  Strong Branching (SB) is a cornerstone of all modern branching rules used in
the Branch-and-Bound (BnB) algorithm, which is at the center of Mixed-Integer
Programming solvers. In its full form, SB evaluates all variables to branch on
and then selects the one producing the best relaxation, leading to small trees,
but high runtimes. State-of-the-art branching rules therefore use SB with
working limits to achieve both small enough trees and short run times. So far,
these working limits have been established empirically. In this paper, we
introduce a theoretical approach to guide how much SB to use at each node
within the BnB. We first define an abstract stochastic tree model of the BnB
algorithm where the geometric mean dual gains of all variables follow a given
probability distribution. This model allows us to relate expected dual gains to
tree sizes and explicitly compare the cost of sampling an additional SB
candidate with the reward in expected tree size reduction. We then leverage the
insight from the abstract model to design a new stopping criterion for SB,
which fits a distribution to the dual gains and, at each node, dynamically
continues or interrupts SB. This algorithm, which we refer to as Probabilistic
Lookahead Strong Branching, improves both the tree size and runtime over MIPLIB
instances, providing evidence that the method not only changes the amount of
SB, but allocates it better.
