cover is being recursively and consistently used throughout the strong-branching based branch-and-bound tree, and finally provide an example of a vertex cover instance where not using strong-branching leads to a tree that has at least exponentially more nodes than the branch-and-bound tree based on strong-branching. On the negative side for strong-branching, we identify another class of instances where strong-branching based branch-and-bound tree has exponentially larger tree in comparison to another branch-and-bound tree for solving these instances. On the computational side, we conduct experiments on various types of instances to understand how much larger is the size of the strong-branching based branch-and-bound tree in comparison to the optimal branch-and-bound tree. The main take-away from these experiments is that for all these instances, the size of the strong-branching based branch-and-bound tree is within a factor of two of the size of the optimal branch-and-bound tree.