Conditions:
 - branch and bound (not branch and cut)
 - suitable LP solver (since it is branch
   and bound, this is likely to be a simplex
   solver)
 - decision at each node is which candidate
   variable to branch on and which direction
   to branch (up or down)

What is the shortest sequence of such branching decisions
that will lead you from the root node to an integer-feasible
leaf node?

A main motivation is practical: I can quite often find
sequences which lead directly from root to integer-feasible
leaf node, but I don't know if this is the shortest
sequence.  I would like to know the shortest sequences for e.g.
all of the MIPLIB2003 instances.  Good heuristic methods that
often return short sequences are welcome (this is really what
the active-constraint branching variable selection methods try
to do).

There are variations.  E.g. suppose I find a sequence of
length k.  A decision version of the problem is "is there
a sequence that is shorter than k?"

As came up in the discussion, there are other variants,
e.g. for binary problems, or for SAT problems, or for
network problems.  Other people may wish to follow up
with statements of some of these variants.