  A key ingredient in branch and bound (B&B) solvers for mixed-integer
programming (MIP) is the selection of branching variables since poor or
arbitrary selection can affect the size of the resulting search trees by orders
of magnitude. A recent article by Le Bodic and Nemhauser [Mathematical
Programming, (2017)] investigated variable selection rules by developing a
theoretical model of B&B trees from which they developed some new, effective
scoring functions for MIP solvers. In their work, Le Bodic and Nemhauser left
several open theoretical problems, solutions to which could guide the future
design of variable selection rules. In this article, we first solve many of
these open theoretical problems. We then implement an improved version of the
model-based branching rules in SCIP 6.0, a state-of-the-art academic MIP
solver, in which we observe an 11% geometric average time and node reduction on
instances of the MIPLIB 2017 Benchmark Set that require large B&B trees.
